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U proteklih dvadesetak godina razvijeno je vi\[SHacek]e razli\ \[CHacek]itih pristupa upravljanju dinamikom molekularnih valnih paketa [1]. \ Uslijed slo\:017eenosti kvantnih sustava i pridru\:017eene im dinamike, \ analiti\[CHacek]ki pristup laserskom upravljanju jo\[SHacek] uvijek nije \ toliko razvijen da bi omogu\[CAcute]io temeljit i potpun uvid u dinami\ \[CHacek]ke procese, a s time i njihovu potpuni nadzor i razumijevanje, tako \ da se neke od danas najupotrebljavanijih metoda zasnivaju na \[CHacek]isto \ numeri\[CHacek]kim optimizacijskim ('geneti\[CHacek]kim') algoritmima [2]. S \ na\[SHacek]eg stanovi\[SHacek]ta su svakako najzanimljivije brze aktivne \ (pikosekundne i femtosekundne) metode. Naime, brzina prijenosa populacije \ koji se posti\:017ee ovakvim metodama omogu\[CAcute]ava potpuno zanemarivanje \ bitno sporijih relaksacijskih procesa. Dinami\[CHacek]ke su jednad\:017ebe \ ovakvih sustava stoga relativno jednostavne i u velikoj mjeri podlo\:017ene \ analiti\[CHacek]kom rje\[SHacek]avanju te izradi u\[CHacek]inkovitih numeri\ \[CHacek]kih simulacijskih algoritama.\ \>", "Text"], Cell[TextData[{ "Pionirski rad u ovom podru\[CHacek]ju na\[CHacek]inili su Paramonov i \ Savva [3], pokazav\[SHacek]i na koji na\[CHacek]in se monokromatskim \ pikosekundnim infra-crvenim laserskim pulsom populacija anharmoni\[CHacek]kog \ oscilatora iz osnovne vibracijske razine mo\:017ee prenijeti u odabranu pobu\ \:0111enu vibracijsku razinu. Optimizacija pulsa koju su oni proveli je \ \[CHacek]isto numeri\[CHacek]ka. Zna\[CHacek]ajna analiti\[CHacek]ka \ nadgradnja ovog pristupa dana je kroz teoriju poop\[CAcute]enih \[Pi]-pulseva \ [4]. Na temelju Floquetovog opisa dinamike [5] sustava pogonjenih periodi\ \[CHacek]kim monokromatskim zra\[CHacek]enjem, autori su poop\[CAcute]ili \ teorem o povr\[SHacek]ini pulsa \[LongDash] poznat iz klasi\[CHacek]ne \ teorije Rabijevih oscilacija u sustavima od samo dvije razine [6] \[LongDash] \ na sustave sa vi\[SHacek]e razina. Ovo poop\[CAcute]enje omogu\[CAcute]ava \ krojenje pogodnog pulsa za prijenos populacije izme\:0111u odabranih razina u \ vi\[SHacek]erazinskom sustavu. Zbog neharmoni\[CHacek]nosti \ me\:0111unuklearnog potencijala ovakav pristup pogonjenja molekulskih \ vibracija ima svoja ograni\[CHacek]enja. Naime, anharmoni\[CHacek]nost \ onemogu\[CAcute]ava u\[CHacek]inkovit prijenos populacije kroz veliki broj \ vibracijskih razina. Ako je frekvencija pogona ugo\:0111ena tako da dobro \ odgovara prijelazima me\:0111u najni\:017eim razinama, tada \[CAcute]e \ prijelazi izme\:0111u vi\[SHacek]ih razina biti zna\[CHacek]ajno izvan \ razonancije sa pogonom. Jedno od rje\[SHacek]enja ovog problema upotrebljava \ vi\[SHacek]e uzastopnih pulseva od kojih je svaki ugo\:0111en na razli\ \[CHacek]itu frekvenciju [7]. Svaki od pulseva prenosi populaciju kroz \ nekoliko razina, nakon \[CHacek]ega prijenos prihva\[CAcute]a \ slijede\[CAcute]i puls. Drugo rje\[SHacek]enje predstavlja upotreba 'pro\ \[SHacek]aranih' (eng. ", StyleBox["chirped", FontSlant->"Italic"], ") pulseva [8,9] koji umjesto monokromatskog zra\[CHacek]enja \ upotrebljavaju pogon sa vremenski promjenjivom frekvencijom \ razvu\[CHacek]enom preko cjelokupnog raspona rezonancijskih frekvencija \ vibracijskih prijelaza. " }], "Text"], Cell[CellGroupData[{ Cell["Ciljevi rada", "Subsubsection"], Cell[TextData[{ "U ovome smo se radu \:017eeljeli pozabaviti daljnjim produbljivanjem \ \[Pi]-puls metoda. Ove se metode u svojoj osnovi sve temelje na mehanizmu \ Rabijevih oscilacija koji je poznat ve\[CAcute] gotovo pola stolje\[CAcute]a \ [10], a analiti\[CHacek]ki je izveden za sustav sa samo dvije razine. Iscrpno \ analiti\[CHacek]ko izu\[CHacek]avanje pojave Rabijevih oscilacija u \ slo\:017eenijim sustavima me\:0111utim do sada nije na\[CHacek]injeno, pa \ uloga cijele teorije (sa stanovi\[SHacek]ta analiti\[CHacek]kog pristupa) \ stoga nije bitno odmakla od jednostavnog didakti\[CHacek]kog primjera. \ \[CapitalSHacek]tovi\[SHacek]e, nisu u potpunosti istra\:017eene \[CHacek]ak \ niti granice valjanosti teorije za sustav od dvije razine. Zbog ovoga niti \ jedna od ranije navedenih u svom analiti\[CHacek]kom dijelu ne \ uklju\[CHacek]uje istra\:017eivanje utjecaja ostatka sustava - razina koje u \ sustavu postoje ali ne sudjeluju bitno u prijenosu populacije. Jasno je da se \ u numeri\[CHacek]kim eksperimantima utjecaj ovih razina svakako \ o\[CHacek]ituje, me\:0111utim nedostatak njihovog uklju\[CHacek]ivanja u \ analiti\[CHacek]ki ra\[CHacek]un onemogu\[CAcute]ava njihovo \ predvi\:0111anje. Prvi cilj kojega \:017eelimo ostvariti jest potanko analiti\ \[CHacek]ki teoriju Rabijevih oscilacija u sustavu od samo dvije razine, te \ ju potom poop\[CAcute]iti pogledav\[SHacek]i na koji na\[CHacek]in bi se \ mogao opisati utjecaj smetnje od dodatnih razina. Pokazat \[CAcute]emo da je \ to mogu\[CAcute]e u\[CHacek]initi, te da nam uklju\[CHacek]ivanje smetnji u \ analiti\[CHacek]ki ra\[CHacek]un daljnje omogu\[CAcute]ava uspostavljanje \ jasne veze izme\:0111u unutra\[SHacek]nje strukture sustava, mogu\[CAcute]e \ brzine prijenosa populacije kroz sustav i u\[CHacek]inkovitosti tog \ prijenosa. Nadalje, svi navedeni \[Pi]-puls pristupi bave se selektivnim pobu\ \:0111ivanjem ", StyleBox["vibracijskih", FontSlant->"Italic"], " razina u sustavu, dok se fina gra\:0111a dinamike uzrokovana postojani \ jem rotacijskih stupnjeva slobode ili u potpunosti zanemaruje, ili se uklju\ \[CHacek]uje putem statisti\[CHacek]kog usrednjavanja. U ovome smo radu tako\ \:0111er \:017eeljeli istra\:017eiti upravo mogu\[CAcute]nost finog \ upravljanja dinamikom unutra\[SHacek]njih stupnjeva slobode dvoatomne \ heteronuklearne molekule, pri tome na umu imaju\[CAcute]i osobito rotacije \ unutar jedne vibracijske razine." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Pregled izlaganja", "Subsubsection"], Cell["\<\ Rad je tematski podjeljen u tri cjeline. U prvoj cjelini potanko \ istra\:017eujemo standardnu teoriju Rabijevih oscilacija u jednostavnom \ kvantnom sustavu gra\:0111enom od samo dvije vlastite razine. Uvodimo pojam \ Rabijevog profila koji \[CAcute]e nam u nastavku poslu\:017eiti kao temeljni \ alat za istra\:017eivanje dinamike prenosa populacije u slo\:017eenijim \ kvantnim sustavima. Kao zavr\[SHacek]ni, i iznimno va\:017ean korak korak u \ ovom poglavlju istra\:017eujemo ograni\[CHacek]enja valjanosti Rabijevih \ rezultata kako bismo dobili osje\[CAcute]aj do koje se mjere na njih \ mo\:017eemo osloniti u kasnijim istra\:017eivanjima.\ \>", "Text"], Cell["\<\ U drugom poglavlju pro\[SHacek]irujemo Rabijevu teoriju na sustave s vi\ \[SHacek]e od dvije vlastite razine. Analiti\[CHacek]ki istra\:017eujemo \ utjeca smetnje koju uzrokuju te dodatne razina na dinamiku prenosa \ populacije izme\:0111u dviju odabranih razina. Ra\[CHacek]un prvo provodimo \ za sustave u kojima postoji samo jedna jedina takva razina 'smetnje', a potom \ poop\[CAcute]avamo dobivene rezultate na sustave sa proizvoljnim brojem \ vlastitih razina.\ \>", "Text"], Cell[TextData[{ "U zavr\[SHacek]nom poglavlju izu\[CHacek]avamo slo\:017eenu metodu prenosa \ populacije temeljenu na upotrebi vi\[SHacek]estruko rezonantnog pogona. \ Analiti\[CHacek]ki izrazi na\:0111eni u drugom poglavlju poslu\:017eit \ \[CAcute]e nam kao temelj za fizi\[CHacek]ku optimizaciju matemati\[CHacek]ki \ utvr\:0111enog postupka ", StyleBox["vi\[SHacek]estruko rezonantnog lan\[CHacek]astog prenosa", FontSlant->"Italic"], ". Rad zalju\[CHacek]ujemo prikazom numeri\[CHacek]ki izra\[CHacek]unate \ dinamike takvog prenosa u 'realnom' sustavu (ro-vibratoru HF) sa analiti\ \[CHacek]ki optimiziranim parametrima vanjskog pogona." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Sustav jedinica", "Subsubsection"], Cell[TextData[{ "Temeljne ", StyleBox["jednad\:017ebe ", FontFamily->"Times New Roman"], "dinamike sustava iznesene su u sustavu atomskih jedinica gdje je ", StyleBox["\[HBar]=1", FontFamily->"Courier"], ". Ve\[CAcute]ina analiti\[CHacek]kih ra\[CHacek]una provedena je u prilago\ \:0111enim sustavima jedinica u kojima jednad\:017ebe imaju najjednostavniji \ oblik. Svaki prelazak iz sustava atomskih jedinica i prilago\:0111enih \ jedinica jasno je nazna\[CHacek]en a same preinake veli\[CHacek]ina su \ podrobno opisane. Za sve numeri\[CHacek]ke vrijednosti i veli\[CHacek]ine na \ grafovima iz konteksta je jasno o kojim se jedinicama radi." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ POGLAVLJE 1 Rabijeva titranja u dvorazinskom kvantnom sustavu\ \>", "Section 1", CellTags->"POGLAVLJE 1"], Cell["\<\ Prvo poglavlje rada posve\[CAcute]eno je potankom istra\:017eivanju osobina \ Rabijevih titranja u dvorazinskim kvantnim sustavima. Raspravu \ zapo\[CHacek]injemo postavljanjem temeljnih dinami\[CHacek]kih jednad\:017ebi \ za op\[CAcute]eniti vi\[SHacek]erazinski kvantni sustave te prikazujemo pod \ kojim se uvjetima njegova dinamika svodi na dinamiku sustava sa samo dvije \ razine. Nakon toga slijedi pregledno izlaganje uobi\[CHacek]ajene Rabijeve \ teorije u sustavima sa samo dvije vezane razine. Poglavlje zatvaramo \ raspravom o uvjetima pod kojim Rabijevo rje\[SHacek]enje dovoljno vjerno \ opisuje stvarnu dinamiku sustava. \ \>", "Text", PageBreakBelow->Automatic], Cell[CellGroupData[{ Cell["\<\ 1.1 Temeljne jednad\:017ebe dinamike vi\[SHacek]erazinskog kvantnog sustava i \ pribli\:017eenje rotiraju\[CAcute]eg vala\ \>", "Subsection", PageBreakAbove->True, CellTags->"1.1"], Cell["\<\ Cjelokupna teorija iznesena u ovom radu zasniva se na rje\[SHacek]avanju \ vremenski ovisne Schroedingerove jednad\:017ebe\ \>", "Text"], Cell[BoxData[ \(\(d\/dt\) \[CapitalPsi][x, t] = \(-\[ImaginaryI]\)\ \((H\_0[x] + V[x, t])\) \[CapitalPsi][x, t]\)], "NumberedEquation"], Cell[TextData[{ "Sa ", Cell[BoxData[ \(x\)]], " ozna\[CHacek]avamo skup svih prostornih koordinata sustava, ", Cell[BoxData[ \(H\_0[x]\)]], " je nesmetani hamiltonijan a ", Cell[BoxData[Cell["V[x,t]"]]], " je potencijal vanjske sile narinut na sustav. Svojstvena stanja ", Cell[BoxData[ \(\[CurlyPhi]\_i[x]\)]], " (njih ", StyleBox["N", FontFamily->"Courier"], ") i pridru\:017eene im svojstvene frekvencije ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_i\)]], " nesmetanog hamiltonijana nalazimo iz jednad\:017ebe za stacionarna \ stanja" }], "Text"], Cell[BoxData[ \(H\_0[x] \[CurlyPhi]\_i[x] = \(\(\[Omega]\_i\) \[CurlyPhi]\_i[x]\ i = 1\), 2, ... , N\)], "NumberedEquation"], Cell[CellGroupData[{ Cell["\<\ Rje\[SHacek]avanje razvojem preko svojstvenih stanja. Matri\[CHacek]ni zapis \ jednad\:017ebe dinamike\ \>", "Subsubsection"], Cell[TextData[{ "Rje\[SHacek]enje jednad\:017ebe (1), valnu funkciju ", Cell[BoxData[ StyleBox[\(\[CapitalPsi][x, t]\), "InlineInput"]]], ", tra\:017eimo u obliku razvoja preko svojstvenih stanja sustava" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox[\(\[CapitalPsi][x, t]\), "jednadzba"], StyleBox["=", "jednadzba"], \(\[Sum]\+i A\_i[t] \[CurlyPhi]\_i[ x]\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\[Omega]\_i\) \ t\)\)}], " ", ";", \(i = 1\)}], ",", "2", ",", "...", ",", "N"}]], "NumberedEquation"], Cell[TextData[{ "gdje su ", Cell[BoxData[ StyleBox[\(A\_i[t]\), "InlineInput"]]], " koeficijenti razvoja \[CHacek]iju dinamiku treba odrediti. \ Uvr\[SHacek]tavanjem razvoja (3) u jednad\:017ebu (1), te zapisivanjem \ rezonantne frekvencije prelaza izme\:0111u razina ", StyleBox["i", "InlineInput"], " i ", StyleBox["j", "InlineInput"], " kao ", Cell[BoxData[ StyleBox[ FormBox[ RowBox[{\(\(\[Omega]\^0\)\_ij\), "\[Congruent]", RowBox[{ SubscriptBox["\[Omega]", StyleBox["i", FontSlant->"Plain"]], "-", SubscriptBox["\[Omega]", StyleBox["j", FontSlant->"Plain"]]}]}], "TraditionalForm"], "InlineInput"]]], ". Rje\[SHacek]avanje dinamike sustava time smo sveli na rje\[SHacek]avanje \ sustava vezanih linearnih diferencijalnih jednad\:017ebi" }], "Text"], Cell[BoxData[ \(\(d\/dt\) A\_j[t] = \(-\[ImaginaryI]\) \(\[Sum]\+i A\_i[ t] \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_ij\ \) t\)\ \(\[Integral]\((\[CurlyPhi]\_i[x] V[x, t] \[CurlyPhi]\_j[x]\ dx)\)\)\)\ ; i, j = 1, 2, ... , N\)], "NumberedEquation"], Cell[TextData[{ "Radi ve\[CAcute]e preglednosti prethodni \[CAcute]emo sustav \ jednad\:017ebi zapisati u matri\[CHacek]nom obliku. S tim ciljem \[CAcute]emo \ ", StyleBox["N", "InlineInput"], " koeficijenata razvoja poslo\:017eiti u vektor ", Cell[BoxData[ \(\(A\& \[Rule] \)[t]\)]] }], "Text"], Cell[BoxData[ RowBox[{\(\(A\& \[Rule] \)[t]\), "\[Congruent]", RowBox[{"(", GridBox[{ {GridBox[{ {\(A\_1[t]\)}, {\(A\_2[t]\)} }]}, {"."}, {"."}, {\(A\_N[t]\)} }], ")"}]}]], "NumberedEquation"], Cell[TextData[{ "dok \[CAcute]emo integrale vanjskom silom pobu\:0111enog vezanja \ izme\:0111u svojstvenih stanja ", Cell[BoxData[ StyleBox[\(\[CurlyPhi]\_i[x]\), "InlineInput"]]], " i ", Cell[BoxData[ StyleBox[\(\[CurlyPhi]\_j[x]\), "InlineInput"]]], " skra\[CAcute]eno pisati kao" }], "Text"], Cell[BoxData[ \(V\_ij = V\_ji \[Congruent] \[Integral]\((\[CurlyPhi]\_i[x]\ V[x, t]\ \[CurlyPhi]\_j[x]\ dx)\)\)], "NumberedEquation"], Cell["\<\ Tada sustav jedna\:017ebi (5) mo\:017eemo zapisati kao:\ \>", "Text"], Cell[BoxData[ StyleBox[ RowBox[{\(\(d\/dt\) \(A\& \[Rule] \)[t]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ { "0", \(\(V\_12\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_12\) t\)\), \(\(V\_13\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_13\) t\)\), ".", ".", \(\(V\_\(1 N\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_\(1 N\)\) t\)\)}, {\(\(V\_12\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_12\) t\)\), "0", \(\(V\_23\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_23\) t\)\), \(\(V\_24\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_24\) t\)\), ".", "."}, {\(\(V\_13\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_13\) t\)\), \(\(V\_23\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_23\) t\)\), "0", ".", \(\(V\_\(N - 3\ N - 1\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_\(N - 3\ N - 1\)\) t\)\), "."}, { ".", \(\(V\_24\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_24\) t\)\), ".", "0", \(\(V\_\(N - 2\ N - 1\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_\(N - 2\ N - 1\)\) t\)\), \(\(V\_\(N - 2\ N\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 2\ N\)\) t\)\)}, {".", ".", \(\(V\_\(N - 2\ N\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 2\ N\)\) t\)\), \(\(V\_\(N - 2\ N - 1\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_\(N - 2\ N - 1\)\) t\)\), "0", \(\(V\_\(N - 1\ N\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 1\ N\)\) t\)\)}, {\(\(V\_\(1 N\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\ \[Omega]\^0\)\_\(1 N\)\) t\)\), ".", ".", \(\(V\_\(N - 2\ N\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 2\ N\)\) t\)\), \(\(V\_\(N - 1\ N\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 1\ N\)\) t\)\), "0"} }], ")"}], \(\(A\& \[Rule] \)[t]\)}]}], FontSize->10]], "NumberedEquation"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Monokromatska vanjska sila. Pribli\:017eenje rotiraju\[CAcute]eg vala \ \>", "Subsubsection"], Cell[TextData[{ "Sila koja djeluje na sustav bit \[CAcute]e uzrokovana homogenim, linearno \ polariziranim, monokromatskim elektri\[CHacek]nim poljem frekvencije ", Cell[BoxData[ FormBox[ StyleBox["\[Omega]", "InlineInput"], TraditionalForm]]], " i amplitude ", Cell[BoxData[ RowBox[{ StyleBox[\(E\_0\), "InlineInput"], ">", "0"}]]] }], "Text"], Cell[BoxData[ \(\(E\& \[Rule] \)[ t] = \(\(\(E\& \[Rule] \)\_0\) Cos[\[Omega]\ t] = \(E\&^\_0\) \(E\_0\/2\) \((\[ExponentialE]\^\(\ \[ImaginaryI]\ \[Omega]\ t\) + \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \[Omega]\ t\))\)\)\)], "NumberedEquation"], Cell["\<\ Uz slijede\[CAcute]u definiciju projekcije dipolnog elektri\[CHacek]nog \ momenta u smjeru paralelnom polarizaciji elektri\[CHacek]nog polja\ \>", "Text"], Cell[BoxData[ \(D\_ij \[Congruent] E\&^\_0\[CenterDot]\(\[Integral]\((\[CurlyPhi]\_i[x]\ D\&^[ x]\ \[CurlyPhi]\_j[x]\ dx)\)\)\)], "NumberedEquation"], Cell[TextData[{ "integrale vezanja ", Cell[BoxData[ StyleBox[\(V\_ij\), "InlineInput"]]], " mo\:017eemo pisati kao" }], "Text"], Cell[BoxData[ \(V\_ij = \(\(D\_ij\ E\_0\)\/2\) \((\[ExponentialE]\^\(\[ImaginaryI]\ \ \[Omega]\ t\) + \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Omega]\ t\))\)\)], \ "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ StyleBox[\(D\&^[x]\), "InlineInput"]]], " prostorni dipolni operator. Nametnutom vanjskom silom bismo \:017eeljeli \ posti\[CAcute]i u\[CHacek]inkovito vezanje izme\:0111u dvaju odabranih \ svojstvenih stanja sustava (ozna\[CHacek]enih sa ", StyleBox["a", FontSlant->"Italic"], " i ", StyleBox["b", FontSlant->"Italic"], "), a istovremeno \[SHacek]to slabije vezanje izme\:0111u svih ostalih \ parova svojstvenih stanja. Dva su uvjeta koji \[CAcute]e nam ovo omogu\ \[CAcute]iti. Prvi je da frekvencija vanjske sile bude bliska rezonantnoj \ frekvenciji prelaza izme\:0111u odabranih stanja ", Cell[BoxData[ FormBox[ StyleBox[\((\[Omega]\_0 \[TildeFullEqual] \[Omega]\_ab)\), "InlineInput"], TraditionalForm]]], ". Drugi je uvjet da amplituda elektri\[CHacek]nog polja lasera ne bude \ prevelika. Ovo se mo\:017ee zaklju\[CHacek]iti promatranjem formalnog rje\ \[SHacek]enja jednad\:017ebe (7):" }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{\(\(A\& \[Rule] \)[t]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"\[Integral]", RowBox[{"(", RowBox[{ RowBox[{"(", GridBox[{ { "0", \(\(V\_12\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_12\) t'\)\), \ \(\(V\_13\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_13\) t'\ \)\), ".", ".", \(\(V\_\(1 N\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(1 N\)\) t\)\)}, {\(\(V\_12\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_12\) t'\)\), "0", \(\(V\_23\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_23\) t'\)\), \ \(\(V\_24\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_24\) t'\ \)\), ".", "."}, {\(\(V\_13\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_13\) t'\)\), \(\(V\_23\) \[ExponentialE]\^\(\(+\ \[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_23\) t'\)\), "0", ".", \(\(V\_\(N - 3\ N - 1\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 3\ N - 1\)\) t'\)\), "."}, { ".", \(\(V\_24\) \ \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_24\) t'\)\), ".", "0", \(\(V\_\(N - 2\ N - 1\)\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 2\ N - 1\)\) t'\)\), \(\(V\_\(N - 2\ N\)\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_\(N - 2\ N\)\) \ t\)\)}, {".", ".", \(\(V\_\(N - 2\ N\)\) \ \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_\(N - 2\ N\)\) t'\)\ \), \(\(V\_\(N - 2\ N - 1\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(N - 2\ N - 1\)\) t'\)\), "0", \(\(V\_\(N - 1\ N\)\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_\(N - 1\ N\)\) \ t\)\)}, {\(\(V\_\(1 N\)\) \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \ \(\(\[Omega]\^0\)\_\(1 N\)\) t'\)\), ".", ".", \(\(V\_\(N - 2\ N\)\) \ \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_\(N - 2\ N\)\) t'\)\ \), \(\(V\_\(N - 1\ N\)\) \ \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(\(\[Omega]\^0\)\_\(N - 1\ N\)\) t'\)\ \), "0"} }], ")"}], \(\(A\& \[Rule] \)[t']\), \(dt'\)}], ")"}]}]}]}], FontSize->10]], "NumberedEquation"], Cell[TextData[{ "Ako je prvi od navedenih dvaju uvjeta zadovoljen, elementi u gornjoj \ matrici mogu se razdijeliti u dvije skupine: sporo rotiraju\[CAcute]e (ili \ \[CHacek]ak stati\[CHacek]ke) i brzo rotiraju\[CAcute]e. U prvu skupinu \ spadaju samo dva \[CHacek]lana koji sadr\:017ee rezonantni ili skoro \ rezonantni produkt ", Cell[BoxData[ StyleBox[\(\[ExponentialE]\^\(\(\[PlusMinus]\[ImaginaryI]\) \((\ \(\(\ \[Omega]\)\(\ \)\(-\)\)\ | \(\[Omega]\^0\)\_ab | )\) t\)\), "InlineInput"]]], ", dok svi ostali spadaju u drugu skupinu. Kako je svaka \ pojedina\[CHacek]na komponenta vektora ", Cell[BoxData[ StyleBox[\(\(A\& \[Rule] \)[t']\), "InlineInput"]]], " po apsolutnoj vrijednosti uvijek manja od ", StyleBox["1", "InlineInput"], ", maksimalna amplituda pojedina\[CHacek]ne komponente ograni\[CHacek]ena \ je integralima oblika:" }], "Text"], Cell[BoxData[ \(\[Integral]\(V\_ij\) \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \(\(\ \[Omega]\^0\)\_ij\) t\)\) dt \[Congruent] \(\(D\_ij\ E\_0\)\/2\) \(\[Integral]\((\ \[ExponentialE]\^\(\(+\[ImaginaryI]\) \((\ \[Omega]\ - \ \ \(\[Omega]\^0\)\_ij)\) t\) + \[ExponentialE]\^\(\(-\[ImaginaryI]\) \((\ \ \[Omega]\ + \(\[Omega]\^0\)\_ij)\) t\))\) dt\)\)], "NumberedEquation"], Cell[TextData[{ "Pojedine komponente vektora stanja sustava ", Cell[BoxData[ StyleBox[\(\(A\& \[Rule] \)[t']\), "InlineInput"]]], " su po svom apsolutnom iznosu uvijek manji od jedan. Ako je vremenski \ raspon promjene vektora ", Cell[BoxData[ StyleBox[\(\(A\& \[Rule] \)[t']\), "InlineInput"]]], " bitno du\:017ea od frekvencije ophodnje oscilatornih \[CHacek]lanova pod \ gornjim integralom, tada \[CAcute]e za ukupni integral unutar tog vremenskog \ raspona vrijediti:" }], "Text"], Cell[BoxData[ \(\(\(|\)\(\[Integral]\(V\_ij\) \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \ \(\(\[Omega]\^0\)\_ij\) t\)\) dt\)\(|\)\(\(\[LessEqual]\)\(\(D\_ij\ \ E\_0\)\/2\)\)\(|\)\((1\/\(\(\ \)\(\[Omega]\ - \ \(\[Omega]\^0\)\_ij\)\) + 1\/\(\(\ \)\(\[Omega]\ + \ \(\[Omega]\^0\)\_ij\)\))\)\(|\)\) = \ \(D\_ij\ \(E\_0\) \[Omega]\)\/\(\(\ \)\(\((\[Omega]\ - \ \ \(\[Omega]\^0\)\_ij)\)\^2\)\)\)], "NumberedEquation"], Cell[TextData[{ "Za sporo rotiraju\[CAcute]e (a osobito za stati\[CHacek]ke) ekponencijalne \ \[CHacek]lanove nazivnik u gornjem integralu biti mnogo manji nego za ostale \ elemente matrice. Zbog toga \[CAcute]e za prve cijeli integral biti bitno \ bitno ve\[CAcute]i nego za druge, pa \[CAcute]e stoga rezonantni elementi u \ matrici gotovo u potpunosti odre\:0111ivati dinamiku cjelokupnog sustava. \ \[CapitalSHacek]to se drugog uvjeta ti\[CHacek]e, uo\[CHacek]imo da iznos \ svih integrala izravno ovisi o produktu ", Cell[BoxData[ StyleBox[\(D\_ij\ E\_0\), "InlineInput"]]], ". Kasnije \[CAcute]emo pokazati kako ovaj umno\:017eak izravno razgrani\ \[CHacek]ava 'jaka' od 'slabih' polja. " }], "Text"], Cell[TextData[{ "Na upravo iznesenom zaklju\[CHacek]ku se temelji ", StyleBox["pribli\:017eenje rotiraju\[CAcute]eg vala", FontSlant->"Italic"], " (RWA - 'Rotating Wave Approximation'), prema kojemu u jednad\:017ebi (8) \ mo\:017eemo zanemariti sve osim dvaju \[CHacek]lanova bliskih rezonanciji" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/dt\) \(A\& \[Rule] \)[t]\), "\[TildeFullEqual]", RowBox[{\(-\[ImaginaryI]\), \(\(\(E\_0\) D\_ab\)\/2\), RowBox[{"(", GridBox[{ {"0", ".", "0", "0", ".", "0"}, {".", ".", ".", ".", ".", "."}, {"0", ".", "0", \(\[ExponentialE]\^\(\(\[PlusMinus]\[ImaginaryI]\) \((\(\ \(\[Omega]\)\(-\)\) | \ \(\[Omega]\^0\)\_ab | )\) t\)\), ".", "0"}, {"0", ".", \(\[ExponentialE]\^\(\(\[MinusPlus]\[ImaginaryI]\) \((\(\ \(\[Omega]\)\(-\)\) | \ \(\[Omega]\^0\)\_ab | )\) t\)\), "0", ".", "0"}, {".", ".", ".", ".", ".", "."}, {"0", ".", "0", "0", ".", "0"} }], ")"}], \(\(A\& \[Rule] \)[t]\)}]}]], "NumberedEquation"], Cell["\<\ Ove \[CAcute]emo \[CHacek]lanove nazvati 'RWA \[CHacek]lanovima', dok \ \[CAcute]emo sve ostale zanemarene \[CHacek]lanove matrice nazvati ne-RWA \ \[CHacek]lanovima. Ako dakle smetnja kojom djelujemo na sustav zadovoljava \ oba ranije navedena uvjeta, sustav od N vezanih svojstvenih stanja pona\ \[SHacek]a se gotovo kao sustav od samo dva vezana stanja. Dinamika ovog \ jednostavnog podsustava biti \[CAcute]e predmetom na\[SHacek]eg \ izu\[CHacek]avanja u slijede\[CAcute]oj cjelini.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["1.2 Rabijeve oscilacije u dvorazinskom kvantnom sustavu ", "Subsection", CellTags->"1.2"], Cell["\<\ Osnovna dinamika Rabijevih oscilacija u sustavu od samo dva svojstvena stanja \ dobro je poznata ve\[CAcute] gotovo pola stolje\[CAcute]a. Ne\[SHacek]to rje\ \:0111e se u literaturi iznosi podrobnija analiti\[CHacek]ka procjena granica \ valjanosti pribli\:017eenja rotiraju\[CAcute]eg vala, a time i Rabijevih \ rezultata. Radi potpunosti u ovom \[CAcute]emo odjeljku izlo\:017eiti opse\ \:017ean pregled ovog jednostavnog, ali za daljnja istra\:017eivanja iznimno \ va\:017enog slu\[CHacek]aja.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Temeljna dinami\[CHacek]ka jednad\:017eba u pribli\:017eenju \ rotiraju\[CAcute]eg vala\ \>", "Subsubsection"], Cell[TextData[{ "Promatramo sustav sa dva svojstvena stanja. Jednad\:017ebu dinamike tog \ sustava nalazimo uvr\[SHacek]tavanjem ", StyleBox["N=2", "InlineInput"], " u relaciju (7)" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/dt\) \(A\& \[Rule] \)[t]\), "=", RowBox[{\(-\[ImaginaryI]\), \(\(\(E\_0\) D\_12\)\/2\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(\(\[Omega]\^0\ \)\_12\) t\)\ \((\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \[Omega]\ t\) + \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Omega]\ t\))\)\)}, {\(\[ExponentialE]\^\(\[ImaginaryI]\ \(\(\[Omega]\^0\)\_12\) \ t\)\ \((\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \[Omega]\ t\) + \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Omega]\ t\))\)\), "0"} }], ")"}], \(\(A\& \[Rule] \)[t]\)}]}]], "NumberedEquation"], Cell[TextData[{ "gdje su definicije vektora stanja sustava ", Cell[BoxData[ StyleBox[\(\(A\& \[Rule] \)[t]\), "InlineInput"]]], " i projekcije dipolnog elektri\[CHacek]nog momenta ", Cell[BoxData[ StyleBox[\(D\_12\), "InlineInput"]]], " analogne onima u izrazima (5) odnosno (9). Ako je frekvencija vanjske \ sile blizu rezonancijske frekvencije prelaza tada dinamiku sustava \ odre\:0111uju samo po jedan od \[CHacek]lanova u svakom od elemenata gornje \ matrice, i to upravo onaj \[CHacek]lan za kojega je razlika izme\:0111u \ frekvencije lasera (za koju po definiciji uzimamo da je ", StyleBox[">0", "InlineInput"], ") i rezonancijske frekvencije prelaza (koja mo\:017ee biti ", StyleBox["<0", "InlineInput"], " ili ", StyleBox[">0", "InlineInput"], ", ovisno o odnosu svojstvenih energija odgovaraju\[CAcute]ih vlastitih \ stanja) bli\:017ea nuli. U RWA se dakle matrica iz jednad\:017ebe (15) \ pojednostavnjuje na:" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/dt\) \(A\& \[Rule] \)[t]\), "=", RowBox[{\(-\[ImaginaryI]\), \(\(\(E\_0\) D\_12\)\/2\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \((\(\(\[Omega]\ \)\(-\)\) | \(\[Omega]\^0\)\_12 | )\) t\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \((\(\(\[Omega]\)\(-\)\) | \(\[Omega]\^0\)\_12 | )\) t\)\), "0"} }], ")"}], \(\(A\& \[Rule] \)[t]\)}]}]], "NumberedEquation"], Cell["\<\ Uvo\:0111enjem zamjenske vremenske veli\[CHacek]ine\ \>", "Text"], Cell[BoxData[ \(\[Tau] \[Congruent] \(\(E\_0 | D\_12 | \)\/2\) t\)], "NumberedEquation"], Cell["\<\ promijenit \[CAcute]emo vremenski raspon problema, \[CHacek]ime se jednad\ \:017eba dinamike daljnje pojednostavnjuje:\ \>", "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{\(\(d\/d\[Tau]\) \(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), " ", \(Sign[D\_12]\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \ \[CapitalDelta]\ \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \[CapitalDelta]\ \ \[Tau]\)\), "0"} }], ")"}], \(\(a\& \[Rule] \)[\[Tau]]\)}]}]}]], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ \(\(a\& \[Rule] \)[\[Tau]]\)]], " zamjenski vektor stanja dobiven iz ", Cell[BoxData[ \(\(A\& \[Rule] \)[t]\)]], " zamjenom ", Cell[BoxData[ \(t \[Rule] \[Tau]\)]], ", a" }], "Text"], Cell[BoxData[ \(\[CapitalDelta] \[Congruent] \ \(\[Omega] - \ \(\[Omega]\^0\)\_12\)\/\(E\_0 | D\_12 | \)\)], "NumberedEquation"], Cell[TextData[{ "Uo\[CHacek]imo da je po\[CHacek]etna jednad\:017eba (15) neosjetljiva na \ obrat u vremenu, a takvi su i po\[CHacek]etni uvjeti koji \[CAcute]emo \ promatrati. Zbog toga u daljnjim ra\[CHacek]unima mo\:017eemo bez smanjenja \ op\[CAcute]enitosti ispustiti predznak ", Cell[BoxData[ StyleBox[\(Sign[D\_12]\), "InlineInput"]]], ". " }], "Text"], Cell[TextData[{ "Primjenom unitarne transformacije ", Cell[BoxData[ StyleBox[\(\(a\& \[Rule] \)[\[Tau]] \[Rule] \(b\& \[Rule] \)[\[Tau]] = \ \(\[ExponentialE]\^\(\(-\[ImaginaryI]\[CapitalLambda]\)\ \[Tau]\)\) \(a\& \ \[Rule] \)[\[Tau]]\), "InlineInput"]]], " na valnu funkciju (predstavljenu vektorom stanja vektor stanja ", Cell[BoxData[ StyleBox[\(\(a\& \[Rule] \)[\[Tau]]\), "InlineInput"]]], "), pri \[CHacek]emu je" }], "Text"], Cell[BoxData[ RowBox[{"\[CapitalLambda]", "=", RowBox[{"(", GridBox[{ {\(\[Phi]\_1\), "0"}, {"0", \(\[Phi]\_2\)} }], ")"}]}]], "NumberedEquation"], Cell["\<\ hermitska matrica, iz matrice u jednad\:017ebi (18) mo\:017eemo ukloniti \ vremensku ovisnost. Ovime hamiltonijan poprima oblik \ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \ \[CapitalDelta]\ \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \[CapitalDelta]\ \ \[Tau]\)\), "0"} }], ")"}], "\[Rule]", RowBox[{"(", GridBox[{ {\(\[Phi]\_1\), \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \((2\ \ \[CapitalDelta] - \((\[Phi]\_1 - \[Phi]\_2)\))\) \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \((2\ \[CapitalDelta] - \ \((\[Phi]\_1 - \[Phi]\_2)\))\)\ \[Tau]\)\), \(\[Phi]\_2\)} }], ")"}]}]], "NumberedEquation"], Cell[TextData[{ "Veli\[CHacek]ine ", Cell[BoxData[ FormBox[ StyleBox[\(\[Phi]\_1\), "InlineInput"], TraditionalForm]]], " i ", Cell[BoxData[ FormBox[ StyleBox[\(\[Phi]\_2\), "InlineInput"], TraditionalForm]]], " su me\:0111usobno potpuno neovisne, te ih mo\:017eemo slobodno odabrati \ tako da bude zadovoljeno ", Cell[BoxData[ FormBox[ StyleBox[\(\[Phi]\_1 - \[Phi]\_2 = 2\ \[CapitalDelta]\), "InlineInput"], TraditionalForm]]], " \[CHacek]ime posti\:017eemo vremensku neovisnost transformiranog \ hamiltonijana. Sa stanovi\[SHacek]ta daljnjeg ra\[CHacek]una, pogodan izbor \ je ", Cell[BoxData[ FormBox[ StyleBox[\(\[Phi]\_1 = 0, \ \ \[Phi]\_2 = \(-2\)\ \[CapitalDelta]\), "InlineInput"], TraditionalForm]]], " kojime (18) prelazi u" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/d\[Tau]\) \(b\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ {"0", "1"}, {"1", \(\(-2\)\ \[CapitalDelta]\)} }], ")"}], \(\(b\& \[Rule] \)[\[Tau]]\)}]}]], "NumberedEquation"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Populacijska dinamika i njezine zna\[CHacek]ajke\ \>", "Subsubsection"], Cell["Svojstvene vrijednosti transformiranog hamiltonijana su", "Text"], Cell[BoxData[ \(\(\[Mu]\_\[PlusMinus]\) = \(-\[CapitalDelta]\) \[PlusMinus] \@\(1 + \ \[CapitalDelta]\^2\)\)], "NumberedEquation"], Cell["\<\ Po\[CHacek]etne uvjete odabrat \[CAcute]emo tako da se u \[Tau]=0 cjelokupna \ populacija sustava nalazi u jednom od vlastitih stanja, recimo prvom\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(a\&\(\(\ \)\(\[Rule]\)\)[\[Tau] = 0]\), "\[Congruent]", RowBox[{"(", GridBox[{ {\(a\_1[\[Tau] = 0]\)}, {\(a\_2[\[Tau] = 0]\)} }], ")"}]}], "=", RowBox[{"(", GridBox[{ {"1"}, {"0"} }], ")"}]}]], "NumberedEquation"], Cell["\<\ Op\[CAcute]e rje\[SHacek]enje jednad\:017ebe (18) uz ove po\[CHacek]etne \ uvjete glasi\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{ RowBox[{"(", GridBox[{ {\(\[ExponentialE]\^\(\[ImaginaryI]\ \[CapitalDelta]\ \ \[Tau]\)\), "0"}, { "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[CapitalDelta]\ \ \[Tau]\)\)} }], ")"}], \(1\/\@\(1 + \[CapitalDelta]\^2\)\), RowBox[{"(", GridBox[{ {\(\(-\@\(1 + \[CapitalDelta]\^2\)\) Cos[\(\@\(1 + \[CapitalDelta]\^2\)\) \[Tau]] - \ \[ImaginaryI]\ \[CapitalDelta]\ Sin[\(\@\(1 + \[CapitalDelta]\^2\)\) \ \[Tau]]\)}, {\(\[ImaginaryI]\ Sin[\(\@\(1 + \[CapitalDelta]\^2\)\) \ \[Tau]]\)} }], ")"}]}]}]], "NumberedEquation"], Cell["\<\ pa je vremenski razvoj populacije dvorazinskog sustava opisan sa\ \>", "Text"], Cell[BoxData[{ \(\(\(\[CapitalPi]\_1[\[Tau]]\)\(\[Congruent]\)\) | \(a\_1\) \((\[Tau])\)\ \( | \^2\) = \(\(\(1\)\(\ \)\)\/\(1 + \[CapitalDelta]\^2\)\) \(1\/2\) \((1 - Cos[2 \(\@\( 1 + \[CapitalDelta]\^2\)\) \[Tau]])\)\), "\ \[IndentingNewLine]", \(\(\(\[CapitalPi]\_2[\[Tau]]\)\(\[Congruent]\)\) | \(a\_2\) \((\[Tau])\)\ \( | \^2\) = 1 - \(1\/\(1 + \[CapitalDelta]\^2\)\) \(1\/2\) \((1 - Cos[2 \(\@\( 1 + \[CapitalDelta]\^2\)\) \[Tau]])\)\)}], \ "NumberedEquation"], Cell[TextData[{ "Iz vremenskog razvoja populacija vidimo da na rezonanciji (", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta] = 0\), "InlineInput"], TraditionalForm]]], ") dolazi do potpunog prijenosa populacije izme\:0111u dvaju svojstvenih \ stanja nakon vremena" }], "Text"], Cell[BoxData[{ \(\[Tau]\_Rabi = \(\(\[Pi]\)\(\ \)\)\/2\), "\[IndentingNewLine]", \(T\_Rabi = \[Pi]\/\(E\_0 | D\_12 | \)\)}], "NumberedEquation"], Cell[TextData[{ "Rezonantne Rabijeve oscilacije prikazane su na ", StyleBox["Slici 1", FontWeight->"Bold"], ". 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GridBoxOptions->{ColumnAlignments->{Center}}]], "Text", TextAlignment->Center], Cell[TextData[{ StyleBox["Slika 1 -", FontWeight->"Bold", FontSlant->"Plain"], " Vremenski razvoj populacija dvorazinskog kvantnog sustava pogonjenog to\ \[CHacek]no rezonantnom vanjskom silom (\[CapitalDelta]=0). Dinamika je odre\ \:0111ena numeri\[CHacek]kim rje\[SHacek]avanjem jednad\:017ebe (15) uz po\ \[CHacek]etne uvjete (24). Dobro slaganje sa analiti\[CHacek]kim RWA \ rezultatom (26) je o\[CHacek]ito." }], "Commentary"], Cell["\<\ Odmicanje frekvencije vanjske sile od rezonancije ima dvojaki u\[CHacek]inak \ na dinami\[CHacek]ke zna\[CHacek]ajke sustava. Kao prvo, smanjuje se vrijeme \ prijenosa populacije\ \>", "Text"], Cell[BoxData[{ \(\[Tau]\_Rabi = \(\[Pi]\/2\) 1\/\@\(1 + \[CapitalDelta]\^2\)\), "\[IndentingNewLine]", \(T\_Rabi = \(\[Pi]\/\(E\_0 | D\_12 | \)\) 1\/\@\(1 + \((\[Omega] - \(\[Omega]\^0\)\_12)\)\^2\/\((\(E\_0\) \ D\_12)\)\^2\)\)}], "NumberedEquation"], Cell[TextData[{ "Drugi je u\[CHacek]inak smanjenje amplitude oscilacija populacija, zbog \ kojega se van rezonancije ne mo\:017ee posti\[CAcute]i potpuni prijenos. U \ trenutku ", Cell[BoxData[ \(TraditionalForm\`\[Tau] = \[Tau]\_Rabi\)]], ", populacija po\[CHacek]etno nenaseljenog vlastitog stanja dosti\:017ee \ svoju najve\[CAcute]u vrijednost koja iznosi" }], "Text"], Cell[BoxData[ \(P[\[CapitalDelta]] \[Congruent] 1\/\(1 + \[CapitalDelta]\^2\)\)], "NumberedEquation"], Cell[TextData[{ "Oba ova u\[CHacek]inka odmaka od rezonancije na dinamiku populacija \ prikazani su na ", StyleBox["Slici 2", FontWeight->"Bold"], ". Funkciju ", Cell[BoxData[ StyleBox[\(P[\[CapitalDelta]]\), "InlineInput"]]], " \[CAcute]emo nazvati 'Rabijevim profilom' a kasnije \[CAcute]emo \ pokazati kako je ona vrlo pogodan alat za izu\[CHacek]avanje utjecaja smetnje \ uzrokovane dodatnim svojstvenim stanjima sustava. Kada je ", StyleBox["\[CapitalDelta]=1", "InlineInput"], ", iznos prijenosa populacije pada na jednu polovinu. Ako Rabijev profil \ promatramo kao funkciju frekvencije ", StyleBox["\[Omega]", "InlineInput"], ", nalazimo da ", StyleBox["jakost vezanja ", FontSlant->"Italic"], StyleBox[Cell[BoxData[ \(I\_12 \[Congruent] E\_0 | D\_12 | \)], "InlineInput"], "InlineInput"], " odre\:0111uje njegovu polu\[SHacek]irinu. Valja uo\[CHacek]iti da je \ Rabijev profil istovremeno funkcija svojstava sustava (projekcije dipolnog \ elektri\[CHacek]nog momenta ", StyleBox[Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["D", FontSlant->"Plain"], "12"], TraditionalForm]], "InlineInput"], "InlineInput"], " i rezonancijske frekvencije prelaza ", StyleBox[Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\^0\)\_12\)], "InlineInput"], "InlineInput"], ") i svojstava vanjske sile (", StyleBox[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], "0"], ",", "\[Omega]"}], TraditionalForm]], "InlineInput"], "InlineInput"], StyleBox[")", "InlineInput"] }], "Text"], Cell[BoxData[ \(P[\[CapitalDelta]] \[Congruent] P[\[Omega], \(\[Omega]\^0\)\_12, I\_12] = 1\/\(1 + \((\(\[Omega] - \(\[Omega]\^0\)\_12\)\/I\_12)\)\^2\)\)], \ "NumberedEquation"], Cell[TextData[{ "Na ", StyleBox["Slici 3", FontWeight->"Bold"], " prikazano je", StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], "nekoliko primjera Rabijevih profila razli\[CHacek]itih polu\[SHacek]irina \ za istu spektralnu liniju. U slijede\[CAcute]em \[CAcute]emo poglavlju \ pokazati kako u sustavu sa vi\[SHacek]e od dva vezana vlastita stanja \ odmakom od rezonancije mo\:017eemo donekle pove\[CAcute]ati amplitudu \ prijenosa populacije izme\:0111u odabranih svojstvenih stanja. 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Redom od gore prema dolje jakosti vezanja (a time i polu\[SHacek]irine \ profila) iznose 0.005, 0 .05 i 0 .2. U sva tri slu\[CHacek]aja \ sredi\[SHacek]te profila nalazi se na rezonantnoj frekvenciji ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\^0\)\_12 = 1\)]], "." }], "Commentary"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1.3 Valjanost pribli\:017eenja rotiraju\[CAcute]eg vala\ \>", "Subsection", CellTags->"1.3"], Cell[TextData[{ "Da bismo istra\:017eili ograni\[CHacek]enja primjenjivosti RWA, moramo \ utvrditi dinami\[CHacek]ku zna\[CHacek]ajnost ne-RWA \[CHacek]lanova ", Cell[BoxData[ StyleBox[\(\[ExponentialE]\^\(\(\[PlusMinus]\[ImaginaryI]\)\ \((\(\(\ \[Omega]\)\(+\)\) | \(\[Omega]\^0\)\_12 | )\) t\)\), "InlineInput"]]], " u matrici hamiltonijana u jednad\:017ebi (15). Dva su parametra o kojima \ moramo voditi ra\[CHacek]una: jakost vezanja i odmak od rezonancije. " }], "Text"], Cell[CellGroupData[{ Cell["Ekstremne jakosti vezanja", "Subsubsection"], Cell[TextData[{ "Iz (27) se jasno uo\[CHacek]ava da \[LongDash] \[CHacek]ak i u \ slu\[CHacek]aju rezonancije (", StyleBox["\[CapitalDelta]=0", "InlineInput"], ") \[LongDash] ekstremna vrijednost jakosti vezanja ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["I", "InlineInput", FontSlant->"Plain"], "12"], TraditionalForm]]], " \[CHacek]ini da vrijeme prijenosa populacije formalno postaje usporedivo \ ili \[CHacek]ak manje od vremena potrebnog da ne-RWA element na\[CHacek]ini \ jedan puni titraj u kompleksnoj ravnini. Analiti\[CHacek]ki ovo se mo\:017ee \ zapisati kao:" }], "Text"], Cell[BoxData[ RowBox[{\(I\_12 \[Congruent] E\_0\), "|", SubscriptBox[ StyleBox["D", FontSlant->"Plain"], "12"], "|", \(\(\[GreaterTilde]\)\(\(\[Omega]\)\(+\)\)\), "|", \(\(\[Omega]\^0\)\_12\), "|", \(\(\[TildeEqual]\)\(2 \( \[Omega]\^0\)\_12\)\)}]], \ "NumberedEquation"], Cell[TextData[{ "U ovom slu\[CHacek]aju ne mo\:017eemo tek tako zanemariti \ dinami\[CHacek]ki utjecaj brzih elemenata u matrici u jednad\:017ebi (15). \ Naime, temeljna pretpostavka RWA da se njihov doprinos integralu (12) poni\ \[SHacek]ti za vrijeme trajanja jednog titraja populacije nije vi\[SHacek]e \ zadovoljena. U slu\[CHacek]aju iznimno jakih polja morali bismo \ ra\[CHacek]unati sa trenutnom vrijedno\[SHacek]\[CAcute]u amplitude lasera, \ koja je gotovo konstantna za vrijeme jednog ciklusa prenosa populacije. Ono \ \[SHacek]to \[CAcute]emo dobiti kao rje\[SHacek]enje za dinamiku sustava su \ ponovno titranja populacije izme\:0111u dviju razina, no ovaj puta njihov \ period ne\[CAcute]e biti konstantan ve\[CAcute] \[CAcute]e se \ periodi\[CHacek]ki mijenjati. Na ", StyleBox["Slici 4", FontWeight->"Bold"], " prikazano je nekoliko primjera dinamika populacije u uvjetima ekstremno \ jakog vezanja. Naravno, mo\:017eemo o\[CHacek]ekivati da \[CAcute]e ovako \ jaka polja bitno izobli\[CHacek]iti kvantne sustave koje razmatramo. Nadalje, \ zbog iznimne ovisnosti reakcijske sile zra\[CHacek]enja o brzini dinami\ \[CHacek]kih promjena u sustavu, a koja u ovom slu\[CHacek]aju postaje \ ekstremna, njezin bi doprinos tako\:0111er trebalo uklju\[CHacek]iti u dinami\ \[CHacek]ke jednad\:017ebe kako bi se dobila to\[CHacek]na rje\[SHacek]enja. \ Tako\:0111er bi i magnetska sastavnica laserskog zra\[CHacek]enja mogla po\ \[CHacek]eti igrati zna\[CHacek]ajnu ulogu. Sve u svemu, pri ovim uvjetima \ cjelokupna dinamika postaje vrlo neure\:0111ena a upravljanje njome prakti\ \[CHacek]ki nemogu\[CAcute]e i stoga neupotrebljivo sa stanovi\[SHacek]ta \ koherentnog upravljanja. Zbog svega navedenog ovaj slu\[CHacek]aj \ ne\[CAcute]emo dalje razmatrati. " }], "Text"], Cell[TextData[{ "Prisjetiv\[SHacek]i se da je sa jako\[SHacek]\[CAcute]u vezanja ", Cell[BoxData[ SubscriptBox[ StyleBox["I", "InlineInput"], "12"]]], " odre\:0111ena polu\[SHacek]irina Rabijevog profila, korisno je \ uo\[CHacek]iti da gore navedeni uvjet odgovara tolikom \[SHacek]irenju \ Rabijevog profila da ona postaje usporediva ili \[CHacek]ak i ve\[CAcute]a od \ udaljenosti sredi\[SHacek]ta linije od ishodi\[SHacek]ta. Mo\:017eemo zaklju\ \[CHacek]iti da \[CAcute]emo se u nastavku ograni\[CHacek]iti na \ izu\[CHacek]avanje samo takve slu\[CHacek]ajeve u kojima je \ polu\[SHacek]irina Rabijevog profila bitno u\:017ea od njegove udaljenosti od \ ishodi\[SHacek]ta" }], "Text"], Cell[BoxData[ \(\(I\_12\) << 2\ \(\[Omega]\^0\)\_12\)], "NumberedEquation"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .33333 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0 0.333333 [ [.2619 -0.0125 -9 -9 ] [.2619 -0.0125 9 0 ] [.5 -0.0125 -3 -9 ] [.5 -0.0125 3 0 ] [.7381 -0.0125 -9 -9 ] [.7381 -0.0125 9 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [1.025 0 0 -6.28125 ] [1.025 0 10 6.28125 ] [.01131 .06667 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Populacijska dinamika i odgovaraju\[CAcute]i Rabijevi profili za \ rezonantnu pobudu i uz ekstremno jaka vezanja. Od gore prema dolje jakosti \ vezanja su redom 0.1, 1 i 10. U prvom je slu\[CHacek]aju uvjet (32) jo\ \[SHacek] uvijek gotovo zadovoljen, ali se ipak uo\[CHacek]ava odstupanje od \ \[CHacek]iste Rabijeve dinamike u vidu sitnih (ali zamijetnih) brzih \ oscilacija superponiranih na Rabijev prijenos. U preostalim dvama \ slu\[CHacek]ajevima odstupanja su o\[CHacek]ita. U zadnjem se slu\[CHacek]aju \ jasno uo\[CHacek]ava periodi\[CHacek]ka modulacija vremena prenosa.", FontWeight->"Plain", FontSlant->"Italic"] }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"] }, Open ]], Cell[CellGroupData[{ Cell["Relativni frekventni odmak", "Subsubsection"], Cell[TextData[{ "Veli\[CHacek]ina ", StyleBox["\[CapitalDelta]", "InlineInput"], " ima ulogu relativnog odmaka od rezonantne frekvencije normaliziranog polu\ \[SHacek]irinom Rabijevog profila. Argument za valjanost RWA u \ slu\[CHacek]aju kona\[CHacek]nog frekventnog odmaka to\[CHacek]no slijedi \ onoga za prethodni, rezonantni slu\[CHacek]aj: vrijeme prijenosa populacije \ mora biti mnogo du\:017ee od vremena jednog titraja ne-RWA elemena. Matemati\ \[CHacek]ki zapisano to glasi" }], "Text"], Cell[BoxData[ \(\(\@\(1 + \((\(\[Omega] - \(\[Omega]\^0\)\_12\)\/I\_12)\)\^2\)\) << \ \(\(\[Omega] + \(\[Omega]\^0\)\_12\)\/I\_12\)\)], "NumberedEquation"], Cell[TextData[{ "Budu\[CAcute]i razmatramo samo profile polu\[SHacek]irina manjih od ", Cell[BoxData[ FormBox[ StyleBox[\(\(\[Omega]\^0\)\_12\), "InlineInput"], TraditionalForm]]], " (vidi (32)) mo\:017eemo postaviti gornju granicu na frekvenciju pogona ", StyleBox["\[Omega]", "InlineInput"], " \[LongDash] u obzir moramo uzeti samo frekvencije ", Cell[BoxData[ FormBox[ StyleBox[\(\[Omega] < 2 \( \[Omega]\^0\)\_12\), "InlineInput"], TraditionalForm]]], ", budu\[CAcute]i da je za ve\[CAcute]e frekvencije visina Rabijevog \ profila zanemarivo mala. Nadalje, za velike relativne odmake od rezonancije \ razlomak pod korjenom u gornjem izrazu je znatno ve\[CAcute]i od jedan, pa \ pribli\:017eno vrijedi" }], "Text"], Cell[BoxData[ \(\(\(|\)\(\[Omega] - \(\[Omega]\^0\)\_12\)\(|\)\(<< 3 \( \[Omega]\^0\)\_12\)\)\)], "NumberedEquation"], Cell[TextData[{ "Ovo je upravo tra\:017eeni kriterij za ovisnost valjanosti RWA o \ relativnom frekventnom odmaku. Uzev\[SHacek]i zajedno rezultate (32) i (34) u \ okviru RWA i za relativno uske profile kakve promatramo, Rabijevo \ rje\[SHacek]enje izvrsno opisuje amplitudu prijenosa populacije u cjelokupnom \ podru\[CHacek]ju frekvencija lasera koje nam je od zna\[CHacek]aja. Ovaj \ zaklju\[CHacek]ak nedvojbeno potvr\:0111uju primjeri dinamike populacija \ prikazani na ", StyleBox["Slici 5", FontWeight->"Bold"], ". " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Dinami\[CHacek]ki u\[CHacek]inci ne-RWA mati\[CHacek]nih elemenata\ \>", "Subsubsection"], Cell[TextData[{ "Posljednji dio rasprave o valjanosti RWA posvetit \[CAcute]emo \ istra\:017eivanju potankosti dinami\[CHacek]kog utjecaja ne-RWA \ \[CHacek]lanova u jednad\:017ebi (15)", ButtonBox[".", ButtonData:>"j15", ButtonStyle->"Hyperlink"], " Probajmo na\[CAcute]i pribli\:017eno potpuno (RWA + ne-RWA) analiti\ \[CHacek]ko rje\[SHacek]enje oblika" }], "Text"], Cell[BoxData[ \(\(a\& \[Rule] \)[\[Tau]] \[Congruent] \(a\& \[Rule] \)\_RWA[\[Tau]] + \ \(a\& \[Rule] \)\_\(ne - RWA\)[\[Tau]]\)], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ \(\(a\& \[Rule] \)\_RWA[\[Tau]]\)]], " RWA rje\[SHacek]enje koje smo ve\[CAcute] ranije na\[SHacek]li, dok je \ ", Cell[BoxData[ \(\(a\& \[Rule] \)\_\(ne - RWA\)[\[Tau]]\)]], " mala popravka. Kada ovo uvrstimo u (15), nalazimo" }], "Text"], Cell[BoxData[{ RowBox[{\(\(d\/d\[Tau]\) \(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \ \[CapitalDelta]\ \[Tau]\) + \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \ \[Delta]\ \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \[CapitalDelta]\ \ \[Tau]\) + \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \[Delta]\ \[Tau]\)\), "0"} }], ")"}], \(\(a\& \[Rule] \)[\[Tau]]\)}]}], "\[IndentingNewLine]", RowBox[{\(\(d\/d\[Tau]\) \(a\& \[Rule] \)\_\(ne - RWA\)[\[Tau]]\), "\[TildeFullEqual]", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \[Delta]\ \ \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \[Delta]\ \[Tau]\)\ \), "0"} }], ")"}], \(\(a\& \[Rule] \)\_RWA[\[Tau]]\)}]}]}], "NumberedEquation"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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}]]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]]], GridBoxOptions->{RowAlignments->Center}]], "Text"], Cell[TextData[{ StyleBox["Slika 5 -", FontWeight->"Bold", FontSlant->"Plain"], " Rabijeve oscilacije za nekoliko slu\[CHacek]ajeva velikih odmaka od \ rezonancije. Polu\[SHacek]irina Rabijevog profila u sva tri slu\[CHacek]aja \ iznosi 0.01 tako da je uvjet (34) u potpunosti zadovoljen. Odmak od \ rezonancije sa lijeva na desno je takav da je ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[CapitalDelta]\^2 = 9\), "TraditionalForm"], ",", \(\[CapitalDelta]\^2 = 99\)}], TraditionalForm]]], " i ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\^2 = 999\)]], ". Jasno se uo\[CHacek]ava izvrsno slaganje sa analiti\[CHacek]kim RWA \ izrazom (29)." }], "Commentary"], Cell[TextData[{ "gdje je ", StyleBox["\[CapitalDelta]", "InlineInput"], " definiran kao i ranije (vidi (19)), dok je" }], "Text"], Cell[BoxData[ \(\[Delta] \[Congruent] \(\[Omega] + \(\[Omega]\^0\)\_12\)\/I\_12\)], \ "NumberedEquation"], Cell["\<\ Formalno rje\[SHacek]enje ne-RWA jednad\:017ebe glasi:\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(a\& \[Rule] \)\_\(ne - RWA\)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"\[Integral]", RowBox[{\(d\[Tau]'\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \[Delta]\ \ \[Tau]'\)\)}, {\(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \[Delta]\ \ \[Tau]'\)\), "0"} }], ")"}], \(\(a\& \[Rule] \)\_RWA[\[Tau]']\)}]}]}]}]], \ "NumberedEquation"], Cell["\<\ Nama zapravo niti ne treba to\[CHacek]no rje\[SHacek]enje ove jednad\:017ebe. \ Ono \[SHacek]to nas zanima jest procjeniti red veli\[CHacek]ine raspona \ ne-RWA popravke, jer sve dok je on bitno manji od RWA doprinosa, ukupno \ \[CAcute]e rje\[SHacek]enje biti dobro opisano pribli\:017enim RWA rje\ \[SHacek]enjem. Dok su god zadovoljeni uvjeti navedeni u prethodnoj cjelini, \ RWA rje\[SHacek]enje mo\:017eemo smatrati sporo promjenjivim parametrom te ga \ mo\:017eemo izvu\[CAcute]i izvan integrala. Tada nam ostaje samo integrirati \ jednostavne eksponencijalne funkcije, pa je kona\[CHacek]no \ rje\[SHacek]enje\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(a\& \[Rule] \)\_\(ne - RWA\)[\[Tau]; \[Tau]\_1 < \[Tau] < \ \[Tau]\_2]\), "\[TildeFullEqual]", RowBox[{\(1\/\[Delta]\), RowBox[{"(", RowBox[{ RowBox[{\(1\/2\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \ \[Delta]\ \[Tau]'\)\)}, {\(-\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \[Delta]\ \ \[Tau]'\)\), "0"} }], ")"}]}], \(\( | \+\(\[Tau]' = \[Tau]\_1\)\)\&\(\[Tau]' = \ \[Tau]\_2\)\)}], ")"}], \(\(a\& \[Rule] \)\_RWA[\(\[Tau]\_2 - \ \[Tau]\_1\)\/2]\)}]}]], "NumberedEquation"], Cell[TextData[{ "Veli\[CHacek]ina ", Cell[BoxData[ FormBox[ StyleBox[\(1\/\[Delta]\), "InlineInput"], TraditionalForm]]], " u potpunosti odre\:0111uje omjer ne-RWA popravke i RWA rje\[SHacek]enja. \ Dva elementa u matrici rotiraju u kompleksnoj ravnini, a njihova apsolutna \ vrijednost je uvijek jednaka jedinici pa cijeli izraz u zagradama ima raspon \ ", StyleBox["1", "InlineInput"], ". Dakle, iznos ne-RWA doprinosa bitno je manji od iznosa RWA doprinosa dok \ god je zadovoljeno: " }], "Text"], Cell[BoxData[ \(\(1\/\[Delta]\) << 1\)], "NumberedEquation"], Cell["ili druga\[CHacek]ije zapisano", "Text"], Cell[BoxData[ \(\(I\_12\) << 3\ \(\[Omega]\^0\)\_12\)], "NumberedEquation"], Cell["\<\ Uo\[CHacek]imo da je ovo ne\[SHacek]to bla\:017ei oblik uvjeta (31) te je \ stoga u okolnostima koje mi uzimamo u razmatranje automatski zadovoljeno.\ \>", "Text"], Cell["\<\ Na poslijetku mo\:017eemo zaklju\[CHacek]iti da RWA rje\[SHacek]enje \ predstavlja izvrsno pribli\:017eenje potpunom rje\[SHacek]enju u cijelom \ podru\[CHacek]ju frekvencija koje nam je od zna\[CHacek]aja dok je god polu\ \[SHacek]irina Rabijevog profila bitno manja od udaljenosti sredi\[SHacek]ta \ linije od ishodi\[SHacek]ta.\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ POGLAVLJE 2 Rabijeva titranja u vi\[SHacek]erazinskim kvantnim sustavima\ \>", "Section 1", CellTags->"POGLAVLJE 2"], Cell["\<\ Nakon \[SHacek]to smo u prethodnom poglavlju potanko ispitali svojstva \ rabijevih oscilacija u sustavu sa samo dvije svojstvene razine, u ovom \ poglavlju izu\[CHacek]avamo utjecaj prisustva dodatnih svojstvenih razina u \ sustavu na dinamiku prijenosa populacije. Prvo u sustav uvodimo jednu dodatnu \ razinu vezanu elektri\[CHacek]nim dipolnim integralom samo sa jednim od po\ \[CHacek]etnih razina, te na temelju spoznaja iz pro\[SHacek]log poglavlja \ ugrubo ispitujemo kakvo bismo pona\[SHacek]anje u takvome sustavu mogli o\ \[CHacek]ekivati. Potom, koriste\[CAcute]i dva razli\[CHacek]ita analiti\ \[CHacek]ka pristupa, potanko istra\:017eujemo utjecaj ove smetnje na \ promjenu zna\[CHacek]ajki \[CHacek]iste Rabijeve dinamike. Na poslijetku poop\ \[CAcute]avamo rezultate dobivene za jednu razinu smetnje na slu\[CHacek]aj \ sa proizvoljnim brojem razina smetnje.\ \>", "Text", PageBreakBelow->Automatic], Cell[CellGroupData[{ Cell["\<\ 2.1 Temeljna jednad\:017eba dinamike trorazinskog sustava i povr\[SHacek]na \ analiza\ \>", "Subsection", CellTags->"2.1"], Cell["\<\ Bez obzira na to\[CHacek]ne pojedinosti dinamike, u sustavu sa samo dvije \ razine cjelokupna je populacija sustava bila nu\:017eno ograni\[CHacek]ena na \ obitavanje u tim dvama vezanim razinama. Uvedemo li u sustav dodatne razine \ vezane sa barem jednom od postoje\[CAcute]ih dviju, postizanje potpunog \ prijenosa populacije izme\:0111u dviju \:017eeljenih razina postaje nemogu\ \[CAcute]e jer \[CAcute]e vanjska sila dio populacije neizbje\:017eno \ prenijeti i u dodanu 'razinu smetnje'. Ono \[SHacek]to ipak mo\:017eemo poku\ \:017eati u\[CHacek]initi jest utvrditi kriterije koji bi nam \ omogu\[CAcute]ili nadzor i ograni\[CHacek]avanje tog 'curenja' populacije van \ \:017eeljene putanje prijenosa. S tim \[CAcute]emo ciljem istra\:017eiti \ kakva je ovisnost curenja o svojstvima vanjske pogonske sile (njeznoj jakosti \ i frekvenciji) za dane parametre sustava (dipolna vezanja i rezonancijske \ frekvencije prelaza).\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Jednad\:017eba dinamike trorazinskog sustava\ \>", "Subsubsection"], Cell[TextData[{ "Dvije razine izme\:0111u kojih \:017eelimo posti\[CAcute]i prijenos \ populacije ozna\[CHacek]it \[CAcute]emo sa '", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], "' (", StyleBox["temeljna", FontSlant->"Italic"], ") i '", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], "' (", StyleBox["posredni\[CHacek]ka", FontSlant->"Italic"], ") a dodatnu razinu ", StyleBox["smetnje", FontSlant->"Italic"], " sa '", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], "'. Temeljnu razinu vanjska sila vezuje samo sa posredni\[CHacek]kom, dok \ je posredni\[CHacek]ku vezuje i sa razinom smetnje. Posredni\[CHacek]ka \ razina 'posreduje' u curenju populacije izvan \:017eeljenog dijela sustava. \ Po\[CHacek]etno se cjelokupna populacija nalazi u jednoj od razina ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " ili ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], ", dok je ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], " prazna. \:017delimo posti\[CAcute]i \[SHacek]to potpuniji prijenos \ populacije izme\:0111u ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " a istovremeno u\[CHacek]initi curenje (populaciju u ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], ") \[SHacek]to manjim. Osim toga, prijenos \:017eelimo u\[CHacek]initi \ \[SHacek]to br\:017eim kako bismo mogli zanemariti spore procese spontane \ emisije koji nu\:017eno postoje u svakom stvarnom sustavu. Iz rezultata (27) \ i (28) vidimo da se vrijeme prenosa mo\:017ee skratiti pove\[CAcute]avanjem \ jakosti vezanja. Budu\[CAcute]i da su projekcije elektri\[CHacek]nog momenta \ ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["D", FontSlant->"Plain"], "ij"], TraditionalForm]], FontFamily->"Courier"], " (definirane analogno onima u (9)) zadane samim sustavom, ostaje nam samo \ mogu\[CAcute]nost pove\[CAcute]anja jakosti vanjskog pogona. No u prethodnom \ smo poglavlju pokazali kako postoji gornja granica jakosti vezanja, a time \ izravno i jakosti pogona, koju ne smijemo prekora\[CHacek]iti ako \:017eelimo \ ostati u podru\[CHacek]ju valjanosti RWA. \[CapitalSHacek]to se ti\[CHacek]e \ frekvencije lasera, mo\:017eemo o\[CHacek]ekivati da \[CAcute]emo ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["-", FontFamily->"Courier"], StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " prijenos posti\[CAcute]i uz frekvenciju koja je, ako ne to\[CHacek]no \ rezonantna za ovaj prelaz, a ono joj je barem bliska. " }], "Text"], Cell[TextData[{ "Potpune dinami\[CHacek]ke jednad\:017ebe sustava ", StyleBox["t-p-s", FontFamily->"Courier", FontSlant->"Italic"], " nalazimo prilagodbom op\[CAcute]enite jednad\:017ebe (7). Radi lak\ \[SHacek]eg algebarskog rukovanja upotrebljavat \[CAcute]emo \ slijede\[CAcute]e skra\[CAcute]ene zapise za svojstvene frekvencije sustava" }], "Text"], Cell[BoxData[{ \(\(\(\[Omega]\^0\)\_\[Alpha]\[Beta] = \(\(\(\[Omega]\^0\)\_\[Beta]\ \[Alpha]\)\(\[Congruent]\)\) | \[Omega]\_\[Alpha] - \[Omega]\_\[Beta] | \ ;\)\), "\[IndentingNewLine]", \(s\_\[Alpha]\[Beta] = \(-s\_\[Beta]\[Alpha]\) \[Congruent] Sign[\[Omega]\_\[Alpha] - \[Omega]\_\[Beta]\ ]\)}], "NumberedEquation"], Cell[TextData[{ "gdje su \[Alpha] i \[Beta] odgovaraju\[CAcute]e oznake stanja vezanih \ pojedinom frekvencijom (", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], ", ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], "). U matri\[CHacek]nom zapisu jednad\:017eba dinamike mo\:017ee se pogodno \ razdijeliti na 'RWA', 'mogu\[CAcute]e-RWA' i 'ne-RWA' dijelove:" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/dt\) \(A\& \[Rule] \)[t]\), "=", RowBox[{\(-\[ImaginaryI]\), \(E\_0\/2\), RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], RowBox[{"(", GridBox[{ {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], ")"}], "t"}]], "0"}, { SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "-", StyleBox[ SubscriptBox[ SuperscriptBox[ StyleBox["\[Omega]", FontSlant->"Plain"], "0"], "pt"], FontSlant->"Italic"]}], ")"}], "t"}]], "0", "0"}, {"0", "0", "0"} }], ")"}]}], "+", RowBox[{ SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], RowBox[{"(", GridBox[{ {"0", "0", "0"}, {"0", "0", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}], ")"}], "t"}]]}, {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}], ")"}], "t"}]], "0"} }], ")"}]}], "+", RowBox[{"(", GridBox[{ {"0", RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "+", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], ")"}], "t"}]]}], "0"}, { RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "+", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], ")"}], "t"}]]}], "0", RowBox[{ SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "+", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}], ")"}], "t"}]]}]}, {"0", RowBox[{ SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], RowBox[{"(", RowBox[{"\[Omega]", "+", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}], ")"}], "t"}]]}], "0"} }], ")"}]}], ")"}], \(\(A\& \[Rule] \)[ t]\)}]}]], "NumberedEquation"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Povr\[SHacek]na analiza temeljena na Rabijevim profilima\ \>", "Subsubsection"], Cell[TextData[{ "Sada se mo\:017eemo okoristiti pojmom Rabijevog profila kako bismo ugrubo \ ispitali dinami\[CHacek]ke zna\[CHacek]ajke trorazinskog sustava. Prvo uo\ \[CHacek]imo sli\[CHacek]nosti i razlike izme\:0111u jednad\:017ebi (43) i \ (15). Obje sadr\:017ee nekoliko sporo rotiraju\[CAcute]ih ('RWA') te nekoliko \ brzo rotiraju\[CAcute]ih ('ne-RWA') \[CHacek]lanova. U jed. (43) oni su \ sakupljeni u prvoj odnosno tre\[CAcute]oj matrici. \[CapitalCHacek]lanovi \ druge matrice u istoj jednad\:017ebi po brzini rotacije su negdje izme\:0111u \ ovih dvaju ekstremnih slu\[CHacek]ajeva, ovisno o odnosu dviju svojstvenih \ frekvencija sustava ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ SuperscriptBox[ StyleBox["\[Omega]", FontWeight->"Plain"], "0"], StyleBox["pt", FontSlant->"Italic"]], "InlineInput", FontFamily->"Courier"], TraditionalForm]]], " i ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]], FontFamily->"Courier"], TraditionalForm]]], ". Rabijev profil svake od dviju linija funkcija je odgovaraju\[CAcute]eg \ relativnog odmaka od rezonancije:" }], "Text"], Cell[BoxData[{ RowBox[{ SubscriptBox["\[CapitalDelta]", StyleBox["t", FontSlant->"Italic"]], "\[Congruent]", FractionBox[ RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]]}], "\[IndentingNewLine]", RowBox[{\(\[CapitalDelta]\_s\), "\[Congruent]", FractionBox[ RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}], RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], "|"}]]}]}], "NumberedEquation"], Cell[TextData[{ "U posljednoj smo cjelini prethodnog poglavlja pokazali da je preduvjet \ mogu\[CAcute]nosti dobrog upravljanja prijenosom populacije da \ polu\[SHacek]irina ciljane linije prelaza bude manja od njene udaljenosti od \ ishodi\[SHacek]ta. Iz rasprave u drugoj cjelini prethodnog poglavlja znamo da \ postizanje zna\[CHacek]ajnog prijenosa populacije izme\:0111u ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " zahtjeva ", Cell[BoxData[ RowBox[{"|", SubscriptBox["\[CapitalDelta]", StyleBox["t", FontSlant->"Italic"]], "|", \(<< 1\)}]]], ". Suprotno tome, da bismo smanjili curenje, frekvencija pogonske sile mora \ biti daleko u krilu Rabijevog profila linije ne\:017eeljenog prelaza ", StyleBox["p-s", FontFamily->"Courier", FontSlant->"Italic"], ", odnosno ", Cell[BoxData[ RowBox[{ RowBox[{"|", SubscriptBox["\[CapitalDelta]", StyleBox["s", FontSlant->"Italic"]], "|"}], ">>", "1"}]]], ". Uvedimo zamjenske veli\[CHacek]ine ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], ", ", StyleBox["\[Sigma]", FontFamily->"Courier"], " i ", StyleBox["R", FontFamily->"Courier"], " odre\:0111ene na slijede\[CAcute]i na\[CHacek]in" }], "Text"], Cell[BoxData[{ RowBox[{"\[CapitalDelta]", "\[Congruent]", SubscriptBox["\[CapitalDelta]", StyleBox["t", FontSlant->"Italic"]], "\[Congruent]", FractionBox[ RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]]}], "\[IndentingNewLine]", RowBox[{"\[Sigma]", "\[Congruent]", FractionBox[ RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], "|"}], RowBox[{ SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]], "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}]]}], "\[IndentingNewLine]", RowBox[{\(\(R\)\(\[Congruent]\)\), "|", FractionBox[ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]]], "|"}]}], "NumberedEquation"], Cell["\<\ Preko njih relativni odmak od rezonancije za liniju smetnje mo\:017eemo \ izraziti kao\ \>", "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["\[CapitalDelta]", StyleBox["s", FontSlant->"Italic"]], "=", RowBox[{ RowBox[{"|", FractionBox[ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]]], "|", RowBox[{ FractionBox[ RowBox[{"\[Omega]", "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]], "+", FractionBox[ RowBox[{ SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]], "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}], RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], "|"}]]}]}], "\[IndentingNewLine]", "=", \(\(1\/R\) \[CapitalDelta] + 1\/\[Sigma]\)}]}]], "NumberedEquation"], Cell[TextData[{ "Ako frekvenciju vanjske sile ugodimo to\[CHacek]no na ciljanu rezonanciju, \ ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " \[CAcute]e biti jednako nuli te \[CAcute]e prvi od dvaju uvjeta biti \ zadovoljen. Veli\[CHacek]inu ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " \[CAcute]emo nazvati ", StyleBox["ugo\:0111enjem", FontSlant->"Italic"], ". Drugi \[CAcute]e uvjet biti zadovoljen ako je", StyleBox[" |\[Sigma]|<<1", FontFamily->"Courier"], ", bez obzira na to koliki je ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], ". Kako je ", Cell[BoxData[ RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]]], " zapravo polu\[SHacek]irina Rabijevog profila linije smetnje, ovo zna\ \[CHacek]i da o\[CHacek]ekujemo zanemarivo curenje dok je god \ polu\[SHacek]irina Rabijevog profila linije smetnje mnogo manja od njene \ udaljenosti do ciljane linije. Kako je ", StyleBox["\[Sigma]", FontFamily->"Courier"], " o\[CHacek]ito razmjeran jakosti vanjske sile, a time izravno povezan sa \ snagom kojom pogonimo sustav, ovu \[CAcute]emo veli\[CHacek]inu nazvati ", StyleBox["parametrom snage", FontSlant->"Italic"], ". Nadalje, primjetimo da veli\[CHacek]ina ", StyleBox["R", "Commentary", FontFamily->"Courier", FontSlant->"Plain"], " odre\:0111uje omjere polu\[SHacek]irina Rabijevih profila linije smetnje \ i ciljane linije. Kona\[CHacek]nost visine Rabijevog profila u cijelom \ frekventnom podru\[CHacek]ju izravno nas upu\[CAcute]uje na zaklju\[CHacek]ak \ da \[CAcute]e u vi\[SHacek]erazinskom sustavu Rabijev prenos biti uvijek \ popra\[CAcute]en nekim kona\[CHacek]nim odljevom. Stoga bismo razliku u polu\ \[SHacek]irinama Rabijevih profila mo\:017eda mogli iskoristiti kako bismo \ donekle pove\[CAcute]ali prijenos populacije. Naime, ako je ", StyleBox["R", FontFamily->"Courier"], " velik, postavljanje ugo\:0111enja (a time izravno i ", Cell[BoxData[ SubscriptBox["\[CapitalDelta]", StyleBox["t", FontSlant->"Italic"]]]], ") na neku malu kona\[CHacek]nu vrijednost istovremeno bi uzrokovalo mnogo \ ve\[CAcute]u promjenu u ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]\_p\), FontFamily->"Courier"], TraditionalForm]]], ". Na ovaj bismo na\[CHacek]in mogli samo malo smanjiti iznos prijenosa, a \ istovremeno bitno smanjiti iznos curenja. Budu\[CAcute]i da cjelokupna \ populacija mora biti razdijeljena izme\:0111u po\[CHacek]etno naseljenog \ stanja, prenosa i curenja, ovako bismo pove\[CAcute]ali omjer izme\:0111u \ prenosa i odljeva, te samim time postigli ne\[SHacek]to potpuniji prenos. U \ nastavku \[CAcute]emo pokazati da ovo uistinu i jest slu\[CHacek]aj, iako je \ to\[CHacek]an mehanizam bitno slo\:017eeniji. " }], "Text"], Cell[TextData[{ "Dakle, crtanje Rabijevih profila obiju linija na\[SHacek]eg trorazinskog \ sustava na isti graf trebalo bi nam pru\:017eiti izravan uvid u osnovne zna\ \[CHacek]ajke dinamike sustava. Primjeri ovakve brze analize prikazani su na \ ", StyleBox["Slici 6", FontWeight->"Bold"], ". Budu\[CAcute]i znamo kako na temelju poznatih iz svojstava sustava i \ pogonske sile odrediti polu\[SHacek]irine Rabijevog profila, odmah \ mo\:017eemo postaviti grubu gornju granicu na dopu\[SHacek]tenu jakost \ lasera. Nadalje, poznavanje veze izme\:0111u jakosti vezanja i vremena \ prijenosa (dane izrazom (27)) populacije omogu\[CAcute]ava nam izravno odre\ \:0111ivanje donje granice brzine prijenosa populacije. Napokon, valja uo\ \[CHacek]iti da ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " mo\:017ee poprimiti i vrijednost 0 (\[SHacek]to se de\[SHacek]ava to\ \[CHacek]no na rezonanciji) a ", StyleBox["\[Sigma]", FontFamily->"Courier"], " ne mo\:017ee, dok god postoji neka kona\[CHacek]na vrijednost jakosti \ pogonske sile. Ova \[CAcute]e \[CHacek]injenica biti va\:017ena u dRWA \ analiti\[CHacek]kom pristupu koji se temelji na razvijanju u red \ veli\[CHacek]ina koje opisuju dinamiku trorazinskog sustava: negativne \ potencije u ", StyleBox["\[Sigma]", FontFamily->"Courier"], " mo\:017eemo uzimati u obzir, no nikako nam se ne smiju javiti negativne \ potencije u ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], "." }], "Text"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 %%ImageSize: 600 300 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.297619 0 0.618034 [ [.17262 -0.0125 -9 -9 ] [.17262 -0.0125 9 0 ] [.32143 -0.0125 -3 -9 ] [.32143 -0.0125 3 0 ] [.47024 -0.0125 -9 -9 ] [.47024 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.45667 .57643 L .45791 .57693 L .45855 .5771 L .45925 .57722 L .46053 .57729 L .46117 .57726 L .46186 .57719 L .46308 .57703 L .46377 .57692 L .46451 .57682 L .46519 .57674 L .46584 .57668 L .46709 .57666 L .46822 .57677 L .46948 .57705 L .47019 .57728 L .47084 .57755 L .473 .57872 L .47537 .58036 L .47772 .58202 L .479 .58281 L .48019 .58342 L .48149 .5839 L .48217 .58407 L .48291 .58419 L .48421 .58424 L .4854 .58412 L .48612 .58397 L .48678 .5838 L .48828 .58332 L .49101 .58233 L .49214 .58199 L .49336 .58171 L .49405 .5816 L .49469 .58155 L .49594 .58156 L .49722 .58173 L .49795 .58189 L .49862 .58208 L .50116 .58302 L .50364 .58399 L .50488 .58437 L .50559 .58453 L Mistroke .50625 .58463 L .50698 .58469 L .50763 .58468 L .5084 .58461 L .50912 .58447 L .51038 .58406 L .51109 .58375 L .51175 .58341 L .51649 .58018 L .51901 .57856 L .5197 .57821 L .52042 .57789 L .52172 .57747 L .52287 .57726 L .52394 .57721 L .52512 .5773 L .52638 .57752 L .52847 .57805 L .52958 .57833 L .53075 .57857 L .53139 .57865 L .53209 .5787 L .53333 .57864 L .53459 .57836 L .53577 .57788 L .53692 .5772 L .53797 .57642 L .54033 .57417 L .58013 .54607 L .59832 .52945 L .60075 .52474 L .60207 .52258 L .6033 .52091 L .60464 .51952 L .60535 .51896 L .60611 .51849 L .6068 .51817 L .60745 .51796 L .60868 .51776 L .60986 .51774 L .61111 .51778 L .61218 .5178 L .61335 .51769 L .61461 .51733 L .61532 .51699 L .61597 .51656 L .61722 .5154 L .61842 .51386 L .63816 .47976 L .65916 .43997 L Mistroke .69736 .38258 L .73801 .30644 L .77715 .22851 L .81874 .16984 L .83961 .13194 L .8407 .13176 L .84188 .13132 L .84299 .13058 L .84403 .12959 L .84523 .12801 L .84651 .12582 L .84876 .1208 L .85396 .10636 L .85629 .10043 L .85752 .09783 L .85881 .09556 L .85996 .09398 L .86121 .0927 L .86235 .09193 L .86341 .09149 L .8645 .09126 L .86549 .09118 L .86664 .09117 L .86772 .09115 L .86892 .09103 L .87018 .09069 L .87126 .09015 L .87244 .08925 L .87372 .08788 L .87509 .08594 L .87756 .08134 L .88287 .06937 L .88549 .06432 L .88676 .06239 L .88791 .06098 L .88904 .05992 L .8901 .0592 L .89123 .05869 L .89244 .05839 L .89369 .05828 L .89501 .05826 L .89613 .05825 L .89737 .05816 L .89868 .05789 L .89989 .05741 L .90052 .05706 L .90122 .05659 L .90263 .05535 L .9052 .05216 L .9076 .04838 L Mistroke .90989 .04455 L .91238 .04068 L .91352 .03916 L .91473 .03777 L .91578 .03678 L .91692 .03594 L .91766 .03551 L .91834 .0352 L .91899 .03496 L .91968 .03478 L .92087 .03459 L .92218 .03452 L .92348 .03451 L .9247 .0345 L .9258 .03444 L .927 .03427 L .92768 .03412 L .92841 .03389 L .92971 .03333 L .93109 .03251 L .93261 .03134 L .93785 .02615 L .94028 .02382 L .94254 .0221 L .94368 .02145 L .94489 .0209 L .94559 .02067 L .94624 .02049 L .94747 .02027 L .94854 .02016 L .94967 .02012 L .95064 .02012 L .9517 .02013 L .95285 .02013 L .95408 .02009 L .95524 .01999 L .9563 .01984 L .95752 .01959 L .95883 .01922 L .96118 .01835 L .96392 .01716 L .9665 .01611 L .96769 .01572 L .96894 .01538 L .9696 .01523 L .97032 .0151 L .97158 .01495 L .97273 .01488 L .97397 .01486 L .97513 .01489 L Mistroke .97619 .01495 L Mfstroke 0 .4 1 r .02381 .60332 m .02499 .60322 L .02605 .60297 L .02729 .60251 L .02846 .60196 L .03056 .6008 L .03279 .59966 L .03395 .5992 L .03524 .59883 L .03641 .59863 L .0375 .59855 L .03879 .59855 L .03997 .59859 L .04062 .5986 L .04132 .59858 L .04262 .59844 L .04326 .59828 L .04396 .59804 L .04523 .59737 L .04646 .59638 L .04759 .59516 L .05017 .59133 L .0529 .5862 L .0551 .58194 L .05745 .578 L .0588 .57625 L .06003 .57505 L .06127 .57423 L .06244 .57378 L .06371 .57358 L .06442 .57355 L .06507 .57357 L .06627 .57362 L .06753 .57359 L .06875 .57334 L .06945 .57306 L .07009 .57269 L .07081 .57214 L .07146 .57149 L .07293 .56953 L .07417 .56729 L .07532 .56479 L .07792 .55798 L .08263 .54507 L .08389 .54232 L .08525 .53997 L .08653 .53837 L .0877 .53744 L .08842 .5371 L .08908 .53693 L Mistroke .0898 .53687 L .09057 .53693 L .09131 .53708 L .092 .53728 L .09329 .53767 L .09395 .53784 L .09467 .53797 L .09542 .538 L .09612 .53791 L .09737 .5374 L .09808 .53689 L .09872 .53627 L .10016 .53433 L .10173 .5314 L .10458 .52436 L .10912 .51226 L .11036 .50959 L .11167 .50732 L .11278 .50587 L .114 .50483 L .11471 .50446 L .11535 .50428 L .11609 .50422 L .11678 .5043 L .11802 .5047 L .11872 .50503 L .11937 .50537 L .12063 .50605 L .12181 .50661 L .12304 .50697 L .12369 .50704 L .12439 .507 L .12507 .50683 L .1258 .50651 L .12709 .50554 L .12781 .50478 L .12849 .50394 L .13003 .50158 L .13538 .49105 L .13773 .4869 L .13902 .4851 L .14021 .48379 L .1415 .48278 L .14219 .48241 L .14292 .48214 L .14423 .48195 L .14541 .48203 L .14791 .4827 L .14914 .4831 L .15025 .48342 L Mistroke .15143 .48365 L .15269 .4837 L .15335 .48364 L .15407 .48348 L .15538 .48297 L .15659 .4822 L .15769 .48128 L .16015 .47846 L .16458 .47162 L .16972 .4635 L .17113 .46171 L .17262 .46014 L .17389 .45908 L .1746 .4586 L .17528 .45823 L .1766 .4577 L .1778 .45742 L .17909 .45729 L .17982 .45726 L .1805 .45726 L .18162 .45726 L .18286 .45719 L .18415 .45695 L .18536 .45649 L .18654 .45574 L .1878 .45456 L .18899 .45304 L .19007 .45129 L .19271 .44561 L .19521 .43883 L .20556 .41611 L .22407 .38562 L .2265 .37534 L .22783 .37018 L .2291 .36588 L .23028 .36252 L .23156 .35963 L .23272 .35771 L .2338 .35649 L .23505 .35567 L .23618 .35534 L .23748 .35526 L .23871 .35525 L .23937 .35519 L .24006 .35505 L .2413 .35446 L .24199 .35388 L .24264 .35315 L .24334 .35213 L .2441 .35074 L Mistroke .24534 .34779 L .24648 .34433 L .24907 .33419 L .25376 .31203 L .25597 .30282 L .25803 .29621 L .25907 .29376 L .26018 .2918 L .26142 .29036 L .26256 .28964 L .26373 .28935 L .26501 .28936 L .26621 .28945 L .26732 .28943 L .26861 .28905 L .26927 .28866 L .26998 .28803 L .27123 .28637 L .2724 .28411 L .27358 .28111 L .2747 .27764 L .27681 .26978 L .28153 .24973 L .28403 .24088 L .28529 .23744 L .28602 .23582 L .28669 .23457 L .28787 .23289 L .28915 .23179 L .29027 .23134 L .29146 .23126 L .29275 .23144 L .29346 .23159 L .29413 .23172 L .29543 .23181 L .29661 .2316 L .2973 .23129 L .29802 .23078 L .29932 .22938 L .30004 .22831 L .30072 .22711 L .30225 .22372 L .32082 .18722 L .322 .18744 L .32325 .18761 L .3239 .18764 L .32461 .18758 L .32587 .18723 L .32706 .18653 L .32833 .18533 L Mistroke .32953 .18376 L .33062 .18197 L .3361 .16937 L .33868 .16298 L .34107 .15796 L .34235 .15579 L .34377 .1539 L .34519 .15256 L .34597 .15204 L .34671 .15168 L .34736 .15145 L .34805 .15129 L .34928 .15118 L .34996 .15118 L .3506 .15121 L .35129 .15125 L .35205 .15128 L .35274 .15128 L .35338 .15123 L .35463 .15099 L .35577 .15053 L .35697 .14972 L .35827 .14845 L .35948 .14685 L .36223 .14185 L .38233 .11464 L .38342 .11444 L .38456 .11401 L .38582 .11318 L .38698 .11204 L .38806 .11062 L .38906 .10899 L .39135 .10427 L .3959 .09241 L .39849 .08602 L .40093 .08135 L .40221 .0796 L .40288 .07888 L .4036 .07826 L .40485 .0775 L .40602 .07712 L .40666 .07701 L .40735 .07696 L .4086 .07696 L .40937 .07697 L .41009 .07696 L .41074 .07691 L .41145 .0768 L .41273 .07636 L .4139 .07565 L Mistroke .41454 .0751 L .41524 .07439 L .4165 .07274 L .41895 .06838 L .42119 .06337 L .42377 .05729 L .42653 .05151 L .42787 .04929 L .42913 .04763 L .43027 .04652 L .43151 .0457 L .43281 .04522 L .43348 .0451 L .4342 .04505 L .43549 .04509 L .4367 .04519 L .4374 .04524 L .43813 .04527 L .43944 .04516 L .44017 .04499 L .44085 .04476 L .4416 .04439 L .4424 .04388 L .44748 .03788 L .45026 .03356 L .45286 .02993 L .454 .02866 L .45508 .02768 L .45632 .02683 L .45746 .02634 L .45871 .02606 L .45988 .02603 L .46115 .02618 L .46186 .02632 L .46251 .02646 L .46375 .02674 L .46505 .02699 L .46573 .02708 L .46648 .02712 L .46779 .02705 L .46903 .02679 L .46973 .02655 L .47036 .02628 L .47276 .02489 L .47534 .023 L .47808 .02112 L .47951 .02038 L .48032 .02007 L .48109 .01986 L .48178 .01974 L Mistroke .48242 .01968 L .48316 .01968 L .48386 .01975 L .4851 .02001 L .4858 .02022 L .48643 .02044 L .48881 .02144 L .49141 .02248 L .49212 .0227 L .49287 .02289 L .49351 .02302 L .49422 .02311 L .49548 .02316 L .49618 .02312 L .49683 .02305 L .49808 .02281 L .49926 .0225 L .50171 .02174 L .50291 .0214 L .50402 .02115 L .50509 .02099 L .50622 .02093 L .50735 .021 L .5086 .02124 L .50986 .02165 L .51122 .02226 L .51366 .02367 L .51619 .02528 L .51752 .02607 L .51893 .02678 L .52016 .02728 L .52086 .0275 L .52152 .02766 L .52267 .02784 L .52391 .0279 L .52514 .02783 L .52584 .02774 L .52649 .02764 L .52894 .02716 L .53009 .02695 L .53118 .0268 L .53242 .02673 L .53355 .02679 L .53462 .02696 L .53575 .02729 L .53671 .02769 L .53777 .02825 L .54012 .0299 L .54232 .03176 L .54475 .03387 L Mistroke .54734 .03575 L .54868 .03646 L .54945 .03677 L .55017 .037 L .55143 .03723 L .55214 .03728 L .55279 .03727 L .55398 .03716 L .55509 .03696 L .55641 .03667 L .55761 .03641 L .55888 .0362 L .56007 .03613 L .56133 .03626 L .56204 .03643 L .56269 .03667 L .56376 .03721 L .56491 .03804 L .56699 .0401 L .57171 .04656 L .57427 .04996 L .57553 .05133 L .5767 .05236 L .57776 .05307 L .57888 .05357 L .58013 .05386 L .58126 .0539 L .58247 .05378 L .58359 .05356 L .58428 .05342 L .58494 .0533 L .58616 .05314 L .58735 .05317 L .58865 .05351 L .58932 .05384 L .59004 .05432 L .59132 .05553 L .59241 .05693 L .59359 .05881 L .59572 .06304 L .60056 .07426 L .60323 .07935 L .60396 .08044 L .60474 .08142 L .60547 .08218 L .60614 .08274 L .60683 .08319 L .60757 .08352 L .60823 .08371 L .60892 .08382 L Mistroke .61012 .08382 L .61144 .08369 L .61274 .08359 L .61393 .08367 L .61459 .08382 L .6153 .08412 L .61659 .08504 L .61781 .08643 L .61909 .08851 L .6214 .09379 L .62637 .10938 L .62879 .11684 L .63099 .12225 L .63219 .12446 L .63333 .12604 L .63452 .1272 L .63582 .12792 L .63645 .12811 L .63715 .12822 L .63788 .12826 L .63856 .12826 L .63978 .12824 L .64109 .12837 L .64229 .12879 L .64341 .12956 L .64463 .13091 L .64593 .13301 L .64717 .13568 L .64834 .13876 L .65053 .14573 L .65472 .16101 L .65701 .16846 L .65915 .17383 L .66026 .1759 L .66146 .17756 L .66214 .17825 L .66279 .17876 L .66399 .17937 L .6651 .17962 L .66613 .17968 L .66725 .17969 L .66845 .17975 L .66963 .18001 L .67091 .18066 L .67212 .18172 L .67322 .18314 L .67445 .18525 L .67579 .18819 L .67849 .19581 L .68372 .21283 L Mistroke .68523 .21708 L .68665 .22048 L .68794 .22297 L .68864 .22409 L .6893 .22497 L .6905 .2262 L .69181 .22705 L .69293 .22744 L .69412 .22762 L .69532 .22765 L .69661 .22765 L .69734 .2277 L .69802 .22779 L .69867 .22794 L .69935 .22819 L .70057 .22888 L .70126 .22945 L .7019 .23009 L .70322 .23178 L .70462 .23413 L .70718 .23972 L .7095 .24581 L .71208 .25284 L .7148 .25956 L .71623 .26245 L .7178 .26499 L .71848 .26589 L .71922 .26669 L .71992 .26731 L .72056 .26777 L .7212 .26812 L .72188 .26839 L .72264 .26857 L .72333 .26865 L .72398 .26866 L .72468 .26863 L .72596 .26849 L .72721 .26838 L .72853 .26842 L .72966 .26867 L .7309 .26928 L .73218 .27034 L .73334 .27175 L .73468 .27392 L .73595 .27649 L .74067 .28927 L .74307 .29629 L .74433 .29966 L .74566 .30279 L .74682 .3051 L Mistroke .7481 .30708 L .74925 .30838 L .75033 .30916 L .75157 .30961 L .75269 .30968 L .75399 .30947 L .75521 .30915 L .75586 .30899 L .75655 .30885 L .75779 .30881 L .75848 .30894 L .75913 .30919 L .75983 .30964 L .76059 .31034 L .76182 .312 L .76296 .31414 L .7642 .31716 L .76553 .32109 L .77022 .33852 L .7726 .3472 L .77377 .35087 L .77484 .35375 L .7758 .35589 L .77685 .35774 L .7779 .35908 L .77902 .35997 L .78033 .36043 L .78154 .36044 L .78217 .36035 L .78287 .36022 L .7836 .36008 L .78428 .36 L .78552 .36007 L .7862 .36029 L .78685 .36067 L .788 .36177 L .78926 .36372 L .79048 .36645 L .79159 .36961 L .79409 .37884 L .79858 .39905 L .80109 .40917 L .80247 .41357 L .80375 .4168 L .80448 .41823 L .80514 .4193 L .80588 .42023 L .80667 .42095 L .80733 .42136 L .80805 .42164 L Mistroke .80935 .42181 L .81057 .42178 L .81126 .42176 L .81191 .4218 L .81322 .42212 L .81395 .42254 L .81463 .42309 L .81576 .42449 L .817 .42674 L .8183 .42994 L .81951 .43367 L .83823 .48175 L .83949 .48181 L .84069 .48209 L .84134 .48239 L .84206 .48285 L .84332 .48408 L .84452 .48579 L .8458 .48819 L .84809 .49387 L .85306 .50898 L .85569 .51599 L .85717 .51907 L .85854 .52125 L .85929 .52219 L .86 .5229 L .86073 .52349 L .86154 .52396 L .86225 .52425 L .8629 .52443 L .86365 .52454 L .86436 .52459 L .86504 .5246 L .86569 .52459 L .86636 .52458 L .8671 .52458 L .86832 .52467 L .86963 .52495 L .87088 .52546 L .87219 .52625 L .87331 .52715 L .87454 .52835 L .87987 .53496 L .88247 .53811 L .88492 .54055 L .88628 .54163 L .88775 .54259 L .88908 .54327 L .89031 .54375 L .89147 .54409 L Mistroke .89256 .54432 L .89375 .54447 L .89503 .54454 L .89567 .54454 L .89637 .5445 L .89764 .54438 L .89887 .54418 L .90002 .54393 L .90226 .54334 L .90463 .54271 L .90587 .54246 L .90658 .54237 L .90723 .54231 L .90839 .54232 L .90962 .5425 L .91078 .54284 L .91205 .5434 L .91324 .54411 L .91435 .5449 L .91865 .54852 L .91983 .5494 L .9211 .55017 L .9223 .55066 L .9234 .55087 L .92469 .55081 L .92541 .55063 L .92609 .55036 L .92732 .54966 L .92864 .54867 L .9315 .5461 L .9329 .54493 L .9342 .54408 L .93537 .5436 L .93662 .54345 L .93786 .54372 L .93857 .54407 L .93923 .54451 L .97619 .57145 L Mfstroke 1 0 .6 r .02381 .01472 m .06244 .03272 L .06484 .03253 L .0655 .03246 L .06619 .03241 L .06744 .03243 L .0686 .03265 L .06969 .03307 L .07088 .03381 L .07216 .03496 L .07362 .03671 L .075 .03873 L .0776 .0431 L .08 .04707 L .08131 .04892 L .08255 .05033 L .08327 .05096 L .08404 .0515 L .08474 .05184 L .08538 .05204 L .08666 .05211 L .0873 .052 L .08801 .05178 L .08878 .05145 L .08949 .05108 L .09108 .05014 L .09258 .04931 L .09323 .04902 L .09396 .04877 L .09522 .04859 L .09592 .04864 L .09657 .04881 L .09728 .04911 L .09794 .04951 L .09866 .05008 L .09941 .0508 L .10188 .05393 L .10458 .0579 L .10586 .05961 L .10654 .0604 L .10725 .06113 L .10856 .06211 L .10977 .06257 L .11093 .06257 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Na lijevim grafovima profili \ ciljane linije iscrtani su zelenom, a linija smetnje crvenom. Uspravnom crtom \ ozna\[CHacek]ena je frekvencija pogonske sile. Na desnim su grafovima \ prikazana numeri\[CHacek]ka rje\[SHacek]enje odgovaraju\[CAcute]e dinamike. \ Populacije t, p i s razina iscrtane su zelenom, plavom odnosno crvenom. Jasno \ se mo\:017ee uo\[CHacek]iti mogu\[CAcute]nost grubog predvi\:0111anja \ dinamike na temelju odnosa Rabijevih profila - dok preklapanja linija nema, \ dinamika je '\[CHacek]ista' Rabijeva. Kada pak profil linije smetnje po\ \[CHacek]ne zadirati u ciljanu liniju, dolazi do bitnih odstupanja. " }], "Commentary"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.2 Analiti\[CHacek]ki pristup I: razvezana dinamika (RaD)\ \>", "Subsection", CellTags->"2.2"], Cell[TextData[{ "U ovom odjeljku zapo\[CHacek]injemo sa opse\:017enim analiti\[CHacek]kim \ izu\[CHacek]avanjem utjecaja razine smetnje na dinamiku prijenosa populacije \ u trorazinskom sustavu. Pristup preko ", StyleBox["pribli\:017eenja", FontSlant->"Italic"], " ", StyleBox["razvezane", FontSlant->"Italic"], " ", StyleBox["dinamike", FontSlant->"Italic"], " temelji se na pretpostavci da dinamika istra\:017eivanog sustava uistinu \ jest bliska \[CHacek]istoj Rabijevoj dinamici s kakvom smo se ve\[CAcute] \ susreli u sustavu sa samo dvije svojstvene razine. Stoga \[CAcute]emo \ unaprijed pretpostaviti rje\[SHacek]enje na\[SHacek]eg dinami\[CHacek]kog \ problema, te \[CAcute]emo uvr\[SHacek]tavanjem tog rje\[SHacek]enja u potpunu \ dinami\[CHacek]ku jednad\:017ebu pokazati da je na\[SHacek]a po\[CHacek]etna \ pretpostavka ispravna. Napomenimo da do pretpostavke oblika rje\[SHacek]enja \ - koja ujedno jest i samo rje\[SHacek]enje - nismo do\[SHacek]li nasumi\ \[CHacek]nim poku\[SHacek]avanjem, ve\[CAcute] smo ga oblikovali na temelju \ rezultata velikog broja numeri\[CHacek]kih eksperimenata i simulacija u \ kojima smo mijenjali sve mogu\[CAcute]e parametre sustava." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Temeljna jednad\:017eba dinamike u pribli\:017eenju razvezane dinamike\ \>", "Subsubsection"], Cell[TextData[{ "Kao po\[CHacek]etni korak ovog istra\:017eivanja razdvojit \[CAcute]emo ", StyleBox["t-p-s", FontFamily->"Courier", FontSlant->"Italic"], " sustav na dva podsustava, od kojih se prvi sastoji od razina ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " a drugi od same razine ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], StyleBox[".", FontSlant->"Italic"], " a upravo zbog ovakvog pristupa smo cijeli postupak nazvali ", StyleBox["razvezanom", FontSlant->"Italic"], " ", StyleBox["dinamikom", FontSlant->"Italic"], ". Dinami\[CHacek]ku jednad\:017ebu rije\[SHacek]it \[CAcute]emo \ rekurzijski tako da pretpostavimo da nam je rje\[SHacek]enje dinamike jednog \ od podsustava poznato te ga kao takvog ugra\:0111ujemo u jednad\:017ebu \ dinamike drugog podsustava kao poznati prarametar. Kao nulti korak \ poslu\:017eit \[CAcute]e nam rje\[SHacek]enje dinamike \[CHacek]istog \ podsustava ", StyleBox["t-p", FontFamily->"Courier", FontSlant->"Italic"], ". Kako mi ba\[SHacek] \:017eelimo dobiti rje\[SHacek]enje koje je vrlo sli\ \[CHacek]no \[CHacek]istim Rabijevim titranjima u sustavu sa dvije razine, \ rezulatat nultog koraka bi trebao ve\[CAcute] biti vrlo dobro \ pribli\:017eenje potpunom rje\[SHacek]enju. Svakako valja imati na umu da se \ cjelokupni postupak mo\:017ee provesti samo dok smetnja uistinu jest samo \ smetnja, odnosno dok je ukupna populacija stanja ", StyleBox["s", FontSlant->"Italic"], " mala (", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalPi]\_p[\[Tau]] << 1\), FontFamily->"Courier"], TraditionalForm]]], " za sve ", StyleBox["\[Tau]", FontFamily->"Courier"], ") . Ovaj zahtjev proizlazi iz \[CHacek]injenice da rekurzivno \ rje\[SHacek]avanje razdvojene dinami\[CHacek]ke jednad\:017ebe \ naru\[SHacek]ava unitarnost valne funkcije." }], "Text"], Cell["\<\ Budu\[CAcute]i da pretpostavljamo da se nalazimo blizu rezonancije sa \ \:017eeljenim prelazom, iz dinami\[CHacek]ke jednad\:017ebe mo\:017eemo \ ispustiti ne-RWA \[CHacek]lanove za ciljani prijelaz. \[CapitalCHacek]lanove \ pridru\:017eene liniji smetnje u potpunosti zadr\:017eavamo jer \[CAcute]e \ nam upravo oni dati popravke u dinamici. Jednad\:017eba koju \ rje\[SHacek]avamo postaje:\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(d\/d\[Tau]\) \(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), " ", RowBox[{"Sign", "[", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "]"}], RowBox[{"(", GridBox[{ {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]], "0"}, { SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]], "0", RowBox[{\(s\_R\), "R", " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((2 \[CapitalGamma] - \[CapitalDelta] - R\/\[Sigma])\), "\[Tau]"}]], "+", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((R\/\[Sigma] + \[CapitalDelta])\), "\[Tau]"}]]}], ")"}]}]}, {"0", RowBox[{\(s\_R\), "R", " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((2 \[CapitalGamma] - \[CapitalDelta] - R\/\[Sigma])\), "\[Tau]"}]], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((R\/\[Sigma] + \[CapitalDelta])\), "\[Tau]"}]]}], ")"}]}], "0"} }], ")"}], \(\(a\& \[Rule] \)[\[Tau]]\)}]}]], "NumberedEquation"], Cell["gdje su", "Text"], Cell[BoxData[{ RowBox[{\(\(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{"(", GridBox[{ { RowBox[{ SubscriptBox["a", StyleBox["t", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}, { RowBox[{ StyleBox[ SubscriptBox[ StyleBox["a", FontSlant->"Plain"], "p"], FontSlant->"Italic"], "[", "\[Tau]", "]"}]}, { RowBox[{ SubscriptBox["a", StyleBox["s", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]} }], ")"}]}], "\[IndentingNewLine]", RowBox[{\(S\_R\), "\[Congruent]", RowBox[{"Sign", "[", FractionBox[ SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]]], "]"}]}], "\[IndentingNewLine]", RowBox[{"\[CapitalGamma]", "\[Congruent]", " ", FractionBox["\[Omega]", RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]]}]}], "NumberedEquation"], Cell[TextData[{ "Predznak ", Cell[BoxData[ RowBox[{"Sign", "[", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "]"}]]], " ponovno mo\:017eemo ispustiti kao \[SHacek]to smo to napravili ranije u \ slu\[CHacek]aju samo dviju razina (jed. (18)), jer su i jednad\:017eba i po\ \[CHacek]etni uvjeti simetri\[CHacek]ni s obzirom na obrat u vremenu. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Rekurzijski postupak", "Subsubsection"], Cell[TextData[{ "Iz jednad\:017ebe (47) nalazimo da je dinamika podsustava ", StyleBox["t-p", FontFamily->"Courier", FontSlant->"Italic"], " opisana jednad\:017ebom" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(d\/d\[Tau]\), RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}], "=", RowBox[{\(-\[ImaginaryI]\), " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", GridBox[{ {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]]}, { SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]], "0"} }], ")"}], RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}], "+", RowBox[{\(s\_R\), "R", " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((2 \[CapitalGamma] - \[CapitalDelta] - R\/\[Sigma])\), "\[Tau]"}]], "+", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((R\/\[Sigma] + \[CapitalDelta])\), "\[Tau]"}]]}], ")"}], RowBox[{"(", GridBox[{ {"0"}, {"1"} }], ")"}], \(a\_s[\[Tau]]\)}]}], ")"}]}]}]], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]]], " vektor podsustava ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["-", FontFamily->"Courier"], StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Courier"] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], "\[Congruent]", " ", RowBox[{"(", GridBox[{ { RowBox[{ SubscriptBox["a", StyleBox["t", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}, { RowBox[{ StyleBox[ SubscriptBox[ StyleBox["a", FontSlant->"Plain"], "p"], FontSlant->"Italic"], "[", "\[Tau]", "]"}]} }], ")"}]}]], "NumberedEquation"], Cell[TextData[{ "Ako je smetnja zbog utjecaja razine ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], " mala, o\[CHacek]ekivali bismo da je rje\[SHacek]enje vrlo sli\[CHacek]no \ rje\[SHacek]enju za slu\[CHacek]aj sustava sa samo dvije vezane svojstvene \ razine. Numeri\[CHacek]kih eksperimenti tu slutnju i potvr\:0111uju, te nazna\ \[CHacek]uju da je to\[CHacek]no rje\[SHacek]enje jednad\:017ebe (43) oblika" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], "=", RowBox[{ RowBox[{"(", GridBox[{ {\(\[ExponentialE]\^\(\[ImaginaryI]\ \[CapitalDelta]'\ \[Tau]\)\ \), "0"}, { "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \[CapitalDelta]'\ \[Tau]\)\)} }], ")"}], \(1\/\@\(1 + \((\[CapitalDelta]')\)\^2\)\), RowBox[{"(", RowBox[{ RowBox[{\(a\_t[0]\), RowBox[{"(", GridBox[{ {"1", "0"}, {"0", "1"} }], ")"}]}], "-", RowBox[{\(a\_p[0]\), RowBox[{"(", GridBox[{ {"0", "1"}, {"1", "0"} }], ")"}]}]}], ")"}], RowBox[{"(", GridBox[{ {\(\(-\@\(1 + \((\[CapitalDelta]')\)\^2\)\) Cos[\(\@\(1 + \((\[CapitalDelta]')\)\^2\)\) \[Tau]] - \ \[ImaginaryI]\ \[CapitalDelta]\ Sin[\(\@\(1 + \((\[CapitalDelta]')\)\^2\)\) \ \[Tau]]\)}, {\(\[ImaginaryI]\ Sin[\(\@\(1 + \((\[CapitalDelta]')\)\^2\)\) \ \[Tau]]\)} }], ")"}]}]}]], "NumberedEquation"], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox[ StyleBox["a", FontFamily->"Courier", FontSlant->"Plain"], "t"], "[", "0", "]"}], FontFamily->"Courier"], TraditionalForm]]], " i ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox[ StyleBox["a", FontFamily->"Courier", FontSlant->"Plain"], "p"], "[", "0", "]"}], FontFamily->"Courier"], TraditionalForm]]], " su po\[CHacek]etne vrijednosti amplituda temeljne odnosno posredni\ \[CHacek]ke razine pri \[CHacek]emu valja imati na umu da je uvijek jedna od \ njih jednaka jedinici, a druga nuli." }], "Text"], Cell[TextData[{ "Ugo\:0111enje ", StyleBox["\[CapitalDelta]'", FontFamily->"Courier"], " razlikuje se od ugo\:0111enja ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " definiranog u jed (19):" }], "Text"], Cell[BoxData[ \(\[CapitalDelta]' \[Congruent] \[CapitalDelta] - \[CapitalDelta]\_0\)], \ "NumberedEquation"], Cell[TextData[{ "t. j. ugo\:0111enje je odmaknuto od izvornog ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " za neki nepoznati iznos ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]\_0\), FontFamily->"Courier"], TraditionalForm]]], ". Ako (51) uistinu jest rje\[SHacek]enje na\[SHacek]eg problema tada bismo \ zatvaranjem rekurzije trebali jednozna\[CHacek]no odrediti optimalnu \ vrijednost ugo\:0111enja ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]\_0\), FontFamily->"Courier"], TraditionalForm]]], ". Uo\[CHacek]imo da je (43) zapravo to\[CHacek]no rje\[SHacek]enje slijede\ \[CAcute]e dinami\[CHacek]ke jednad\:017ebe:" }], "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{\(d\/d\[Tau]\), RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "'"}], "[", "\[Tau]", "]"}]}], "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ { "0", \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2\ \ \[CapitalDelta]'\ \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2\ \[CapitalDelta]'\ \ \[Tau]\)\), "0"} }], ")"}], RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "'"}], "[", "\[Tau]", "]"}]}]}]}]], "NumberedEquation"], Cell[TextData[{ "pri \[CHacek]emu je novo ugo\:0111enje ", StyleBox["\[CapitalDelta]'", FontFamily->"Courier"], " odre\:0111eno kao" }], "Text"], Cell[BoxData[ RowBox[{\(\[CapitalDelta]'\), "\[Congruent]", FractionBox[ RowBox[{"\[Omega]", "-", RowBox[{ SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]], "'"}]}], RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]]}]], "NumberedEquation"], Cell["\<\ Pretpostavljeni oblik rje\[SHacek]enja jed. (43) zapravo nam ka\:017ee da o\ \[CHacek]ekujemo kako \[CAcute]e uvo\:0111enje smetnje u sustav dviju razina \ za posljedicu imati pomicanje Rabijevog profila ciljane linije bez mijenjanja \ njegovog oblika. \ \>", "Text"], Cell[TextData[{ "Sada iz jed. (47) o\[CHacek]itavamo jednad\:017ebu za dinamiku podsustava \ ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/d\[Tau]\) a\_s[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), " ", \(s\_R\), "R", " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((2 \[CapitalGamma] - \[CapitalDelta] - R\/\[Sigma])\), "\[Tau]"}]], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((R\/\[Sigma] + \[CapitalDelta])\), "\[Tau]"}]]}], ")"}], RowBox[{ SubscriptBox["a", StyleBox["p", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}]}]], "NumberedEquation"], Cell[TextData[{ "Prema na\[SHacek]oj po\[CHacek]etnoj pretpostavci ", Cell[BoxData[ \(a\_p[\[Tau]]\)]], " nam je poznat kao jedna od komponenti rje\[SHacek]enja dinamike \ podsustava ", StyleBox["t-p", "InlineInput", FontSlant->"Italic"], ", pa je jednad\:017eba (54) stoga jednostavna diferencijalna \ jednad\:017eba za funkciju ", Cell[BoxData[ \(a\_s[\[Tau]]\)]], ". Rje\[SHacek]enje glasi" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["a", StyleBox["s", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], "\[TildeFullEqual]", RowBox[{\(\[Alpha]\_0\), "-", RowBox[{\(s\_R\), SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], \(\[Sigma]\/2\), SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((R\/\[Sigma] + \[CapitalDelta])\), "\[Tau]"}]], RowBox[{"(", RowBox[{"1", "+", RowBox[{\(1\/\(1 - \(2 \[CapitalGamma]\ \[Sigma]\)\/R\)\), SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "4", " ", "\[CapitalGamma]", " ", "\[Tau]"}]]}]}], ")"}], RowBox[{ SubscriptBox["a", StyleBox["p", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}]}]}]], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_0\)]], " konstanta integracije za koju se mo\:017ee lako pokazati da je po iznosu \ vrlo mala. " }], "Text"], Cell[TextData[{ "Napokon zatvaramo rekurziju uvr\[SHacek]tavanjem rje\[SHacek]enja (55) za \ podsustav ", StyleBox["s", FontSlant->"Italic"], " u jednad\:017ebu dinamike podsustava ", StyleBox["t-p", "InlineInput", FontSlant->"Italic"], " (49). Zadr\:017eavanjem samo najsporije rotiraju\[CAcute]ih \ \[CHacek]lanova nalazimo" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(d\/\(d\ \[Tau]\)\), RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}], "\[TildeFullEqual]", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], " ", "2", "\[CapitalDelta]", " ", "\[Tau]"}]]}, { SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], " ", "2", "\[CapitalDelta]", " ", "\[Tau]"}]], \(-\[Chi]\)} }], ")"}], RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}]}]], "NumberedEquation"], Cell[TextData[{ "gdje je veli\[CHacek]ina ", StyleBox["\[Chi]", FontFamily->"Courier"], " odre\:0111ena sa" }], "Text"], Cell[BoxData[ RowBox[{"\[Chi]", "\[Congruent]", RowBox[{ SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], \(\(R\ \[Sigma]\)\/2\), \((1 + 1\/\(1 - \(2\ \[CapitalGamma]\ \[Sigma]\)\/R\))\)}]}]], \ "NumberedEquation"], Cell[TextData[{ "Da bismo odredili pomak u uga\:0111anju ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\_0\)]], " moramo na\[CAcute]i unitarnu transformaciju ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "'"}], "[", "\[Tau]", "]"}], "=", RowBox[{\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ B\&^\ \[Tau]\)\), RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]}]}]]], " koja povezuje ", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "'"}], "[", "\[Tau]", "]"}]]], " iz jed. (53) sa ", Cell[BoxData[ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "[", "\[Tau]", "]"}]]], " iz jed. (57), pri \[CHacek]emu je " }], "Text"], Cell[BoxData[ RowBox[{\(B\&^\), "\[Congruent]", RowBox[{"(", GridBox[{ {\(\[Beta]\_1\), "0"}, {"0", \(\[Beta]\_2\)} }], ")"}]}]], "NumberedEquation"], Cell[TextData[{ "hermitska matrica. Uvo\:0111enjem ove transformacije jednad\:017eba \ dinamike podsustava ", StyleBox["t-p", "InlineInput", FontSlant->"Italic"], " prelazi u" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(d\/\(d\ \[Tau]\)\), RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "'"}], "[", "\[Tau]", "]"}]}], "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ {\(\[Beta]\_1\), SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], " ", RowBox[{"(", RowBox[{\(2 \[CapitalDelta]\), "-", RowBox[{ SubscriptBox["s", StyleBox["pt", FontSlant-> "Italic"]], \((\[Beta]\_1 - \[Beta]\_2)\)}]}], ")"}], " ", "\[Tau]"}]]}, { SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], " ", RowBox[{"(", RowBox[{\(2 \[CapitalDelta]\), "-", RowBox[{ SubscriptBox["s", StyleBox["pt", FontSlant-> "Italic"]], \((\[Beta]\_1 - \[Beta]\_2)\)}]}], ")"}], " ", "\[Tau]"}]], \(\(-\[Chi]\) + \[Beta]\_2\)} }], ")"}], RowBox[{ RowBox[{ SubscriptBox[\(a\& \[Rule] \), StyleBox["tp", FontSlant->"Italic"]], "'"}], "[", "\[Tau]", "]"}]}]}]], "NumberedEquation"], Cell[TextData[{ "Zahtijevanjem jednakosti izme\:0111u jednad\:017ebi (53) i (60) jednozna\ \[CHacek]no nalazimo vrijednost ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]\_0\), "InlineInput"], TraditionalForm]]], ":" }], "Text"], Cell[BoxData[ RowBox[{\(\[CapitalDelta]\_0\), "=", RowBox[{ RowBox[{"-", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]]}], SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], \(\(R\ \[Sigma]\)\/4\), \((1 + 1\/\(1 - \(2\ \[CapitalGamma]\ \[Sigma]\)\/R\))\)}]}]], \ "NumberedEquation"], Cell["\<\ Iz ovoga izravno nalazimo da optimalna frekvencija pogonske sile za \ \[SHacek]to potpuniji prenos iznosi\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[Omega]", StyleBox["pt", FontSlant->"Italic"]], "'"}], "=", RowBox[{ RowBox[{ RowBox[{ SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]], "+", \(E\_0\)}], "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|", \(\[CapitalDelta]\_0\)}], "\[IndentingNewLine]", "=", RowBox[{ SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]], "-", RowBox[{ SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], \(\[Sigma]\/4\), " ", \(E\_0\), " ", SubscriptBox["D", StyleBox["ps", FontSlant->"Italic"]], \((1 + 1\/\(1 - \(2\ \[CapitalGamma]\ \[Sigma]\)\/R\))\)}]}]}]}]], \ "NumberedEquation"], Cell["\<\ dok je vrijeme prenosa odre\:0111eno sa\ \>", "Text"], Cell[BoxData[ \(\[Tau]\_RaD = \(\(\[Pi]\/2\) 1\/\@\(1 + \[CapitalDelta]'\^2\) = \(\[Pi]\/2\) 1\/\@\(1 + \((\[CapitalDelta] - \[CapitalDelta]\_0)\)\^2\)\)\)], \ "NumberedEquation"], Cell[TextData[{ "Ispravnost ovih rezultata mo\:017ee se provjeriti numeri\[CHacek]kim rje\ \[SHacek]avanjem jednad\:017ebe (43). 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.20177 L .59907 .20318 L .59977 .20445 L .60099 .2061 L Mistroke .60213 .20709 L .60337 .20771 L .60451 .20803 L .60576 .2082 L .6064 .20825 L .60709 .20827 L .60826 .20822 L .60951 .20797 L .61069 .20745 L .61177 .20665 L .61289 .20547 L .61392 .20407 L .61627 .20023 L .61733 .19853 L .61833 .19721 L .61937 .19629 L .62053 .19602 L .6217 .19667 L .62295 .19841 L .62413 .20091 L .62521 .20368 L .6265 .20725 L .62788 .21074 L .62905 .21302 L .63034 .21441 L .63166 .21443 L .63289 .21319 L .63359 .21205 L .63422 .2108 L .63566 .2077 L .63689 .2053 L .63758 .20432 L .63823 .20377 L .63942 .20394 L .64068 .20609 L .64193 .21038 L .64308 .21609 L .64568 .23323 L .64792 .2488 L .64911 .2558 L .65038 .26172 L .65154 .26548 L .65259 .2676 L .65366 .26868 L .65466 .26898 L .65572 .26889 L .65683 .26878 L .65806 .26914 L .65918 .27031 L .66031 .27257 L Mistroke .66155 .27638 L .6627 .28109 L .66377 .28626 L .66643 .30037 L .66884 .31145 L .67006 .31555 L .67075 .31733 L .67139 .31865 L .67256 .32032 L .6738 .32122 L .6751 .32154 L .67632 .32154 L .67702 .3215 L .67766 .32146 L .6784 .32144 L .6791 .32144 L .68026 .32147 L .68131 .32144 L .68253 .3212 L .68367 .32065 L .68474 .31977 L .68571 .31862 L .68791 .31514 L .68908 .31313 L .69033 .31128 L .69151 .31013 L .69259 .3098 L .6938 .31042 L .69494 .31195 L .69618 .3145 L .69752 .31793 L .69863 .32086 L .69981 .32365 L .70111 .3258 L .70232 .32659 L .70355 .32603 L .70471 .3243 L .70575 .32196 L .70688 .31897 L .70821 .31549 L .70894 .31393 L .70962 .31286 L .71082 .31222 L .71214 .31383 L .71336 .31771 L .71446 .32318 L .71696 .3405 L .71967 .36134 L .72121 .37092 L .72194 .37452 L Mistroke .72263 .37726 L .72387 .38066 L .72453 .38173 L .72524 .38239 L .72594 .38265 L .72668 .38264 L .72732 .38251 L .72801 .38237 L .72871 .38234 L .72937 .38254 L .7306 .38381 L .7313 .38516 L .73193 .38686 L .73263 .38925 L .73338 .39235 L .73599 .407 L .73837 .42213 L .73951 .4286 L .74076 .43446 L .74194 .43861 L .74303 .44122 L .74431 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.8136 .53876 L .81431 .54059 L .81506 .54198 L .81576 .54278 L .81642 .54318 L .81766 .54321 L .81837 .54298 L .81902 .54269 L .82019 .5422 L .82143 .54196 L .82249 .54211 L .82365 .5426 L .8249 .54333 L .82559 .54371 L .82624 .54398 L .8275 .54405 L Mistroke .82864 .54336 L .82974 .54188 L .8309 .53941 L .83298 .53316 L .83424 .5289 L .83544 .52512 L .83651 .52232 L .83768 .52019 L .83895 .51916 L .84013 .51936 L .8414 .52053 L .84211 .52143 L .84275 .5223 L .84393 .52368 L .8452 .5244 L .84639 .52386 L .84749 .52209 L .84879 .5184 L .85 .51363 L .85144 .50701 L .85275 .50101 L .8534 .49833 L .85412 .49587 L .85542 .49302 L .85614 .49254 L .85691 .49298 L .85756 .49414 L .85829 .49623 L .86085 .50929 L .8636 .52693 L .86493 .53418 L .86618 .53927 L .86731 .54218 L .86853 .54361 L .86983 .54359 L .87049 .54317 L .87121 .54259 L .87249 .54166 L .87368 .54151 L .87441 .54202 L .8751 .54301 L .87639 .54635 L .87711 .54907 L .87778 .55209 L .8793 .56035 L .88143 .57343 L .88373 .58584 L .88505 .59086 L .88626 .59386 L .88741 .59538 L Mistroke .88863 .59584 L .88981 .59557 L .89108 .59499 L .89227 .59457 L .89336 .59453 L .89464 .59499 L .89536 .59547 L .89603 .59598 L .89734 .59702 L .89854 .59761 L .89969 .59747 L .90077 .59644 L .90192 .59423 L .90317 .59051 L .90541 .5812 L .9075 .57179 L .90879 .56685 L .91 .5634 L .91109 .56146 L .91227 .56058 L .91356 .56078 L .91421 .56117 L .91492 .56168 L .91618 .56242 L .91736 .56239 L .91859 .56104 L .91929 .55954 L .91994 .55765 L .92113 .55293 L .92239 .54641 L .9246 .53314 L .92576 .52654 L .92698 .52087 L .9283 .51698 L .92951 .51591 L .93064 .51714 L .93188 .5207 L .93259 .52359 L .93323 .52659 L .93468 .53399 L .93603 .54073 L .93729 .546 L .93837 .54929 L .93954 .55133 L .94083 .55183 L .94201 .55099 L .94317 .54946 L .9444 .5477 L .94511 .54689 L .94575 .54641 L Mistroke .947 .54649 L .94814 .54795 L .94937 .55111 L .95053 .55547 L .9516 .56033 L .95371 .57089 L .95486 .57632 L .95596 .58068 L .95723 .58451 L .95839 .58673 L .95948 .5878 L .96066 .58811 L .96196 .58783 L .96261 .58761 L .96334 .58739 L .96461 .58725 L .96581 .58757 L .96645 .58792 L .96715 .58841 L .96842 .58945 L .96966 .59026 L .97081 .59044 L .97151 .59011 L .97217 .58943 L .97341 .58701 L .97411 .58499 L .97475 .58271 L .97619 .57627 L Mfstroke 1 0 .6 r .02381 .01472 m .04262 .04549 L .04498 .06014 L .04622 .06836 L .04754 .07629 L .04869 .08185 L .04996 .08595 L .05112 .0876 L .0522 .08738 L .05289 .08645 L .05355 .0851 L .05478 .08172 L .05622 .07733 L .05693 .07538 L .05756 .07388 L .05869 .07212 L .0599 .07179 L .06121 .07339 L .06244 .07654 L .06488 .08516 L .06552 .08738 L .06621 .08948 L .06745 .09211 L .06815 .09273 L .06879 .09266 L .06955 .09176 L .07026 .09006 L .0715 .08531 L .07286 .07802 L .07507 .06362 L .07747 .04936 L .07813 .04624 L .07883 .04347 L .08009 .03989 L .08127 .03802 L .08255 .03722 L .08322 .03713 L .08393 .03716 L .08524 .03714 L .08602 .0369 L .08674 .0364 L .0874 .03568 L .0881 .03461 L .08885 .03312 L .08956 .03143 L .09115 .02682 L .09261 .02234 L .09399 .0186 L .09527 .01612 L .09599 .01526 L Mistroke .09666 .01484 L .09784 .01498 L .0991 .01618 L .09981 .01722 L .10058 .01852 L .10197 .02101 L .10458 .02449 L .10571 .02516 L .10691 .02537 L .10822 .02536 L .10944 .02554 L .11057 .02632 L .11163 .02788 L .11279 .03078 L .11404 .03542 L .11539 .04204 L .11681 .05028 L .11934 .06529 L .12045 .07071 L .12165 .07501 L .12295 .07762 L .12417 .07806 L .12491 .07749 L .1256 .07651 L .12636 .07507 L .12716 .0733 L .12787 .0717 L .12862 .07012 L .12933 .06889 L .12999 .06806 L .13131 .06764 L .13251 .06889 L .1332 .07031 L .13383 .07203 L .13528 .07703 L .13652 .08194 L .13786 .0869 L .13913 .09037 L .1403 .09189 L .14153 .09136 L .14283 .08832 L .14353 .08574 L .14417 .08282 L .14564 .07486 L .1481 .06022 L .1494 .05357 L .15011 .05056 L .15078 .0482 L .15198 .04513 L .15262 .04409 L Mistroke .15331 .04337 L .15453 .04289 L .15564 .04296 L .1568 .04309 L .15788 .04288 L .15905 .04194 L .16031 .0399 L .16151 .03694 L .16264 .03342 L .16474 .02598 L .16603 .02176 L .16723 .01859 L .16848 .0164 L .16915 .01575 L .16986 .01543 L .17112 .01576 L .17232 .01687 L .17456 .01977 L .17584 .0212 L .17649 .02173 L .1772 .02214 L .17846 .02242 L .17965 .02231 L .18032 .0222 L .18103 .02212 L .18231 .02235 L .18302 .02283 L .18368 .02358 L .1844 .0248 L .18517 .02661 L .19016 .04843 L .19147 .0549 L .19219 .05806 L .19288 .06067 L .19352 .06269 L .19421 .06442 L .19543 .06619 L .19612 .06651 L .19686 .06637 L .19755 .06585 L .19818 .06514 L .19964 .06308 L .2004 .06203 L .20121 .06115 L .20189 .06071 L .20264 .06063 L .20395 .0617 L .2046 .06281 L .20529 .06442 L .20654 .06823 L Mistroke .20914 .0779 L .20985 .08032 L .2106 .08251 L .21125 .08397 L .21196 .08505 L .21324 .08539 L .21395 .08464 L .21461 .08335 L .21588 .07942 L .21709 .07429 L .21931 .06338 L .22053 .05773 L .22168 .05327 L .22233 .05121 L .22304 .04941 L .22428 .04735 L .22554 .04648 L .22669 .04642 L .22795 .04668 L .22915 .04675 L .23022 .04632 L .23139 .04505 L .23266 .04259 L .2333 .04094 L .234 .03883 L .23642 .03033 L .23868 .02288 L .23994 .01987 L .24111 .01815 L .24213 .01747 L .24325 .01752 L .24442 .01826 L .24551 .0193 L .2468 .02061 L .24748 .02122 L .2482 .02173 L .24952 .02224 L .25074 .02221 L .25191 .02185 L .253 .02144 L .2542 .0212 L .2555 .02159 L .25621 .02222 L .25686 .02311 L .25759 .02446 L .25836 .02633 L .25978 .03085 L .26107 .03582 L .2635 .04562 L .26479 .05001 L Mistroke .2655 .05196 L .26617 .05344 L .2668 .0545 L .26749 .05527 L .2687 .05576 L .26992 .05535 L .27105 .05451 L .27235 .05349 L .27356 .05296 L .27482 .05325 L .27549 .05384 L .27621 .05486 L .27733 .05717 L .27856 .06056 L .28103 .06846 L .28214 .07158 L .28332 .074 L .28443 .07514 L .28544 .07504 L .28675 .07333 L .2874 .07185 L .28812 .06982 L .29059 .06104 L .29198 .05602 L .29275 .05358 L .29347 .05157 L .29415 .05002 L .29488 .0487 L .29616 .04728 L .30627 .03849 L .30866 .03087 L .30996 .02699 L .31117 .02404 L .31248 .02183 L .31318 .02113 L .31392 .02072 L .3146 .02063 L .31525 .02076 L .31645 .02144 L .31868 .02327 L .3198 .024 L .32104 .0244 L .32168 .02442 L .32237 .02431 L .32363 .0238 L .32487 .02313 L .32602 .02262 L .32729 .02251 L .32797 .02275 L .32869 .02327 L Mistroke .32937 .02406 L .33002 .02507 L .33122 .02761 L .33354 .03422 L .33599 .04156 L .3373 .04468 L .33805 .04603 L .33875 .04698 L .33997 .04795 L .34128 .04813 L .34202 .04795 L .34282 .04763 L .34353 .04731 L .34428 .04703 L .34496 .04688 L .3457 .0469 L .34701 .04755 L .34823 .04897 L .34936 .05095 L .35193 .05698 L .35322 .06001 L .35443 .06231 L .35511 .06321 L .35584 .06384 L .35711 .06395 L .35822 .06301 L .35945 .06097 L .36073 .05802 L .36192 .05488 L .36405 .04956 L .36525 .04723 L .36636 .0457 L .36767 .04471 L .3684 .04447 L .36906 .04441 L .37038 .04454 L .37162 .04466 L .37282 .04448 L .37393 .04384 L .37514 .04248 L .37642 .0403 L .37933 .03339 L .38076 .02988 L .38207 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.45314 .02683 L .45378 .02617 L .45492 .02545 L .45616 .02533 L .45687 .02554 L .45754 .0259 L .4588 .02686 L .46026 .02817 L .46091 .02871 L .46161 .02924 L .46281 .02989 L .46407 .03018 L .46473 .03016 L .46544 .03005 L .46673 .02967 L .46804 .02926 L .46928 .0291 L .46996 .0292 L .4707 .02949 L .47199 .03054 L .47323 .0322 L .47454 .03455 L .47687 .03952 L .47819 .0422 L .47892 .04348 L .4796 .04452 L .48024 .04531 L .48093 .04598 L .48215 .04663 L .48284 .04674 L .4836 .04668 L .4844 .04645 L .48515 .04616 L .48792 .04527 L .48907 .0454 L .49032 .04604 L .49149 .04711 L .49258 .0484 L .49384 .05008 L .49518 .05176 L .4963 .05283 L .49754 .05339 L .49825 .05337 L .49889 .05311 L Mistroke .49964 .05252 L .50035 .0517 L .50161 .04969 L .50298 .04695 L .50538 .04196 L .50675 .03965 L .50803 .0381 L .50929 .03718 L .51 .03691 L .51064 .03677 L .51129 .03671 L .512 .0367 L .51273 .0367 L .51342 .03665 L .51468 .03636 L .51583 .03571 L .51647 .03516 L .51716 .03441 L .51841 .03268 L .5209 .02841 L .52227 .02618 L .52354 .02456 L .52463 .0237 L .5258 .0234 L .52709 .02381 L .52827 .02477 L .5468 .0421 L .54908 .04785 L .55036 .05033 L .55154 .05185 L .55258 .05254 L .55372 .05265 L .5549 .05219 L .55601 .05143 L .55729 .05047 L .5585 .04979 L .55916 .04959 L .55989 .04958 L .56116 .05011 L .56233 .05121 L .56358 .05288 L .56582 .0563 L .56709 .05781 L .5678 .05834 L .56846 .05857 L .56967 .05824 L .57094 .0568 L .57161 .0556 L .57233 .05403 L .57363 .05066 L Mistroke .57505 .04659 L .57658 .04237 L .57798 .03918 L .57927 .03708 L .5805 .03585 L .58181 .03525 L .58298 .03512 L .58407 .03512 L .58527 .03498 L .58655 .03443 L .58777 .03335 L .58846 .03247 L .58911 .03149 L .59183 .02623 L .59297 .02396 L .59418 .02194 L .5955 .02044 L .59671 .01991 L .59803 .0203 L .59877 .02091 L .59944 .02166 L .60199 .0254 L .60341 .02737 L .60422 .02826 L .60496 .02887 L .60561 .02925 L .60632 .0295 L .60707 .02959 L .60777 .02956 L .60901 .02939 L .60971 .02933 L .61036 .02936 L .61156 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L .94131 .03138 L .94254 .03114 L .94366 .03106 L .94473 .03105 L .94587 .03105 L .94711 .03099 L .94827 .03081 L .94935 .03051 L .95037 .03009 L .95149 .02946 L .95269 .02861 L Mistroke .95688 .02479 L .95811 .02366 L .95943 .0226 L .96064 .02177 L .96179 .02116 L .96308 .02068 L .96426 .02042 L .96555 .02028 L .96628 .02026 L .96696 .02025 L .96806 .02026 L .96909 .02027 L .97023 .02022 L .97144 .0201 L .97272 .01985 L .97392 .01949 L .97619 .01856 L Mfstroke 0 .4 1 r .02381 .60332 m .04262 .5702 L .04498 .55431 L .04622 .54545 L .04754 .53689 L .04869 .53086 L .04996 .52632 L .05112 .52432 L .0522 .52423 L .05289 .52495 L .05355 .52607 L .05478 .52895 L .05622 .53261 L .05693 .53411 L .05756 .53516 L .05869 .53604 L .0599 .53531 L .06121 .53247 L .06244 .52814 L .06488 .51739 L .06552 .51469 L .06621 .51213 L .06745 .50877 L .06815 .50778 L .06879 .50754 L .06955 .5081 L .07026 .50947 L .0715 .51361 L .07221 .51678 L .07286 .52012 L .07528 .53398 L .07647 .54028 L .07756 .54512 L .0787 .54895 L .07993 .55157 L .08121 .55281 L .08185 .553 L .08255 .55298 L .08327 .55284 L .08404 .55265 L .08477 .55255 L .08544 .55262 L .08672 .55336 L .08736 .55409 L .08807 .55518 L .08954 .55826 L .09112 .56229 L .09248 .56568 L .0932 .5672 L .09396 .56849 L Mistroke .09474 .5694 L .09544 .56982 L .09615 .56985 L .09681 .56954 L .09811 .56809 L .09884 .5669 L .09951 .56567 L .10458 .55874 L .10574 .55862 L .107 .55881 L .10819 .55887 L .10929 .55835 L .11058 .55651 L .11122 .55493 L .11194 .5526 L .11318 .54702 L .11434 .54014 L .11878 .50726 L .11986 .50035 L .12101 .49451 L .12227 .4902 L .12344 .4882 L .12412 .48782 L .12484 .48791 L .12617 .48897 L .12735 .4902 L .12864 .49097 L .12984 .49043 L .13112 .48803 L .1323 .48394 L .1334 .47863 L .13605 .46229 L .13736 .45425 L .13857 .44802 L .13924 .44531 L .13996 .44315 L .14068 .44179 L .14147 .44126 L .14213 .44156 L .14284 .44259 L .14412 .446 L .14649 .45527 L .14766 .45987 L .14874 .46348 L .14998 .4665 L .15063 .46752 L .15132 .46818 L .15244 .4684 L .15367 .46769 L .15502 .46632 L Mistroke .15577 .46552 L .15647 .46488 L .15713 .46444 L .15784 .4642 L .15909 .4645 L .16036 .4658 L .16107 .46691 L .16171 .4681 L .16414 .47321 L .16546 .47561 L .16612 .47653 L .16686 .47726 L .16815 .4778 L .16936 .47748 L .17009 .47699 L .17078 .47638 L .17208 .4751 L .17276 .47447 L .17348 .4739 L .17427 .47346 L .175 .47324 L .17621 .47329 L .17751 .47378 L .17863 .47428 L .17986 .47449 L .18057 .47425 L .18123 .4737 L .18192 .47269 L .18266 .47106 L .18335 .46897 L .18401 .46648 L .18523 .46051 L .18765 .44473 L .19026 .4264 L .19152 .41894 L .19271 .41343 L .19379 .40997 L .19496 .40775 L .19619 .40687 L .1975 .40692 L .19874 .40714 L .19986 .40681 L .20056 .40611 L .2012 .40503 L .2019 .40328 L .20264 .40076 L .20389 .39491 L .20526 .38644 L .20763 .36878 L .20987 .35306 L Mistroke .21111 .34652 L .21181 .34378 L .21246 .3419 L .2136 .34013 L .21481 .34024 L .21586 .34166 L .217 .34408 L .21905 .34907 L .2203 .35143 L .22148 .35265 L .22255 .35275 L .22371 .35179 L .22626 .34697 L .2276 .34396 L .22834 .34249 L .22902 .34131 L .23026 .33986 L .23092 .3395 L .23163 .33944 L .23288 .3401 L .23402 .34142 L .23632 .34501 L .23754 .34679 L .23883 .34822 L .23995 .34896 L .24119 .34927 L .24232 .34918 L .24338 .34895 L .2446 .3487 L .24529 .34864 L .24594 .34866 L .24725 .34897 L .24798 .34928 L .24864 .34962 L .24983 .35021 L .25112 .35054 L .25234 .35015 L .25347 .34889 L .25477 .34607 L .25598 .34201 L .25732 .33609 L .25875 .3285 L .26435 .30186 L .26925 .29747 L .26993 .29738 L .27066 .29708 L .27135 .29647 L .27199 .29555 L .2732 .29266 L .27447 .28774 L Mistroke .27588 .28006 L .2772 .27119 L .2802 .24888 L .28152 .2402 L .28276 .23362 L .28345 .23082 L .2842 .22849 L .28551 .22614 L .28676 .22569 L .28792 .22639 L .2892 .2278 L .28986 .22857 L .29058 .22929 L .29129 .2298 L .29195 .23002 L .29264 .22993 L .29339 .22943 L .29408 .22858 L .29473 .22745 L .29596 .22456 L .29836 .21716 L .29969 .21306 L .30095 .20979 L .30211 .20759 L .30339 .20625 L .30454 .206 L .30561 .20645 L .30668 .20739 L .30786 .20876 L .31025 .2115 L .31152 .21252 L .31223 .21292 L .31288 .21317 L .31408 .21341 L .31535 .21346 L .31658 .21347 L .3179 .21356 L .31914 .21378 L .32027 .21405 L .32091 .2142 L .3216 .21431 L .32235 .21434 L .32304 .21423 L .32428 .21357 L .32499 .21286 L .32565 .21194 L .34412 .17838 L .36266 .12222 L .36392 .1215 L .36462 .12068 L Mistroke .36528 .11965 L .36647 .11708 L .36774 .11357 L .37058 .10444 L .37182 .10078 L .37251 .09904 L .37315 .09762 L .37441 .09558 L .37556 .09456 L .37663 .09427 L .3778 .09456 L .37908 .09536 L .38045 .09644 L .38164 .09736 L .3829 .09811 L .3842 .09856 L .38539 .09869 L .38668 .09859 L .3874 .09846 L .38808 .09832 L .38928 .09806 L .39058 .09781 L .39173 .09765 L .39295 .09748 L .39364 .09736 L .39441 .09717 L .3951 .09694 L .39576 .09664 L .39699 .09582 L .39831 .09452 L .39984 .09248 L .4013 .09017 L .40404 .08583 L .42413 .05509 L .43374 .04675 L .43439 .04649 L .43507 .04607 L .43582 .04542 L .43651 .04464 L .43775 .04282 L .43912 .04031 L .44166 .03505 L .44398 .03092 L .44511 .02947 L .44634 .02842 L .44751 .0279 L .44858 .02778 L .44972 .02792 L .45097 .02827 L .45229 .0287 L Mistroke .4535 .02902 L .45463 .0292 L .45571 .02924 L .45685 .02913 L .45809 .02888 L .46238 .02769 L .46494 .0272 L .46765 .02676 L .46906 .02645 L .47061 .026 L .47334 .02502 L .47467 .02457 L .47592 .02422 L .47661 .02407 L .47737 .02396 L .47869 .02389 L .47994 .02393 L .48058 .02398 L .48128 .02405 L .48252 .02416 L .48368 .0242 L .48481 .02411 L .48604 .02383 L .48733 .0233 L .48853 .02258 L .49297 .0189 L .49417 .01794 L .49547 .01705 L .49669 .01639 L .49782 .01595 L .49914 .01562 L .4998 .01552 L .50053 .01545 L .50181 .0154 L .503 .01541 L .50425 .01545 L .50542 .01549 L .50669 .01554 L .50734 .01557 L .50805 .01562 L .50935 .01574 L .51057 .0159 L .51175 .01612 L .51285 .01636 L .52185 .01787 L .52309 .01783 L .52375 .0178 L .52445 .01777 L .52573 .01775 L .52691 .01781 L Mistroke .5276 .01788 L .52826 .01798 L .52897 .01812 L .52974 .0183 L .53233 .01913 L .53482 .01997 L .53617 .02036 L .53743 .02063 L .53851 .02082 L .53967 .02096 L .54211 .0212 L .54437 .02144 L .54561 .02158 L .54676 .0217 L .54801 .02179 L .54867 .0218 L .54939 .02179 L .55065 .02168 L .5518 .0215 L .55405 .02098 L .55532 .02073 L .5565 .02064 L .55763 .02074 L .55866 .02106 L .55984 .02174 L .56095 .02267 L .57179 .03272 L .58162 .04235 L .62025 .07991 L .62525 .07787 L .62594 .07751 L .62669 .07718 L .6274 .07697 L .62806 .0769 L .62929 .07719 L .6306 .0782 L .63175 .07973 L .63297 .082 L .63519 .0872 L .63761 .09288 L .63827 .09416 L .63898 .09533 L .64025 .0968 L .64098 .09728 L .64166 .0975 L .64294 .09741 L .64417 .09695 L .64481 .0967 L .6455 .09648 L .64672 .09651 L Mistroke .64801 .09745 L .6492 .09945 L .65031 .10244 L .65174 .10787 L .6531 .11437 L .65563 .1282 L .65696 .13507 L .65836 .14125 L .6591 .14387 L .65991 .14621 L .66064 .14785 L .66134 .14898 L .6626 .15015 L .6638 .15048 L .66446 .15049 L .66517 .15046 L .66643 .15057 L .66762 .15123 L .66889 .15281 L .66997 .15502 L .67116 .15834 L .67611 .1771 L .67745 .18151 L .6782 .18351 L .6789 .18508 L .67955 .18625 L .68027 .18721 L .68151 .18819 L .68277 .18845 L .68395 .18827 L .68502 .18794 L .68616 .18757 L .68742 .18729 L .68805 .18722 L .68874 .1872 L .68997 .18725 L .69112 .18733 L .69227 .18732 L .69337 .18714 L .69435 .1868 L .69541 .18624 L .69659 .18544 L .69782 .18454 L .69887 .18391 L .70002 .18354 L .70123 .18371 L .70236 .18451 L .70358 .18613 L .70488 .18861 L .70707 .19377 L Mistroke .70822 .19644 L .70944 .19879 L .71014 .1998 L .7109 .20057 L .71159 .20094 L .71224 .20101 L .71352 .2004 L .71423 .19972 L .71489 .19896 L .7161 .19744 L .71742 .19617 L .71867 .19594 L .71983 .19702 L .72103 .19977 L .72232 .20471 L .72355 .2111 L .72467 .21811 L .72722 .23564 L .72857 .24418 L .72931 .24826 L .73001 .25162 L .7307 .25438 L .73145 .25677 L .73275 .25951 L .73407 .26073 L .73529 .26097 L .73596 .26096 L .73668 .261 L .73799 .26157 L .7387 .26235 L .73935 .26344 L .74012 .26526 L .74085 .26755 L .74579 .29415 L .74697 .30125 L .74821 .30793 L .74937 .31297 L .75044 .31652 L .75167 .31916 L .75232 .31999 L .75302 .32052 L .75377 .32072 L .75448 .32066 L .75511 .32046 L .7558 .32016 L .75703 .3196 L .75833 .31925 L .75944 .3193 L .76066 .31974 L .76183 .32043 L Mistroke .76305 .32126 L .76409 .32187 L .76523 .32227 L .76646 .32225 L .76778 .32167 L .76902 .3207 L .77015 .31964 L .77122 .31869 L .77235 .31795 L .77359 .3177 L .77473 .31809 L .7758 .31902 L .77679 .32029 L .77906 .32381 L .7803 .32545 L .78147 .32633 L .78212 .32646 L .78282 .32625 L .78404 .32504 L .78475 .32386 L .78551 .3223 L .78686 .31904 L .78812 .31599 L .78881 .31457 L .78947 .31353 L .79075 .31271 L .79191 .31371 L .793 .31635 L .79419 .3211 L .79549 .32828 L .79687 .33738 L .79935 .35425 L .80048 .36074 L .80155 .36569 L .80278 .36968 L .80391 .37174 L .80497 .37249 L .80595 .37247 L .80703 .37208 L .80818 .3718 L .80944 .37237 L .81015 .37334 L .8108 .37475 L .812 .37878 L .81327 .3851 L .81548 .40011 L .8179 .41882 L .81911 .42736 L .8204 .43492 L .82108 .43809 L Mistroke .82182 .44093 L .82313 .44432 L .82436 .44583 L .82505 .44613 L .82569 .44617 L .82686 .44589 L .82809 .44554 L .8292 .44551 L .83039 .44604 L .83151 .44713 L .83253 .44856 L .83489 .4526 L .83618 .45453 L .83739 .45571 L .83867 .45609 L .83935 .45591 L .84008 .45544 L .84137 .45405 L .84255 .45242 L .84392 .45052 L .84465 .44968 L .84542 .44898 L .84616 .44855 L .84683 .44837 L .84815 .44856 L .84946 .44926 L .85066 .44999 L .85198 .45042 L .85265 .45032 L .85338 .44988 L .85461 .44822 L .85593 .44508 L .85717 .44098 L .8583 .43664 L .8595 .43193 L .86079 .42744 L .86201 .42446 L .86312 .42328 L .8643 .42398 L .86555 .42694 L .86683 .43208 L .86823 .43942 L .87053 .45242 L .8718 .45847 L .87299 .46269 L .8741 .46508 L .87531 .46598 L .8764 .46552 L .87741 .46437 L .87861 .46269 L Mistroke .8799 .46136 L .88111 .46134 L .88222 .46291 L .88352 .46705 L .88425 .47053 L .88493 .47443 L .88751 .49357 L .89017 .5149 L .89148 .52349 L .89268 .52955 L .89394 .53389 L .8946 .53538 L .89529 .53644 L .89601 .53708 L .89667 .53735 L .8974 .53743 L .89818 .53739 L .89941 .53743 L .90011 .53767 L .90076 .53812 L .90208 .53979 L .90282 .54119 L .9035 .54274 L .90604 .54976 L .90728 .55304 L .9084 .55539 L .90972 .55703 L .91044 .55736 L .91112 .55729 L .91186 .55682 L .91254 .55608 L .91329 .55499 L .91409 .55364 L .91949 .54671 L .92921 .52982 L .93177 .51611 L .93313 .50983 L .93381 .50732 L .93456 .50521 L .93522 .50401 L .93592 .50342 L .9372 .50416 L .93847 .50698 L .93963 .51087 L .94084 .51552 L .94213 .52032 L .94334 .52391 L .94445 .52589 L .9457 .52636 L .94687 .52511 L Mistroke .94814 .52225 L .94952 .51816 L .9507 .51469 L .9518 .51219 L .95295 .51096 L .95421 .51182 L .9553 .51465 L .9563 .51889 L .95745 .52549 L .95854 .53299 L .96065 .54874 L .96263 .56186 L .96373 .56742 L .96492 .57174 L .96599 .57419 L .96698 .5754 L .96809 .5759 L .96931 .57593 L .97058 .57602 L .97176 .57668 L .97288 .57812 L .9739 .5802 L .97508 .58346 L .97619 .58716 L Mfstroke 1 0 .6 r .02381 .01472 m .04262 .04635 L .04498 .06149 L .04622 .06979 L .04754 .07763 L .04869 .083 L .04996 .0868 L .05112 .08815 L .0522 .08769 L .05289 .08667 L .05355 .08528 L .05478 .08202 L .05622 .07808 L .05693 .0765 L .05756 .07541 L .05869 .0745 L .0599 .07524 L .06121 .07806 L .06244 .08229 L .06488 .0924 L .0661 .0967 L .0668 .09845 L .06745 .0995 L .06816 .09992 L .06881 .09957 L .06955 .09828 L .07026 .0962 L .0715 .09076 L .07286 .08284 L .0753 .06663 L .07664 .05867 L .07789 .05274 L .07896 .04914 L .08011 .04673 L .08138 .04557 L .08255 .04547 L .08384 .04574 L .08456 .04582 L .08521 .04572 L .08648 .04479 L .08768 .04281 L .08836 .04119 L .08908 .0391 L .09037 .03459 L .09177 .02909 L .09328 .02339 L .09403 .02095 L .09474 .01896 L .09542 .0174 L .09606 .01631 L Mistroke .09731 .01518 L .09796 .01508 L .09866 .01529 L .09943 .01582 L .10026 .01665 L .10177 .01837 L .10248 .01912 L .10323 .01977 L .10388 .02018 L .10458 .02046 L .10589 .02056 L .1071 .02034 L .10773 .02022 L .10843 .02018 L .10916 .02033 L .10984 .02073 L .11109 .02236 L .11223 .02507 L .11349 .02951 L .11481 .03567 L .11713 .04855 L .11932 .05986 L .12053 .06448 L .12184 .06763 L .12303 .06877 L .12413 .06856 L .12533 .06736 L .12665 .06559 L .12731 .06479 L .12803 .06417 L .12931 .06404 L .13061 .06562 L .13183 .06887 L .13293 .07312 L .13413 .07881 L .13688 .09249 L .13755 .09515 L .13827 .09746 L .13906 .09916 L .1398 .09988 L .14044 .09979 L .14114 .09892 L .14186 .0972 L .14254 .09492 L .14376 .08953 L .14506 .08258 L .14764 .06895 L .14894 .06383 L .14968 .06172 L .15037 .0603 L Mistroke .15161 .05905 L .15274 .05907 L .15404 .05992 L .15528 .06089 L .15598 .06123 L .15661 .0613 L .15735 .06101 L .15803 .0603 L .15926 .05787 L .15995 .05584 L .16059 .05359 L .16358 .03964 L .16504 .03251 L .16638 .02692 L .16754 .02317 L .16863 .02073 L .16979 .01927 L .17105 .01879 L .17234 .01907 L .17299 .01935 L .17371 .01967 L .17498 .02006 L .17618 .02001 L .17744 .01944 L .17808 .01898 L .17877 .01839 L .18002 .01731 L .18116 .0166 L .18185 .01643 L .1825 .01653 L .18319 .01696 L .18394 .01785 L .18459 .01899 L .18528 .02058 L .18652 .02431 L .19597 .04861 L .19717 .04828 L .19846 .0479 L .1992 .04792 L .19987 .0482 L .20117 .04974 L .2024 .05261 L .20372 .0573 L .20611 .06884 L .20854 .08079 L .20976 .08523 L .21046 .08698 L .2111 .08805 L .21226 .08853 L .21332 .08741 L Mistroke .21453 .08455 L .21569 .08076 L .21772 .07323 L .21883 .06959 L .21988 .06697 L .221 .06522 L .22224 .06466 L .2233 .06519 L .22443 .0665 L .22573 .06841 L .22696 .06997 L .22764 .0705 L .22838 .07069 L .22966 .0698 L .2309 .06734 L .2322 .06307 L .23453 .05241 L .23706 .04011 L .23849 .03458 L .23978 .03096 L .24052 .02954 L .24132 .02848 L 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rezonantnom frekvencijom \:017eeljenog prijelaza. Desni stupac su dinamike uz \ optimalnu frekvenciju odre\:0111enu prema (62). U srednjem stupcu su \ Rabijevi profili linija sustava, ciljana linija je zelena dok je linija \ smetnje crvena. Vertikalnom crtom ozna\[CHacek]ena je optimalna frekvencija.U \ prvom retku optimalno ugo\:0111enje je u smjeru suprotnom od linije smetnje \ (I-raspored), u drugom retku u smjeru linije smetnje \ (\[CapitalLambda]-raspored). U tre\[CAcute]em retku je prikazan \ zna\[CHacek]ajan u\[CHacek]inak gotovo jedva zamjetnog ugo\:0111enja - na \ rezonantnoj frekvenciji je amplituda oscilacija populacije gotovo \ zanemariva, na optimalnoj je gotovo jednaka jedinici. U sva tri \ slu\[CHacek]aja se tako\:0111er uo\[CHacek]ava i valjanost rezultata (63), \ jer se na optimalnoj frekvenciji vrijeme prijenosa ne razlikuje primjetno od \ \[Pi]/2 dok je ista razlika na rezonantnoj frekvenciji uo\[CHacek]liva.", FontWeight->"Plain", FontSlant->"Italic"] }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"], Cell[TextData[{ "Rezultat (62) nedvojbeno i analiti\[CHacek]ki potvr\:0111uje slutnju \ navedenu u prethodnoj cjelini, o tome da bi u sustavima sa vi\[SHacek]e od \ dvije razine kona\[CHacek]an odmak od rezonantne frekvencije ciljane linije \ mogao pove\[CAcute]ati amplitutu prijenosa populacije. Istovremeno \ mo\:017eemo uo\[CHacek]iti da na\[SHacek] naivni argument nije potupuno to\ \[CHacek]an. Naime, ovisno o energijskom rasporedu svojstvenih stanja uklju\ \[CHacek]enih u dinamiku pogodan pomak mo\:017ee biti ", StyleBox["od", FontSlant->"Italic"], " linije smetnje, ali tako\:0111er i ", StyleBox["prema", FontSlant->"Italic"], " njoj. Prvi se slu\[CHacek]aj javlja ako je energija razine ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " izme\:0111u energija razina ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], ", pa su ", Cell[BoxData[ SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]]]], " i ", Cell[BoxData[ SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]]]], " suprotnog predznaka. Ako skiciramo dijagram u kojemu se na apscisi nalazi \ redom razine ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], ", ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], ", a na ordinati njihove odgovaraju\[CAcute]e energije, u ovom \ slu\[CHacek]aju to\[CHacek]ke grafa nalazit \[CAcute]e se u vrhovima nako\ \[SHacek]enog slova I. zbog toga \[CAcute]emo ovakav energijski raspored \ nazvati ", StyleBox["I-rasporedom", FontSlant->"Italic"], ". Ako je pak energija razine ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], " ve\[CAcute]a ili manja od obiju energija razina ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["s", FontFamily->"Courier", FontSlant->"Italic"], ", ", Cell[BoxData[ SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]]]], " i ", Cell[BoxData[ SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]]]], " su istog suprotnog predznaka i pogodno ugo\:0111enje je ", StyleBox["prema", FontSlant->"Italic"], " liniji smetnje. Ranije opisani dijegram tada ima obliku uspravnog ili \ obrnutog slova '\[CapitalLambda]', zbog \[CHacek]ega \[CAcute]emo ovakav \ energijski raspored nazvati ", StyleBox["\[CapitalLambda]-rasporedom", FontSlant->"Italic"], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Nedostaci pribli\:017eenja razvezane dinamike\ \>", "Subsubsection"], Cell[TextData[{ "Prvi problem rezultata dobivenih ra\[CHacek]unom u pribli\:017eenju \ razvezane dinamike ti\[CHacek]e se iznosa optimalnog ugo\:0111enja iskazanog \ izrazom (61). Valja naime uo\[CHacek]iti da je", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]\_0\), FontFamily->"Courier"], TraditionalForm]]], " kroz veli\[CHacek]inu ", StyleBox["\[CapitalGamma]", FontFamily->"Courier"], " (odre\:0111enu u (48)) zapravo funkcija frekvencije vanjske sile. Ovo je \ nezgodno, jer bi ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]\_0\), FontFamily->"Courier"], TraditionalForm]]], " trebala biti konstanta ovisna samo o parametrima sustava i amplitudi \ vanjske sile, no ne i o njezinoj frekvenciji. Pogledajmo me\:0111utim je li \ to uistinu toliki problem. Promotrimo bolje izraz u kojemu se ", StyleBox["\[CapitalGamma]", FontFamily->"Courier"], " javlja:" }], "Text"], Cell[BoxData[ RowBox[{\(\(2\ \[CapitalGamma]\ \[Sigma]\)\/R\), "=", RowBox[{ RowBox[{"2", FractionBox["\[Omega]", RowBox[{ SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]], "-", SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]]}]]}], "=", RowBox[{ FractionBox["2", RowBox[{"1", "-", FractionBox[ SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]], SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]]}]], FractionBox["\[Omega]", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]]}]}]}]], "NumberedEquation"], Cell[TextData[{ "U zavisnosti o spektralnim polo\:017eajima linija i omjeru ", StyleBox["R", FontFamily->"Courier"], " polu\[SHacek]irina njihovih Rabijevh profila optimalna frekvencija mo\ \:017ee biti poprili\[CHacek]no odmaknuta od rezonancijske. U svakom slu\ \[CHacek]aju, raspon frekvencija koje nas zanimaju prote\:017ee se od \ rezonantne do optimalne, i izme\:0111u krajnjih to\[CHacek]aka ovog raspona \ iznos izraza (64) mo\:017ee biti vrlo razli\[CHacek]it. No, u stvarnim \ sustavima uz liniju kojom \:017eelimo posti\[CAcute]i prijenos populacije \ redovito postoji i \[CHacek]itavo mno\[SHacek]tvo linija smetnje. Neke od \ ovih linija su svakako spektralno relativno bliske ciljanoj liniji, odnosno \ njihova je udaljenost od rezonantne linije bitno manja od njihove udaljenosti \ do ishodi\[SHacek]ta. Jasno je da ugo\:0111enje svakako mora biti manje od te \ najmanje me\:0111ulinijske udaljenosti, pa je stoga promatrani frekventni \ raspon relativno uzak. U tom slu\[CHacek]aju (64) je gotovo konstantan preko \ cijelog tog raspona, pa problem time nestaje. Tada je ", Cell[BoxData[ RowBox[{"\[Omega]", "\[TildeEqual]", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}]]], " pa izraz (64) iznosi pribli\:017eno ", Cell[BoxData[ RowBox[{"2", "/", RowBox[{"(", RowBox[{"1", "-", FractionBox[ SubscriptBox[\(\[Omega]\^0\), StyleBox["ps", FontSlant->"Italic"]], SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]]}], ")"}]}]]], ". \[SHacek]tovi\[SHacek]e, i u slu\[CHacek]aju da ovaj uvjet nije \ ispunjen, optimalnu frekvenciju mo\:017eemo odrediti rekurzivnim postupkom \ \[LongDash] kao nulti korak uzimamo ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"\[Omega]", "=", SubscriptBox[\(\[Omega]\^0\), StyleBox["pt", FontSlant->"Italic"]]}], FontFamily->"Courier"], TraditionalForm]]], ", a zatim iterativni postupak nastavimo sa tako dobivenom vrijednos\ \[CAcute]u optimalne frekvencije ", Cell[BoxData[ RowBox[{"\[Omega]", "=", RowBox[{ SubscriptBox["\[Omega]", StyleBox["pt", FontSlant->"Italic"]], "'"}]}]]], " sve dok se vrijednost ne prestane mijenjati." }], "Text"], Cell[TextData[{ "Vratimo se naposlijetku problemu neo\[CHacek]uvanja unitarnosti valne \ funkcije. U prvom smo odjeljku ove cjeline spomenuli kako su rezultati \ dobiveni metodom razvezane dinamike 'slijepi' s obzirom na curenje. Za vrlo \ mala curenja teorijski rezultati izvrsno se poklapaju sa numeri\[CHacek]kim \ simulacijama, a i za ve\[CAcute]a curenja se dio teorijskih \ dinami\[CHacek]kih zna\[CHacek]ajki jo\[SHacek] uvijek lijepo podudara sa \ teorijskim rezultatima. Me\:0111utim, s porastom curenja opa\:017eeno vrijeme \ prijenosa sve vi\[SHacek]e odstupa od svoje teorijske vrijednosti. Primjeri \ ovog odstupanja su prikazani na ", StyleBox["Slici 8", FontWeight->"Bold"], ". Ovaj \[CAcute]emo nedostatak djelimi\[CHacek]no razrje\[SHacek]iti druga\ \[CHacek]ijim analiti\[CHacek]kim pristupom, ", StyleBox["dvostrukim pribli\:017eenjem rotiraju\[CAcute]eg vala", FontSlant->"Italic"], ", kod kojega je unitarnost valne funkcije sa\[CHacek]uvana." }], "Text"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 %%ImageSize: 600 400 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0606305 0.0147151 0.588604 [ [.11905 .00222 -8.09375 -18.1875 ] [.11905 .00222 8.09375 0 ] [.21429 .00222 -5 -12.5625 ] [.21429 .00222 5 0 ] [.30952 .00222 -12.4063 -20.125 ] [.30952 .00222 12.4063 0 ] [.40476 .00222 -9.3125 -12.5625 ] [.40476 .00222 9.3125 0 ] [.5 .00222 -12.4063 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Vremenski raspon je razvu\ \[CHacek]ena kako bi se bolje uo\[CHacek]ilo odstupanje 'eksperimentalnih' \ numeri\[CHacek]kih rezultata od teorijske vrijednosti odre\:0111ene iz (63). \ Nasuprot skra\[CAcute]enju kojeg uzrokuje odmak od optimalne frekvencije \ (ispravno opisanom rezultatom (63)), vrijeme prijenosa na optimalnoj \ frekvenciji ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["pove\[CAcute]ava ", FontWeight->"Plain"], StyleBox["se s porastom curenja, ", FontWeight->"Plain", FontSlant->"Italic"] }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.3 Analiti\[CHacek]ki pristup II: dvostruko pribli\:017eenje rotiraju\ \[CAcute]eg vala (dRWA)\ \>", "Subsection", CellTags->"2.3"], Cell["\<\ Ispravnost rezultata koje \[CAcute]emo dobiti pomo\[CAcute]u dvostrukog \ pribli\:017eenja rotiraju\[CAcute]eg vala je ograni\[CHacek]ena na relativno \ mali broj slu\[CHacek]ajeva. Naime, temeljna je pretpostavka ovog pristupa \ da se ciljana linija i linija smetnje mogu istovremeno tretirati u pribli\ \:017eenju rotiraju\[CAcute]eg vala. Da bi ovo bilo istina, te dvije linije \ moraju spektralno biti izuzetno bliske, \[SHacek]to mo\:017ee biti ispunjeno \ samo za mali broj linija spektra. Ipak, rezultati koje \[CAcute]emo ovako \ dobiti ukazat \[CAcute]e nam na koji na\[CHacek]in mo\:017eemo upotpuniti \ nedostatke u bitno sveobuhvatnijim rezultatima na\:0111enim ra\[CHacek]unom u \ pribli\:017eenju razvezane dinamike.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Temeljna jednad\:017eba dinamike u dRWA\ \>", "Subsubsection"], Cell["\<\ Jednad\:017ebu dinamike trorazinskog sustava u dRWA dobivamo iz \ jednad\:017ebe (43) ispu\[SHacek]taju\[CAcute]i zadnju, ne-RWA matricu:\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(d\/d\[Tau]\) \(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), " ", RowBox[{"Sign", "[", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "]"}], RowBox[{"(", GridBox[{ {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]], "0"}, { SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]], "0", RowBox[{\(s\_R\), "R", " ", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((\[CapitalDelta] + R\/\[Sigma])\), "\[Tau]"}]]}]}, {"0", RowBox[{\(s\_R\), "R", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]], "2", \((\[CapitalDelta] + R\/\[Sigma])\), "\[Tau]"}]]}], "0"} }], ")"}], \(\(a\& \[Rule] \)[t]\)}]}]], "NumberedEquation"], Cell[TextData[{ "Kao i ranije i iz istog razloga mo\:017eemo ispustiti predznak ", Cell[BoxData[ RowBox[{"Sign", "[", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "]"}]]], ". Primjenom unitarne transformacije, sli\[CHacek]no kako smo to napravili \ u (18), iz hamiltonijana uklanjamo ovisnost o vremenu te dobivamo novu jednad\ \:017ebu " }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/d\[Tau]\) \(b\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ {"0", \(\(1\)\(\ \)\), "0"}, {\(\(1\)\(\ \)\), RowBox[{\(-2\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "\[CapitalDelta]"}], \(\(s\_R\) R\)}, {"0", \(\(s\_R\) R\), RowBox[{\(-2\), " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "-", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]]}], ")"}], "\[CapitalDelta]"}], " ", "-", RowBox[{ SubscriptBox["s", StyleBox["ps", FontSlant-> "Italic"]], \(\(\(R\)\(\ \)\)\/\[Sigma]\)}]}], ")"}]}]} }], ")"}], \(\(b\& \[Rule] \)[\[Tau]]\)}]}]], "NumberedEquation"], Cell[TextData[{ "Op\[CAcute]e rje\[SHacek]enje ove jednad\:017ebe, izra\:017eeno preko \ zasada neodre\:0111enih svojstvenih vrijednosti ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], ";", \(i = 1\)}], ",", "2", ",", "3"}], TraditionalForm]]], " preina\[CHacek]enog vremenski nezavisnog hamiltonijana glasi:" }], "Text"], Cell[BoxData[ RowBox[{\(\(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{ RowBox[{"(", GridBox[{ {"1", "0", "0"}, {"0", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], " ", "2", " ", "\[CapitalDelta]", " ", "\[Tau]"}]], "0"}, {"0", "0", SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", "2", " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "-", SubscriptBox["s", StyleBox["ps", FontSlant->"Italic"]]}], ")"}], "\[CapitalDelta]"}], " ", "-", RowBox[{ SubscriptBox["s", StyleBox["ps", FontSlant-> "Italic"]], \(\(\(R\)\(\ \)\)\/\[Sigma]\)}]}], ")"}], "\[Tau]"}]]} }], ")"}], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["i", FontSlant->"Italic"], "=", "1"}], "3"], RowBox[{ SubscriptBox["c", StyleBox["i", FontSlant->"Italic"]], RowBox[{"(", GridBox[{ {"1"}, { SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]]}, { RowBox[{\(1\/R\), RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], RowBox[{"(", RowBox[{ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], "+", RowBox[{"2", " ", SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "\[CapitalDelta]"}]}], ")"}]}], "-", "1"}], ")"}]}]} }], ")"}], SuperscriptBox["\[ExponentialE]", RowBox[{\(-\[ImaginaryI]\), " ", SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], "\[Tau]"}]]}]}]}]}]], "NumberedEquation"], Cell["\<\ Razmatrat \[CAcute]emo dvije vrste po\[CHacek]etnih uvjeta\ \>", "Text"], Cell[BoxData[ RowBox[{GridBox[{ { RowBox[{\(\(a\& \[Rule] \)[\[Tau] = 0]\), "\[Congruent]", RowBox[{ RowBox[{"(", GridBox[{ {GridBox[{ {"1"}, {"0"} }]}, {"0"} }], ")"}], "-", RowBox[{ StyleBox["po\[CHacek]etno", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["naseljena", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["razina", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["t", FontSlant->"Italic"]}]}]}]}, { RowBox[{\(\(a\& \[Rule] \)[\[Tau] = 0]\), "\[Congruent]", RowBox[{ RowBox[{"(", GridBox[{ {"0"}, {"1"}, {"0"} }], ")"}], "-", RowBox[{ StyleBox["po\[CHacek]etno", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["naseljena", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["razina", FontFamily->"Times New Roman"], " ", StyleBox["p", FontSlant->"Italic"]}]}]}]} }], " "}]], "NumberedEquation"], Cell[TextData[{ "Budu\[CAcute]i je novi hamiltonijan simetri\[CHacek]an, svojstvene su mu \ vrijednosti realne a zbog oblika po\[CHacek]etnih uvjeta realne su i \ konstante ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["c", StyleBox["i", FontSlant->"Italic"]], ";", " ", \(i = 1\)}], ",", "2", ",", "3"}], TraditionalForm]]], ". Vremenski razvoj populacije opisan je sa" }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{\(\(\[CapitalPi]\_\[Alpha][\[Tau]]\)\(\[Congruent]\)\), "|", RowBox[{ SubscriptBox["a", StyleBox["\[Alpha]", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], \( | \^2\)}], "=", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["i", FontSlant->"Italic"], "=", "1"}], "3"], RowBox[{ SuperscriptBox[ RowBox[{"(", SubscriptBox["d", StyleBox[\(i, \[Alpha]\), FontSlant->"Italic"]], ")"}], "*"], SubscriptBox["d", StyleBox[\(i, \[Alpha]\), FontSlant->"Italic"]]}]}], "+", RowBox[{"2", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["i", FontSlant->"Italic"], "=", "1"}], "3"], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["j", FontSlant->"Italic"], ">", StyleBox["i", FontSlant->"Italic"]}], "3"], RowBox[{ SuperscriptBox[ RowBox[{"(", SubscriptBox["d", StyleBox[\(i, \[Alpha]\), FontSlant->"Italic"]], ")"}], "*"], SubscriptBox["d", StyleBox[\(j, \[Alpha]\), FontSlant->"Italic"]], RowBox[{"Cos", "[", RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], "-", SubscriptBox["\[Mu]", StyleBox["j", FontSlant->"Italic"]]}], ")"}], "\[Tau]"}], "]"}]}]}]}]}]}]}], "\[IndentingNewLine]", \(\[Alpha] \ \[Congruent] t, p, s\)}], "NumberedEquation"], Cell["\<\ gdje smo koristili skra\[CAcute]eni zapis\ \>", "Text"], Cell[BoxData[ RowBox[{ SubscriptBox[\(d\& \[Rule] \), StyleBox["i", FontSlant->"Italic"]], "=", RowBox[{ RowBox[{"(", GridBox[{ { SubscriptBox["d", StyleBox[\(i, t\), FontSlant->"Italic"]]}, { SubscriptBox["d", StyleBox[\(i, p\), FontSlant->"Italic"]]}, { SubscriptBox["d", StyleBox[\(i, s\), FontSlant->"Italic"]]} }], ")"}], "\[Congruent]", RowBox[{ SubscriptBox["c", StyleBox["i", FontSlant->"Italic"]], RowBox[{"(", GridBox[{ {"1"}, { SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]]}, { RowBox[{\(1\/R\), RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], RowBox[{"(", RowBox[{ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], "+", RowBox[{ SubscriptBox["s", StyleBox["pt", FontSlant->"Italic"]], "2", " ", "\[CapitalDelta]"}]}], ")"}]}], "-", "1"}], ")"}]}]} }], ")"}]}]}]}]], "NumberedEquation"] }, Open ]], Cell[CellGroupData[{ Cell["Svojstvene vrijednosti ", "Subsubsection"], Cell[TextData[{ "Kako vidimo iz (69) na\[SHacek] je dinami\[CHacek]ki problem \ rije\[SHacek]en onoga \[CHacek]asa kada odredimo svojstene vrijednosti novog \ hamiltonijana. Ovo lako mo\:017eemo napraviti pomo\[CAcute]u nekog od \ programskih paketa za simboli\[CHacek]ku matematiku kao \[SHacek]to su \ 'MATHEMATICA', 'MATLAB' ili 'MAPLE'. Nama zapravo niti ne trebaju potpuni \ analiti\[CHacek]ki izrazi za ove veli\[CHacek]ine, ve\[CAcute] \[CAcute]emo \ se zadovoljiti sa njihovim razvojem po malim parametrima ", StyleBox["\[Sigma]", FontFamily->"Courier"], " i ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], ". Usput napomenimo da smo mi ovo napravili u 'MATHEMATICI' te da smo pri \ tome rabili ne\[SHacek]to malo druga\[CHacek]iji skup parametara razvoja: " }], "Text"], Cell[BoxData[{ \(\(P \[Congruent] 1\/R;\)\), "\[IndentingNewLine]", \(\(x \[Congruent] 2 R\ \[CapitalDelta];\)\), "\[IndentingNewLine]", \(\[Epsilon] \[Congruent] 2 \[Sigma]\)}], "NumberedEquation"], Cell["\<\ jer je sa ovima, iz nama nepoznatih razloga, softver bio vremenski bitno u\ \[CHacek]inkovitiji.. \ \>", "Text"], Cell[TextData[{ "Iz matrice vremenski neovisnog hamiltonijana mo\:017ee se uo\[CHacek]iti \ da svakom paru energijskih rasporeda (uzlazni i silazni I odnosno uspravni i \ obrnuti \[CapitalLambda]) odgovara razli\[CHacek]ita karakteristi\[CHacek]na \ jednad\:017eba. Svojstvene vrijednosti ", Cell[BoxData[ FormBox[ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], TraditionalForm]], FontFamily->"Courier"], StyleBox["; ", FontFamily->"Courier"], StyleBox["i", FontFamily->"Courier", FontSlant->"Italic"], StyleBox["=1,2,3", FontFamily->"Courier"], " za svaki od rasporeda, razvijene do drugog reda u ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " i ", StyleBox["\[Sigma]", FontFamily->"Courier"], " glase:" }], "Text"], Cell[BoxData[{ \(\[CapitalLambda] - rasporedi\), "\[IndentingNewLine]", \(\[Mu]\_1 \[TildeFullEqual] \(\[PlusMinus]\((\((1 - \(R\/4\) \[Sigma] + 1\/32\ \((\(-4\) + R\^2)\)\ \[Sigma]\^2)\) - \((1 - \(R\/4\) \[Sigma])\) \ \[CapitalDelta] + \(1\/2\) \[CapitalDelta]\^2)\)\)\), "\n", \(\[Mu]\_2 \[TildeFullEqual] \(\[PlusMinus]\(\(2 R\)\/\[Sigma]\)\) \((1 + \(1\/4\) \[Sigma]\^2)\)\), "\n", \(\(\(\[Mu]\_3 \[TildeFullEqual] \(\[MinusPlus]\((\((1 + \(R\/4\) \ \[Sigma] + 1\/32\ \((\(-4\) + R\^2)\)\ \[Sigma]\^2)\) + \((1 + R\/4\ \[Sigma])\) \[CapitalDelta]\ + \(1\/2\) \ \[CapitalDelta]\^2)\)\)\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\[CapitalIota] - rasporedi\), "\[IndentingNewLine]", \(\[Mu]\_1 \[TildeFullEqual] \(\[PlusMinus]\((\((1 - \(R\/4\) \[Sigma] + 1\/32\ \((\(-4\) + R\^2)\)\ \[Sigma]\^2)\) + \((1 - \(R\/4\) \[Sigma])\) \ \[CapitalDelta]\ + \(1\/2\) \[CapitalDelta]\^2)\)\)\), "\n", \(\[Mu]\_2 \[TildeFullEqual] \(\[PlusMinus]\(\(2\ R\)\/\[Sigma]\)\) \ \((\((1 + \(1\/4\) \[Sigma]\^2)\) + \(2\/R\) \[Sigma]\ \[CapitalDelta]\ \ )\)\), "\n", \(\[Mu]\_3 \[TildeFullEqual] \(\[MinusPlus]\((\((1 + \(R\/4\) \[Sigma] + 1\/32\ \((\(-4\) + R\^2)\)\ \[Sigma]\^2)\) - \((1 + \(R\/4\) \[Sigma])\) \ \[CapitalDelta]\ + \(1\/2\) \[CapitalDelta]\^2)\)\)\)}], "NumberedEquation"], Cell[TextData[{ "Dva razli\[CHacek]ita predznaka ispred svakog od rje\[SHacek]enja \ odgovaraju dvama razli\[CHacek]itim energijskim podrasporedima. U \ slu\[CHacek]aju iznimno slabe vanjske sile (", StyleBox["\[Sigma]<<1", FontFamily->"Courier"], ") ovi se rezultati svode na one koje smo dobili u slu\[CHacek]aju dviju \ razina (rezultat (23))" }], "Text"], Cell[BoxData[ \(\(\[Mu]\_\[PlusMinus]\) \[Congruent] \(-\[CapitalDelta]\) \[PlusMinus] \ \@\(1 + \[CapitalDelta]\^2\)\( \[TildeFullEqual] \&\(\[CapitalDelta] << 1\)\)\(-\[CapitalDelta]\) \[PlusMinus] \((1\ + \(1\/2\) \ \[CapitalDelta]\^2)\) \[Congruent] \[Mu]\_\(1, 3\)\)], "NumberedEquation"] }, Open ]], Cell[CellGroupData[{ Cell["Zna\[CHacek]ajke populacijske dinamike", "Subsubsection"], Cell[TextData[{ "Iz izraza (69) mo\:017ee se uo\[CHacek]iti da svojstvene vrijednosti ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["\[Mu]", StyleBox["i", FontSlant->"Italic"]], FontFamily->"Courier"], TraditionalForm]]], " u dinamiku ulaze samo preko svojih razlika, ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox[ StyleBox["\[Mu]", FontFamily->"Courier", FontSlant->"Plain"], "i"], "-", SubscriptBox[ StyleBox["\[Mu]", FontFamily->"Courier", FontSlant->"Plain"], "j"]}], "Commentary", FontFamily->"Courier"], TraditionalForm]]], ". Uz uvo\:0111enje novih frekvencija ", StyleBox["\[CapitalOmega]", FontFamily->"Courier"], " i ", StyleBox["\[Rho]", FontFamily->"Courier"], " odre\:0111enih sa" }], "Text"], Cell[BoxData[{ \(\[CapitalOmega] \[Congruent] \(\[Mu]\_1 - \[Mu]\_3\)\/2\), "\n", \(\[Rho] \[Congruent] \(\[Mu]\_1 + \[Mu]\_3\)\/2 - \[Mu]\_2\)}], \ "NumberedEquation"], Cell["\<\ algebarskim se preoblikovanjem rje\[SHacek]enje (69) mo\:017ee izraziti u \ zgodnom obliku \ \>", "Text"], Cell[BoxData[ RowBox[{\(\[CapitalPi]\_\[Alpha][\[Tau]]\), "=", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{ StyleBox["i", FontSlant->"Italic"], "=", "1"}], "3"], RowBox[{ SuperscriptBox[ RowBox[{"(", SubscriptBox["d", StyleBox[\(i, \[Alpha]\), FontSlant->"Italic"]], ")"}], "*"], SubscriptBox["d", StyleBox[\(i, \[Alpha]\), FontSlant->"Italic"]]}]}], "+", RowBox[{"2", SubscriptBox["d", RowBox[{ StyleBox["1", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]], SubscriptBox["d", RowBox[{ StyleBox["3", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]], \(Cos[ 2 \[CapitalOmega]\ \[Tau]]\)}], "+", RowBox[{"2", SubscriptBox["d", RowBox[{ StyleBox["2", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]], RowBox[{"(", RowBox[{ SubscriptBox["d", RowBox[{ StyleBox["1", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]], "+", SubscriptBox["d", RowBox[{ StyleBox["3", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]]}], ")"}], \(Cos[\[CapitalOmega]\ \[Tau]]\), \(Cos[\[Rho]\ \ \[Tau]]\)}], "-", RowBox[{"2", SubscriptBox["d", RowBox[{ StyleBox["2", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]], RowBox[{"(", RowBox[{ SubscriptBox["d", RowBox[{ StyleBox["1", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]], "-", SubscriptBox["d", RowBox[{ StyleBox["3", FontSlant->"Plain"], StyleBox[",", FontSlant->"Italic"], StyleBox["\[Alpha]", FontSlant->"Italic"]}]]}], ")"}], \(Sin[\[CapitalOmega]\ \[Tau]]\), \(Sin[\[Rho]\ \ \[Tau]]\)}]}]}]], "NumberedEquation"], Cell[TextData[{ "gdje je ", StyleBox["\[Alpha]\[Congruent]t,p,s", FontFamily->"Courier"], StyleBox[".", FontFamily->"Times New Roman"] }], "Text"], Cell["\<\ Vrijednosti ovih frekvencija pridru\:017eene dvama rasporedima su\ \>", "Text"], Cell[BoxData[{ StyleBox[\(\[CapitalLambda] - rasporedi\), FontFamily-> "Times New Roman"], "\[IndentingNewLine]", \(\[CapitalOmega] \ \[Congruent] \(\[Mu]\_1 - \[Mu]\_3\)\/2 \[TildeFullEqual] \(\[PlusMinus]\((1 \ - \[Sigma]\^2\/8 + \(1\/2\) \((\[CapitalDelta] + \(R\ \ \[Sigma]\)\/4)\)\^2)\)\) \[TildeFullEqual] \(\[PlusMinus]1\)\), "\n", RowBox[{\(\[Rho] \[Congruent] \(\[Mu]\_1 + \[Mu]\_3\)\/2 - \[Mu]\_2 \ \[TildeFullEqual] \(\[MinusPlus]\(\(2\ R\)\/\[Sigma]\)\) \((1 + \ \(\(\(\[Sigma]\)\(\ \)\)\/\(2\ R\)\) \((\[CapitalDelta] + 3\ \(R\ \[Sigma]\)\/4)\))\) \[TildeFullEqual] \(\ \[MinusPlus]\(\(2\ R\)\/\[Sigma]\)\)\), "\[IndentingNewLine]"}], "\[IndentingNewLine]", StyleBox[\(\[CapitalIota]\ - rasporedi\), FontFamily-> "Times New Roman"], "\[IndentingNewLine]", \(\[CapitalOmega] \ \[Congruent] \(\[Mu]\_1 - \[Mu]\_3\)\/2 \[TildeFullEqual] \(\[PlusMinus]\((1 \ - \[Sigma]\^2\/8 + 1\/2\ \((\[CapitalDelta] - \(R\ \[Sigma]\)\/4)\)\^2)\)\) \ \[TildeFullEqual] \(\[PlusMinus]1\)\), "\n", \(\[Rho] \[Congruent] \(\[Mu]\_1 \ + \[Mu]\_3\)\/2 - \[Mu]\_2 \[TildeFullEqual] \(\[MinusPlus]\(\(2\ R\)\/\ \[Sigma]\)\) \((1 + \(\(\(\[Sigma]\)\(\ \)\)\/\(2\ R\)\) \((3 \ \[CapitalDelta] + 3 \( R\ \[Sigma]\)\/4)\))\) \[TildeFullEqual] \ \(\[MinusPlus]\(\(2\ R\)\/\[Sigma]\)\)\)}], "NumberedEquation"], Cell["\<\ Rje\[SHacek]avanjam jednad\:017ebe (66) uz po\[CHacek]etne uvjete (68) \ nalazimo da prevladavaju\[CAcute]i doprinos dinamici daju prva dva \ \[CHacek]lana u izrazu (75). Dakle, nalazimo Rabijeve oscilacije slijede\ \[CAcute]eg perioda:\ \>", "Text"], Cell[BoxData[ \(\[Tau]\_dRWA = \(\[Pi]\/\(2\ \[CapitalOmega]\) = \(\[Pi]\/2\) 1\/\(1 + \(1\/2\) \((\((\[CapitalDelta] \[PlusMinus] \(R\ \ \[Sigma]\)\/4)\)\^2 - \[Sigma]\^2\/4)\)\)\)\)], "NumberedEquation"], Cell[TextData[{ "Usporedimo ovaj rezultat sa rezultatom (63) dobivenim u okviru RaD. Kada \ (63) razvijemo u red po ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " i ", StyleBox["\[Sigma]", FontFamily->"Courier"], ", nalazimo" }], "Text"], Cell[BoxData[ RowBox[{\(\[Tau]\_RaD\), "=", RowBox[{\(\(\[Pi]\/2\) 1\/\@\(1 + \((\[CapitalDelta] - \[CapitalDelta]\_0)\)\^2\)\), OverscriptBox["\[TildeFullEqual]", GridBox[{ {\(\[Sigma] << 1\)}, {\(\[CapitalDelta] << 1\)} }]], \(\(\[Pi]\/2\) 1\/\(1 + \(1\/2\) \((\[CapitalDelta] \[PlusMinus] \(\(R\ \[Sigma]\)\ \/4\) \((1 + 1\/\(1 - \(2\ \[CapitalGamma]\ \ \[Sigma]\)\/R\))\))\)\^2\)\)}]}]], "NumberedEquation"], Cell[TextData[{ "Kako su u slu\[CHacek]aju valjanosti dRWA ciljana linija i linija smetnje \ vrlo bliske, odgovaraju\[CAcute]i izraz (64) je mnogo ve\[CAcute]i od 1 pa \ razlomak u zagradi uz ", Cell[BoxData[ \(\(R\ \[Sigma]\)\/4\)]], " is\[CHacek]ezava, \[CHacek]ime se (77) svodi na" }], "Text"], Cell[BoxData[ RowBox[{\(\[Tau]\_RaD\), OverscriptBox["\[TildeFullEqual]", GridBox[{ {\(\[Sigma] << 1\)}, {\(\[CapitalDelta] << 1\)} }]], \(\(\[Pi]\/2\) 1\/\(1 + \(1\/2\) \((\[CapitalDelta] \[PlusMinus] \(R\ \[Sigma]\)\/4)\ \)\^2\)\)}]], "NumberedEquation"], Cell[TextData[{ "Usporedbom ovog izraza sa izrazom (77) odre\:0111enim u okviru dRWA \ nalazimo da u ovom kasnijem postoji dodatni \[CHacek]lan,", Cell[BoxData[ \(\[Sigma]\^2\/4\)]], ", i to je upravo onaj \[CHacek]lan kojim \[CAcute]emo popuniti nedostatke \ rezulata (63). Da bismo vidjeli kako to to\[CHacek]no u\[CHacek]initi, moramo \ utvrditi \[SHacek]to taj dodatni \[CHacek]lan predstavlja. Odgovor na to \ pitanje na\[CAcute]i \[CAcute]emo ispitivanjem dinamike populacija \ trorazinskog sustava. " }], "Text"], Cell[TextData[{ "Iznosi pojedinih populacija u trenutku ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["\[Tau]", FontFamily->"Courier"], "dRWA"], FontFamily->"Courier"], TraditionalForm]]], " i do drugog reda u ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " i ", StyleBox["\[Sigma]", FontFamily->"Courier"], " su" }], "Text"], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ StyleBox[\(\[CapitalPi]\_0\), FontFamily->"Courier New"], RowBox[{"(", StyleBox[\(\[Tau] = \[Pi]\/\(2 \[CapitalOmega]\)\), FontFamily->"Courier New"], StyleBox[")", FontFamily->"Courier New"]}]}], StyleBox["\[TildeFullEqual]", FontFamily->"Courier New"], StyleBox[ RowBox[{"1", "-", \(\[Sigma]\^2\/4\), "-", SuperscriptBox[ RowBox[{"(", RowBox[{"\[CapitalDelta]", "\[PlusMinus]", FractionBox[ RowBox[{ StyleBox["R", FontSlant->"Plain"], " ", "\[Sigma]"}], "4"]}], ")"}], "2"]}], FontFamily->"Courier New"]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ StyleBox[ RowBox[{\(\(\[CapitalPi]\_1\)(\[Tau] = \[Pi]\/\(2 \[CapitalOmega]\))\ \), "\[TildeFullEqual]", SuperscriptBox[ RowBox[{"(", RowBox[{"\[CapitalDelta]", "\[PlusMinus]", FractionBox[ RowBox[{ StyleBox["R", FontSlant->"Plain"], " ", "\[Sigma]"}], "4"]}], ")"}], "2"]}], FontFamily->"Courier New"], TraditionalForm], "\[IndentingNewLine]", FormBox[ StyleBox[\(\(\[CapitalPi]\_s\)(\[Tau] = \[Pi]\/\(2 \[CapitalOmega]\)) \ \[TildeFullEqual] \[Sigma]\^2\/4\), FontFamily->"Courier New"], TraditionalForm]}], "NumberedEquation"], Cell[TextData[{ "pri \[CHacek]emu su ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[CapitalPi]\_0\), FontFamily->"Courier New"], "[", "\[Tau]", "]"}], TraditionalForm]]], " odnosno ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[CapitalPi]\_1\), FontFamily->"Courier New"], "[", "\[Tau]", "]"}], TraditionalForm]]], " populacije po\[CHacek]etno nenaseljene odnosno po\[CHacek]etno naseljene \ ciljane razine. Razli\[CHacek]iti predznaci ponovno odgovaraju \ razli\[CHacek]itim energijskim rasporedima sustava. Iz ovoga nalazimo da se \ po\[CHacek]etno nenaseljeno stanje naseljava do vrijednosti:" }], "Text"], Cell[BoxData[ \(1 - \[Sigma]\^2\/4 - \((\[CapitalDelta] \[PlusMinus] \(R\ \ \[Sigma]\)\/4)\)\^2\)], "NumberedEquation"], Cell[TextData[{ "Ovaj izraz se mo\:017ee maksimizirati (do vrijednosti ", Cell[BoxData[ \(TraditionalForm\`1 - \[Sigma]\^2\/4\)]], ") odmakom pogona od rezonancije za iznos ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta] = \(\[MinusPlus]\(\(R\ \[Sigma]\)\/4\ \)\)\)]], ". Ponovno nalazimo da razli\[CHacek]iti energijski rasporedi imaju razlli\ \[CHacek]it odziv na smetnju." }], "Text"], Cell["\<\ \ \>", "Text", TextAlignment->Left], Cell[TextData[{ Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 %%ImageSize: 600 400 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0579021 0.0147151 0.588604 [ [.11476 .00222 -8.09375 -18.1875 ] [.11476 .00222 8.09375 0 ] [.20571 .00222 -5 -12.5625 ] [.20571 .00222 5 0 ] [.29667 .00222 -12.4063 -20.125 ] [.29667 .00222 12.4063 0 ] [.38762 .00222 -9.3125 -12.5625 ] [.38762 .00222 9.3125 0 ] [.47857 .00222 -12.4063 -20 ] [.47857 .00222 12.4063 0 ] [.56952 .00222 -9.3125 -12.5625 ] [.56952 .00222 9.3125 0 ] [.66048 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sustava \[CHacek]ije linije zadovoljavaju \ dRWA. Vremenski raspon iznosi 10 ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[\(\[Tau]\_dRWA\), FontWeight->"Plain", FontSlant->"Plain"], TraditionalForm]], FontSlant->"Italic"], StyleBox[",", FontSlant->"Italic"], StyleBox[" gdje je ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[\(\[Tau]\_dRWA\), FontWeight->"Plain"], TraditionalForm]]], StyleBox[" ", FontWeight->"Plain"], StyleBox["odre\:0111en prema rezultatu (77). Iako je curenje \ prili\[CHacek]no veliko (oko 0.1), teorijski rezultat za vrijeme prijenosa se \ izvrsno poklapa sa numeri\[CHacek]kom simulacijom. Tako\:0111er se jasno uo\ \[CHacek]ava ispravnos rezultata (80) koji nam za maksimalnu vrijednost \ curenja ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[CapitalPi]\_s\)(\[Tau] = \[Pi]\/\(2 \ \[CapitalOmega]\))\)], FontWeight->"Plain"], StyleBox[" predvi\:0111a vrijednost 0.09. ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"] }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"], Cell["\<\ Po\[CHacek]etno naseljeno stanje prazni se do vrijednosti\ \>", "Text"], Cell[BoxData[ \(\((\[CapitalDelta] \[PlusMinus] \(R\ \[Sigma]\)\/4)\)\^2\)], \ "NumberedEquation"], Cell["\<\ koje ima istovjetnu ovisnost o ugo\:0111enju kao i rezultat (81), te ga se \ odgovaraju\[CAcute]im ugo\:0111enjem mo\:017ee minimizirati na vrijednost 0, \ odnosno mo\:017ee se posti\[CAcute]i potpuna depopulacija.\ \>", "Text"], Cell[TextData[{ "Iznos je curenja do drugog reda razvoja neovisan o ugo\:0111enju, smjeru \ prijenosa ili energijskom rasporedu a iznosi ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^2\/4\)]], ". Na ", StyleBox["Slici 9", FontWeight->"Bold"], " prikazan je primjer dinamike trorazinskog sustava \[CHacek]ije spektralne \ linije zadovoljavaju kriterij valjanosti dRWA." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.4 Potpuna slika u\[CHacek]inka smetnje: RaD+dRWA\ \>", "Subsection", CellTags->"2.4"], Cell["\<\ Napokon mo\:017eemo razultate dobivene dvama iznesenim analiti\[CHacek]kim \ pristupima stopiti u jedinstvenu cjelinu. Kona\[CHacek]an rezultat imat \ \[CAcute]e zna\[CHacek]ajke obaju 'roditelja': op\[CAcute]enitost kao RaD i \ preciznost kao dRWA. \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Ograni\[CHacek]enja valjanosti RaD i dRWA rezultata \ \>", "Subsubsection"], Cell["\<\ Za po\[CHacek]etak prou\[CHacek]imo malo potankije razlike \ ograni\[CHacek]enjima valjanosti rezultata dobivernih u okviru pojedinog \ pribli\:017eenja. Oba se pribli\:017eenja temelje na pretpostavci da \ rezonancijska linija zadovoljava RWA uvjet. Kako smo u prvom poglavlju \ temeljito istra\:017eili ograni\[CHacek]enja samog RWA ovdje se nemamo \ potrebe ponovno upu\[SHacek]tati u istu analizu. Ono \[SHacek]to nam je va\ \:017enije jest pogledati \[SHacek]to nam donose ostali uvjeti koje je \ pojedino pribli\:017eenje zahtjevalo. \ \>", "Text"], Cell[TextData[{ "Osim navedenog uvjeta valjanosti RWA na ciljanoj spektralnoj liniji, \ jedini dodatni uvjet kojega RaD tra\:017ei je maleno curenje. Znamo da ovo \ zapravo zna\[CHacek]i da polu\[SHacek]irina linija smetnje bude manja od \ njene udaljenosti do ciljane linije. Iz \[CHacek]injenice da se dinami\ \[CHacek]ka matrica kroz RaD mo\:017ee preoblikovati u matricu sustava sa \ dvije razine za koju potpuno vrijede sve primjedbe navedene za obi\[CHacek]ni \ RWA a valjanost rezultat je vrlo op\[CAcute]enita ba\[SHacek] kao \[SHacek]to \ je to slu\[CHacek]aj u sustavu sa samo dvije razine. Dodatni uvjet nam pri \ tome jedino ka\:017ee da odmak od prilago\:0111ene rezonancije (62) ne smije \ biti toliki da visina Rabijevog profila linije smetnje postane usporediva ili \ ve\[CAcute]a od visine Rabijevog profila ciljane linije. Vidjeli smo da RaD \ rezultat izvrsno odre\:0111uje promjenu rezonancijske frekvencije, no ne \ ukazuje ni na kakvu promjenu ", StyleBox["oblika", FontSlant->"Italic"], " (polu\[SHacek]irine) Rabijevog profila ciljane linije." }], "Text"], Cell[TextData[{ "dRWA dodatno zahtjeva da i linija smetnje tako\:0111er bude u RWA \ re\:017eimu, te da ", StyleBox["\[CapitalDelta]", FontFamily->"Courier"], " bude mnogo manji od jedinice. Stoga je dRWA rezultat prili\[CHacek]no \ krut sa stanovi\[SHacek]ta '\[SHacek]etanja' po spektru van stroge blizine \ rezonancije. No uz bitno op\[CAcute]enitije RaD rezultate jedini uistinu zna\ \[CHacek]ajan rezultat u dRWA je onaj koji se odnosi na curenjem uzrokovanu \ ", StyleBox["popravku vremena prenosa", FontSlant->"Italic"], ". U nastavku \[CAcute]emo pokazati da nam ova zna\[CHacek]ajka dRWA \ rezultata omogu\[CAcute]ava odrediti i promjenu u ", StyleBox["obliku", FontSlant->"Italic"], " Rabijevog profila ciljane linije, odnosno njenoj efektivnoj \ \[SHacek]irini." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Popravka vremena prenosa i promjena polu\[SHacek]irine Rabijevog profila \ ciljane linije\ \>", "Subsubsection"], Cell["\<\ Ugradimo sada popravku prona\:0111enu u dRWA u rezultate dobivene u pribli\ \:017eenju RaD. Dodatni \[CHacek]lan koji se pojavio u izrazu za vrijeme \ prenosa populacije u dRWA (77) jednak je iznosu curenja. Stoga \[CAcute]emo \ pretpostaviti da rezultat (63) mo\:017eemo nadopuniti time da u njega 'rukom' \ umetnemo takav sli\[CHacek]an \[CHacek]lan. U pribli\:017eenju razvezane \ dinamike curenje nalazimo iz rezultata (56)\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], "\[Congruent]", SuperscriptBox[ RowBox[{"(", RowBox[{"|", RowBox[{ SubscriptBox["a", StyleBox["s", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], "|"}], ")"}], "2"], "\[TildeFullEqual]", RowBox[{\(\[Sigma]\^2\/4\), \((1 + \((1\/\(1 - \(2 \[CapitalGamma]\ \ \[Sigma]\)\/R\))\)\^2 + \(2\/\(1 - \(2 \[CapitalGamma]\ \[Sigma]\)\/R\)\) Cos \((4\ \[CapitalGamma]\ \[Tau])\))\), SuperscriptBox[ RowBox[{"(", RowBox[{"|", RowBox[{ SubscriptBox["a", StyleBox["p", FontSlant->"Italic"]], "[", "\[Tau]", "]"}], "|"}], ")"}], "2"]}]}]], "NumberedEquation"], Cell["\<\ Curenje je o\[CHacek]ito najve\[CAcute]e dok je populacija \ posredni\[CHacek]kog stanja najve\[CAcute]a. Odmah uo\[CHacek]avamo da \ curenje, osim o relativno sporo promjenjivoj populaciji posredni\[CHacek]kog \ stanja, ima i dodatnu 'brzu' vremensku ovisnost kroz kosinusni \[CHacek]lan u \ zagradi, pa ne znamo to\[CHacek]no koju vrijednost curenja treba ugraditi kao \ popravku u izraz (63). Ra\[CHacek]un u dRWA nam tu na\:017ealost ne mo\:017ee \ pomo\[CAcute]i, jer u njemu \[CHacek]itava zagrada poprima vrijednost 1 pa se \ dodatna vremenska ovisnost ne primje\[CAcute]uje Na ra\[CHacek]un niza numeri\ \[CHacek]kih eksperimenata i simulacija do\[SHacek]li smo do \ zaklju\[CHacek]ka kako najbolje rezultate posti\:017eemo ako za popravku \ uzmemo\ \>", "Text"], Cell[BoxData[ \(P\_s \[Congruent] \(\[Sigma]\^2\/4\) \((1 + 1\/\(1 - \(2 \ \[CapitalGamma]\ \[Sigma]\)\/R\))\)\^2\)], "NumberedEquation"], Cell["uz koju popravljeno vrijeme prijenosa iznosi", "Text"], Cell[BoxData[{ RowBox[{\(\[Tau]\_prijenos\), "=", RowBox[{\(\[Pi]\/2\), FractionBox["1", SqrtBox[ RowBox[{ "1", "+", \(\((\[CapitalDelta] - \[CapitalDelta]\_0)\)\^2\), "-", SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]]}]]]}]}], "\[IndentingNewLine]", RowBox[{\(T\_prijenos\), "=", RowBox[{ RowBox[{ FractionBox["\[Pi]", RowBox[{\(E\_0\), "|", SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "|"}]], FractionBox["1", SqrtBox[ RowBox[{"1", "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"\[Omega]", "-", RowBox[{ SubscriptBox["\[Omega]", StyleBox["pt", FontSlant->"Italic"]], "'"}]}], ")"}], "2"], SuperscriptBox[ RowBox[{"(", RowBox[{\(E\_0\), SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]]}], ")"}], "2"]], "-", SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]]}]]]}], "=", RowBox[{ FractionBox["\[Pi]", RowBox[{\(E\_0\), "|", RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "'"}], "|"}]], FractionBox["1", SqrtBox[ RowBox[{"1", "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"\[Omega]", "-", RowBox[{ SubscriptBox["\[Omega]", StyleBox["pt", FontSlant->"Italic"]], "'"}]}], ")"}], "2"], SuperscriptBox[ RowBox[{"(", RowBox[{\(E\_0\), RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "'"}]}], ")"}], "2"]]}]]]}]}]}]}], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox[ StyleBox["D", FontSlant->"Plain"], StyleBox["pt", FontSlant->"Italic"]], "'"}], "InlineInput"], TraditionalForm]]], " ", StyleBox["efektivna vrijednost ", FontSlant->"Italic"], "projekcije dipolnog elektri\[CHacek]nog momenta u smjeru polarizacije \ polja pogonske sile, povezana sa po\[CHacek]etnom vrijedno\[SHacek]\[CAcute]u \ ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["D", FontSlant->"Plain"], "12"], "InlineInput"], TraditionalForm]]], " preko " }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "'"}], "\[Congruent]", RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], \(\@\(1 - P\_s\)\)}]}]], "NumberedEquation"], Cell[TextData[{ "Ovaj nam rezultat jasno ukazuje da smetnja, uz pomicanje rezonantne \ frekvencije ciljane linije, tako\:0111er uzrokuje i promjenu oblika njezinog \ Rabijevog profila. Uz postojanje smetnje, polu\[SHacek]irina profila postaje \ ", Cell[BoxData[ RowBox[{\(E\_0\), "|", RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "'"}], "|"}]]], ". " }], "Text"], Cell["\<\ Ne smijemo zaboraviti da smo popravku u (85) umetnuli rukom. Budu\[CAcute]i \ da nam je po\[CHacek]atna zamisao za takvu popravku iznikla iz dRWA ra\ \[CHacek]una, jasno je da bi rezultat (85) svakako trebao vrijediti barem \ onda kada su zna\[CHacek]ajne linije smetnje dosta blizu ciljane linije, pa \ zadovoljavaju uvjete dRWA. Me\:0111utim, kada se iscrpno numeri\[CHacek]ki \ istra\:017ei pona\[SHacek]anje sustava koji uistinu zadovoljavaju dRWA, \ nalazi se da \[CHacek]ak niti u njima ova popravka nije savr\[SHacek]ena. \ Naime, ovisno o energijskom rasporedu (I ili \[CapitalLambda]) i obliku po\ \[CHacek]etnih uvjeta, mo\:017ee se desiti da se maksimalno pra\:017enjenje \ po\[CHacek]etno naseljene razine i maksimalno naseljavanje ciljane razine ne \ posti\:017eu istovremeno. Rezultati (85) - (87) tada daju neko srednje \ vrijeme izme\:0111u ovih dvaju, a jedino \[SHacek]to nam \[CHacek]vrsto jam\ \[CHacek]e jest da je u slu\[CHacek]aju vrlo malih curenja popravka mala ili \ \[CHacek]ak zanemariva. Naravno, budu\[CAcute]i \[CAcute]e nas zanimati \ \[SHacek]to potpuniji prenos izme\:0111u ciljanih razina i vrlo malo \ odgovaraju\[CAcute]e curenje, ovaj \[CAcute]e uvjet redovito biti zadovoljen.\ \ \>", "Text"], Cell["\<\ Na koncu ovog odjeljka ka\:017eimo jo\[SHacek] kako daleko najbolju popravku \ vremena prijenosa na optimalnoj pogonskoj frekvenciji dobivamo ako umjesto \ rezultata (86) koristimo\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], "'"}], "\[Congruent]", FractionBox[ SubscriptBox["D", StyleBox["pt", FontSlant->"Italic"]], \(\@\(1 + P\_s\)\)]}]], "NumberedEquation"], Cell[TextData[{ "Do ovog rezultata do\[SHacek]li smo sasvim slu\[CHacek]ajno, te za sada \ nismo uspjeli izvesti \[CHacek]vrsti analiti\[CHacek]ki dokaz. Napomenimo \ samo da je on toliko robustan da daje ispravnu popravku u vremenu prijenosa \ \[CHacek]ak i u slu\[CHacek]ajevima estremno jakih smetnji, kada je ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["P", FontSlant->"Plain"], StyleBox["s", FontSlant->"Plain"]], FontFamily->"Courier New", FontSlant->"Italic"], TraditionalForm]]], " tek ne\[SHacek]to malo manji od jedinice." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2.5 Vi\[SHacek]estruke linije smetnje", "Subsection", CellTags->"2.5"], Cell[TextData[{ "Poop\[CAcute]imo sada razmatranje na slu\[CHacek]aj proizvoljnog broja \ razina smetnje. U op\[CAcute]enitom \[CAcute]e slu\[CHacek]aju na svaku od \ dviju razina izme\:0111u kojih \:017eelimo posti\[CAcute]i prijenos \ populacije biti vezan \[CHacek]itav niz razina smetnje. Tada \[CAcute]e obje \ \[CAcute]e ciljane razine istovremeno imati ulogu temeljne i \ posredni\[CHacek]ke razine, ovisno o tome koju od razina smetnje promatramo. \ Zbog toga nema vi\[SHacek]e mnogo smisla rabiti indekse ", StyleBox["t", FontFamily->"Courier", FontSlant->"Italic"], " i ", StyleBox["p", FontFamily->"Courier", FontSlant->"Italic"], ", ve\[CAcute] \[CAcute]emo ciljane razine ozna\[CHacek]avati sa ", StyleBox["\[Gamma]", FontFamily->"Courier"], " i ", StyleBox["\[Delta]", FontFamily->"Courier"], ". Koeficijente razvoja pridru\:017eene razinama smetnje vezanim sa \ ciljanom razinom ", StyleBox["\[Gamma]", FontFamily->"Courier"], " ozna\[CHacek]it \[CAcute]emo sa ", Cell[BoxData[ \(TraditionalForm\`\(c\_i\^\[Gamma]\)[\[Tau]]; \ i = 1, ... , n\)], FontFamily->"Courier"], " dok \[CAcute]emo one pridru\:017eene razinama smetnje vezanim sa ciljanom \ razinom ", StyleBox["\[Delta]", FontFamily->"Courier"], " ozna\[CHacek]iti sa ", Cell[BoxData[ FormBox[ StyleBox[\(\(c\_j\^\[Delta]\)[\[Tau]]; \ j = 1, ... , m\), FontFamily->"Courier"], TraditionalForm]]], ". " }], "Text"], Cell[CellGroupData[{ Cell["Jednad\:017eba dinamike", "Subsubsection"], Cell[TextData[{ "Jednad\:017eba dinamke za sustav u kojemu prelaz ", StyleBox["\[Gamma]-\[Delta]", FontFamily->"Courier"], " zadovoljava uvjet RWA glasi" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/\(d\ \[Tau]\)\) \(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(\(s\^R\)\_\[Delta]\[Gamma]\), \((\(-\[ImaginaryI]\))\), RowBox[{"(", GridBox[{ {"0", ".", "0", \(g\_\(\[Gamma], n\)[\[Tau]]\), "0", ".", ".", "0"}, {".", ".", ".", ".", ".", ".", ".", "."}, {"0", ".", "0", \(g\_\(\[Gamma], 1\)[\[Tau]]\), "0", ".", ".", "."}, {\(f\_\(\[Gamma], n\)[\[Tau]]\), ".", \(f\_\(\[Gamma], 1\)[\[Tau]]\), "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(s\_\[Gamma]\ \[Delta]\) 2 \[CapitalDelta]\ \[Tau]\)\), "0", ".", "0"}, {"0", ".", "0", \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(s\_\[Delta]\ \[Gamma]\) 2 \[CapitalDelta]\ \[Tau]\)\), "0", \(f\_\(\[Delta], 1\)[\[Tau]]\), ".", \(f\_\(\[Delta], m\)[\[Tau]]\)}, {".", ".", ".", "0", \(g\_\(\[Delta], 1\)[\[Tau]]\), "0", ".", "0"}, {".", ".", ".", ".", ".", ".", ".", "."}, {"0", ".", ".", "0", \(g\_\(\[Delta], m\)[\[Tau]]\), "0", ".", "0"} }], ")"}], \(\(a\& \[Rule] \)[\[Tau]]\)}]}]], "NumberedEquation"], Cell[TextData[{ "pri \[CHacek]emu je vektor ", Cell[BoxData[ \(\(a\& \[Rule] \)[\[Tau]]\)]], " stanja sustava opisan sa" }], "Text"], Cell[BoxData[ RowBox[{\(\(a\& \[Rule] \)[\[Tau]]\), "\[Congruent]", RowBox[{"(", GridBox[{ {\(\(c\_n\^\[Gamma]\) \([\[Tau]]\)\)}, {"."}, {\(\(c\_1\^\[Gamma]\)[\[Tau]]\)}, {\(a\_\[Gamma][\[Tau]]\)}, {\(a\_\[Delta][\[Tau]]\)}, {GridBox[{ {\(\(c\_1\^\[Delta]\) \([\[Tau]]\)\)}, {"."}, {\(\(c\_m\^\[Delta]\) \([\[Tau]]\)\)} }]} }], ")"}]}]], "NumberedEquation"], Cell["\<\ dok su elementi matrice odre\:0111eni sa\ \>", "Text"], Cell[BoxData[{ \(f\_\(\[Gamma], i\)[\[Tau]] \[Congruent] \(s\^R\)\_\(\[Gamma], i\)\ \ \(R\_\(\[Gamma], i\)\) \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \(s\_\(\[Gamma], i\)\) 2 \((R\_\(\[Gamma], i\)\/\[Sigma]\_\(\[Gamma], i\) + \ \[CapitalDelta])\) \[Tau]\)\) \((\[ExponentialE]\^\(\[ImaginaryI]\ \(s\_\(\ \[Gamma], i\)\) 4\ \(\[CapitalGamma]\_\(\[Gamma], i\)\) \[Tau]\) + 1)\)\), "\[IndentingNewLine]", \(g\_\(\[Delta], j\)[\[Tau]] \[Congruent] \(s\^R\)\_\(\[Delta], j\)\ \ \(R\_\(\[Delta], j\)\) \(\[ExponentialE]\^\(\(+\[ImaginaryI]\)\ \(s\_\(\[Delta], j\)\) 2 \((R\_\(\[Delta], j\)\/\[Sigma]\_\(\[Delta], j\) + \ \[CapitalDelta])\) \[Tau]\)\) \((\[ExponentialE]\^\(\[ImaginaryI]\ \(s\_\(\ \[Delta], j\)\) 4\ \(\[CapitalGamma]\_\(\[Delta], j\)\) \[Tau]\) + 1)\)\)}], "NumberedEquation"] }, Open ]], Cell[CellGroupData[{ Cell["Ukupno curenje", "Subsubsection"], Cell["\<\ Dok je god dinamika sustava sli\[CHacek]na \[CHacek]istom Rabijevom prenosu \ izme\:0111u dviju odabranih razina, dinamiku svake od linija smetnje \ mo\:017eemo rje\[SHacek]avati zasebno, neovisno od ostalih. Iz \ dinami\[CHacek]ke matrice u gornjoj jednad\:017ebi mo\:017eemo uo\[CHacek]iti \ da svaka od dviju ciljanih linija 'pogoni' samo one razine smetnje koje su na \ nju vezane ali nema utjecaja na razine smetnje vezane sa drugom ciljanom \ razinom. Mo\:017eemo dakle o\[CHacek]ekivati da \[CAcute]e za svaku od razina \ smetnje vrijediti relacija istovjetna relaciji (56):\ \>", "Text"], Cell[BoxData[{\(\(c\_i\^\[Gamma]\) \([\[Tau]]\) \[TildeFullEqual] \ \[Alpha]\_\(\[Gamma], i\) - \(\(s\^R\)\_\(\[Gamma], i\)\) \(s\_\(\[Gamma], i\)\) \(\[Sigma]\_\(\[Gamma], i\)\/2\) \(\[ExponentialE]\^\(\ \[ImaginaryI]\ \(s\_\(\[Gamma], i\)\) 2 \((R\_\(\[Gamma], i\)\/\[Sigma]\_\(\[Gamma], i\) + \ \[CapitalDelta])\) \[Tau]\)\) \((1 + \(1\/\(1 - \(2 \[CapitalGamma]\ \ \[Sigma]\_\(\[Gamma], i\)\)\/R\_\(\[Gamma], i\)\)\) \[ExponentialE]\^\(\ \[ImaginaryI]\ \(s\_\(\[Gamma], i\)\) 4\ \[CapitalGamma]\ \[Tau]\))\) a\_\[Gamma][\[Tau]]\), "\[IndentingNewLine]", RowBox[{\(\(c\_j\^\[Delta]\)[\[Tau]]\), " ", "-", " ", StyleBox["analogno", FontFamily->"Times New Roman"]}]}], "NumberedEquation"], Cell["\<\ Maksimalno ukupno curenje nalazimo zbrajanjem maksimalnih vrijednosti \ pojedinih curenja \ \>", "Text"], Cell[BoxData[ RowBox[{" ", RowBox[{ SuperscriptBox[ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "max"], "=", \(\[Sum]\+\(i = 1\)\%n\ Max[\(|\)\(\(c\_i\^\[Gamma]\)[\[Tau]]\)\( | \^2\)] + \ \[Sum]\+\(j = 1\)\%m Max[\(|\)\(\(c\_j\^\[Delta]\)[\[Tau]]\)\( | \^2\)] = E\_0\^2\ \[Lambda]\^2\)}]}]], "NumberedEquation"], Cell["\<\ gdje smo koristili skra\[CAcute]eni zapis\ \>", "Text"], Cell[BoxData[{\(\[Lambda]\^2 \[Congruent] \[Sum]\+\(i = 1\)\%n\((\[Lambda]\_i\ \^\[Gamma])\)\^2 + \[Sum]\+\(j = 1\)\%m\((\[Lambda]\_j\^\[Delta])\)\^2\), "\ \[IndentingNewLine]", \(\[Lambda]\_i\^\[Gamma] = \(1\/2\) \(\(\(|\)\(D\_\(\ \[Gamma], i\)\)\(|\)\)\/\(\(\[Omega]\^0\)\_\[Gamma]\[Delta] - \ \[Omega]\_\(\[Gamma], i\)\)\) \((1 + 1\/\(\(|\)\(1 - \(2\/\(1 - \[Omega]\_\(\[Gamma], \ i\)\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\)\) \ \[Omega]\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\)\(|\)\))\)\), "\ \[IndentingNewLine]", RowBox[{\(\[Lambda]\_j\^\[Delta]\), "-", StyleBox["analogno", FontFamily->"Times New Roman"]}]}], "NumberedEquation"], Cell[TextData[{ "Sa ", Cell[BoxData[ StyleBox[\(\(\[Omega]\^0\)\_\[Gamma]\[Delta]\), FontFamily->"Courier"]]], " smo ozna\[CHacek]ili rezonantnu frekvenciju prelaza izme\:0111u vlastitih \ razina ", StyleBox["\[Gamma]", "InlineInput"], " i ", StyleBox["\[Delta]", "InlineInput"], ", dok je ", Cell[BoxData[ \(\[Omega]\_\(\[Gamma], i\)\)]], " rezonancijska frekvencija za prelaz izme\:0111u razine ", StyleBox["\[Gamma]", "InlineInput"], " i ", StyleBox["i", "InlineInput"], "-te razine smetnje vezane na razinu ", StyleBox["\[Gamma]", FontFamily->"Courier"], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Optimalno ugo\:0111enje i popravljeno vrijeme prijenosa\ \>", "Subsubsection"], Cell["\<\ Uvr\[SHacek]tavanjem doprinosa od svih razina smetnje u (88) i zatvaranjem \ rekurzije vodi nas do nove jednad\:017ebe za ciljani podsustav:\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(d\/\(d\ \[Tau]\)\) \(a\& \[Rule] \)\_\[Gamma]\[Delta][\[Tau]]\ \), "\[TildeFullEqual]", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ {\(-\[Chi]\_\[Gamma]\), \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ s\_\[Gamma]\[Delta]\ 2 \[CapitalDelta]\ \[Tau]\)\)}, {\(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ s\_\[Delta]\[Gamma]\ \ 2 \[CapitalDelta]\ \[Tau]\)\), \(-\[Chi]\_\[Delta]\)} }], ")"}], \(\(a\& \[Rule] \)\_\[Gamma]\[Delta][\[Tau]]\)}]}]], \ "NumberedEquation"], Cell["pri \[CHacek]emu je", "Text"], Cell[BoxData[{\(\[Chi]\_\[Gamma] \[Congruent] \[Sum]\+\(i = 1\)\%n \ \[Chi]\_i\^\[Gamma] = \(E\_0\/\(2 | D\_\[Gamma]\[Delta] | \)\) \(\[Sum]\+\(i = 1\)\%n\( s\_\(\[Gamma], i\)\) \(\((D\_\(\[Gamma], i\))\)\^2\/\(\(\[Omega]\^0\)\_\ \[Gamma]\[Delta] - \[Omega]\_\(\[Gamma], i\)\)\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\[Gamma], \ i\)\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\)\) \ \[Omega]\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\))\)\)\), "\[IndentingNewLine]", RowBox[{\(\[Chi]\_\[Delta]\), "-", StyleBox["analogno", FontFamily->"Times New Roman"]}]}], "NumberedEquation"], Cell[TextData[{ "Odavde nalazimo da optimalno ugo\:0111enje ", Cell[BoxData[ \(\[CapitalDelta]\_0\)]], " i optimalna frekvencija ", Cell[BoxData[ \(\[Omega]\_\[Gamma]\[Delta]'\)]], " iznose" }], "Text"], Cell[BoxData[{ \(\[CapitalDelta]\_0 = \(-s\_\[Gamma]\[Delta]\) \(\[Chi]\_\[Gamma] - \ \[Chi]\_\[Delta]\)\/2\), "\[IndentingNewLine]", \(\[Omega]\_\[Gamma]\[Delta]' = \(\[Omega]\^0\)\_\[Gamma]\[Delta] - \(s\_\ \[Gamma]\[Delta]\) \(\((E\_0\/2)\)\^2\) \((\[Sum]\+\(i = 1\)\%n\( s\_\(\[Gamma], i\)\) \(\((D\_\(\[Gamma], \ i\))\)\^2\/\(\(\[Omega]\^0\)\_\[Gamma]\[Delta] - \[Omega]\_\(\[Gamma], \ i\)\)\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\[Gamma], i\)\/\(\ \[Omega]\^0\)\_\[Gamma]\[Delta]\)\) \[Omega]\/\(\[Omega]\^0\)\_\[Gamma]\ \[Delta]\))\) - \[Sum]\+\(j = 1\)\%m\( s\_\(\[Delta], j\)\) \(\((D\_\(\[Delta], \ j\))\)\^2\/\(\(\[Omega]\^0\)\_\[Gamma]\[Delta] - \[Omega]\_\(\[Delta], \ j\)\)\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\[Delta], j\)\/\(\ \[Omega]\^0\)\_\[Gamma]\[Delta]\)\) \[Omega]\/\(\[Omega]\^0\)\_\[Gamma]\ \[Delta]\))\))\)\)}], "NumberedEquation"], Cell["\<\ Budu\[CAcute]i je odljev u svaku od razina smetnje neovisan o ostalima \ (naravno, dok su svi skupa mali), na temelju rezultata (85) i (86) nalazimo \ da popravljeno vremena prenosa iznosi\ \>", "Text"], Cell[BoxData[ \(T\_prijenos = \(\[Pi]\/\(\(|\)\(\(E\_0\) D\_\[Gamma]\[Delta]'\)\(|\)\)\) 1\/\@\(1 + \((\[Omega] - \[Omega]\_\[Gamma]\[Delta]')\)\^2\/\((\(E\_0\ \) D\_\[Gamma]\[Delta]')\)\^2\)\)], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ \(\[Omega]\_\[Gamma]\[Delta]'\)]], " popravljena rezonancijska frekvencija odre\:0111ena ugo\:0111enjem (96), \ dok je ", Cell[BoxData[ RowBox[{ StyleBox[\(D\_\[Gamma]\[Delta]'\), "InlineInput"], StyleBox["\[Congruent]", "InlineInput"], FractionBox[ StyleBox[\(D\_\[Gamma]\[Delta]\), "InlineInput"], \(\@\(1 + P\_s\)\)]}]]], " pri \[CHacek]emu je ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["P", FontSlant->"Plain"], "s"], "InlineInput"], TraditionalForm]]], " popravka uslijed curenja populacije u sve razine smetnje izra\:017eena \ kao " }], "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "\[Congruent]", \(\(\((E\_0\/2)\)\^2\) \((\[Sum]\+\(i = \ 1\)\%n\(\((D\_\(\[Gamma], i\))\)\^2\/\((\(\[Omega]\^0\)\_\[Gamma]\[Delta] - \ \[Omega]\_\(\[Gamma], i\))\)\^2\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\ \[Gamma], i\)\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\)\) \ \[Omega]\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\))\)\^2 + \[Sum]\+\(j = \ 1\)\%m\(\((D\_\(\[Delta], j\))\)\^2\/\((\(\[Omega]\^0\)\_\[Gamma]\[Delta] - \ \[Omega]\_\(\[Delta], j\))\)\^2\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\ \[Delta], j\)\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\)\) \ \[Omega]\/\(\[Omega]\^0\)\_\[Gamma]\[Delta]\))\)\^2)\)\)}]], \ "NumberedEquation"], Cell[TextData[{ "Na ", StyleBox["Slici 10 ", FontWeight->"Bold"], "i", StyleBox[" Slici 11 ", FontWeight->"Bold"], "prikazani su primjeri usporedbe numeri\[CHacek]kih rje\[SHacek]enja \ dinamike populacija i odgovaraju\[CAcute]ih analiti\[CHacek]kih \ predvi\:0111anja za molekulu HF u osnovnom elektronskom stanju." }], "Text"], Cell[TextData[{ "\n", Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 23.8095 0 0.618034 [ [.2619 -0.0125 -12 -9 ] [.2619 -0.0125 12 0 ] [.5 -0.0125 -12 -9 ] [.5 -0.0125 12 0 ] [.7381 -0.0125 -12 -9 ] [.7381 -0.0125 12 0 ] [.97619 -0.0125 -12 -9 ] [.97619 -0.0125 12 0 ] [1.025 0 0 -6.28125 ] [1.025 0 10 6.28125 ] [.01131 .12361 -18 -4.5 ] [.01131 .12361 0 4.5 ] [.01131 .24721 -18 -4.5 ] [.01131 .24721 0 4.5 ] [.01131 .37082 -18 -4.5 ] [.01131 .37082 0 4.5 ] [.01131 .49443 -18 -4.5 ] [.01131 .49443 0 4.5 ] [.01131 .61803 -6 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Gornji odnosno donji graf \ odgovara vezanjima razine (4,3,0) odnosno (2,4,0) sa ostatkom sustava. \ Vertikalnim linijama ozna\[CHacek]ena su sredi\[SHacek]ta pojedinih profila. \ Na oba je grafa ciljana linija druga sa desne strane. Jakost \ elektri\[CHacek]nog polja vanjskog pogona iznosi ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], "0"], "=", \(0.012\ au\)}], TraditionalForm]], FontWeight->"Plain"], ". ", StyleBox["U tablicama su redom navedeni identifikacijski broj linije", FontWeight->"Plain", FontSlant->"Italic"], ",", StyleBox[" kona\[CHacek]na razina danog prelaza, odgovaraju\[CAcute]a \ frekvencija prelaza, teorijski iznos curenja za pojedini prelaz te iznos \ projekcije integrala dipolnog momenta u smjeru osi elektri\[CHacek]nog polja \ za dani prelaz. Sve veli\[CHacek]ine su dane u atomskim jedinicama (au). \ Teorijski o\[CHacek]ekivano maksimalno ukupno curenje iznosi oko 10 %. \ Zanimljivo je uo\[CHacek]iti da najve\[CAcute]a smetnja ne dolazi od \ spektralno najbli\:017eih linija (8., 10. i 17.), ve\[CAcute] od onih \ najudaljenijih (1.,2., 11. i 12.), \[SHacek]to je posljedica mnogo \ ve\[CAcute]e vrijednosti projekcije integrala dipolnog momenta za ove \ kasnije.", FontWeight->"Plain", FontSlant->"Italic"] }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"], Cell[TextData[{ Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0=WIf@8000000P3IfMT00080fMWI0P00000H0=WIf@03 0000003IfMT0fMWI02d0fMWI1P00002=0=WIf@L00000D03IfMT200000080 fMWI00020=WIf@800000603IfMT00`000000fMWI0=WIf@0_0=WIf@D00000 RP3IfMT6000005<0fMWI0P0000020=WIf@000P3IfMT2000001P0fMWI0P00 000a0=WIf@H00000QP3IfMT6000005D0fMWI0P0000020=WIf@000P3IfMT2 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Lijevi graf u prvom retku prikazuje dinamiku populacija ciljanih razina u \ slu\[CHacek]aju optimalnog ugo\:0111enja ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((\[Omega] = 0.329654\ 10\^\(-1\)\ au)\)\)], FontWeight->"Plain"], StyleBox[" odre\:0111enog prema (94), za jakost pogonskog \ elektri\[CHacek]nog polja ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], "0"], "=", \(0.012\ au\)}], TraditionalForm]], FontWeight->"Plain"], StyleBox[".", FontWeight->"Plain"], StyleBox[" Desni grafovi u prvom retku prikazuju potankosti oko trenutka ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["T", FontFamily->"Courier", FontSlant->"Plain"], "prenos"], FontFamily->"Courier", FontWeight->"Plain"], TraditionalForm]]], StyleBox[" odre\:0111enog prema (97). Ostali grafovi prikazuju dinamiku \ pojedinih ja\[CHacek]e pobu\:0111enih razina smetnje. Vrp\[CHacek]ast izgled \ ovih grafova posljedica je brzih oscilacija populacija odre\:0111enih \ kosinusnim \[CHacek]lanom kao u (83). Iako se one ne mogu jasno razabrati na \ gornjem vremenskom rasponu, jasno su uo\[CHacek]ljive na detaljnim grafovima \ za populacije ciljanih razina \[CHacek]iji je vremenski raspon za red veli\ \[CHacek]ine manji. Slaganje sa predvi\:0111anjima navedenim u tablicama na ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["Slici 10a", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" je vrlo dobro. Ipak, teorija i numeri\[CHacek]ka simulacija \ razilaze se u vremenu prenosa, \[SHacek]to smo mogli i o\[CHacek]ekivati radi \ relativno velikog curenja - teorija daje ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox[ StyleBox["T", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "prijenosa"], "=", \(86878\ au\)}], FontFamily->"Courier", FontWeight->"Plain"], TraditionalForm]]], ",", StyleBox[" dok iz gornjih grafova mo\:017eemo o\[CHacek]itati ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox[ StyleBox["T", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "prijenosa"], "\[TildeEqual]", \(86400\ au\)}], FontFamily->"Courier", FontWeight->"Plain"], TraditionalForm]]], "." }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 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ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}]]], TextAlignment->Left] }], "Text", TextAlignment->Center], Cell[TextData[{ "Slika 11b -", StyleBox[" Numeri\[CHacek]ko rje\[SHacek]enje dinamike populacija za prelaz \ (\[Nu],L,m)\[Rule](\[Nu]',L',m') \[Congruent] (2,6,0)\[LeftRightArrow](1,5,0) \ . Lijevi graf u prvom retku prikazuje dinamiku populacija ciljanih razina u \ slu\[CHacek]aju optimalnog ugo\:0111enja ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((\[Omega] = 0.01904 a . u . )\)\)], FontWeight->"Plain"], StyleBox[" odre\:0111enog na temelju rezultata (96), za jakost pogonskog \ elektri\[CHacek]nog polja ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], "0"], "=", \(0.012\ au\)}], TraditionalForm]], FontWeight->"Plain"], StyleBox[".", FontWeight->"Plain"], StyleBox[" Razabire se da ugo\:0111enje nije savr\[SHacek]eno, \[SHacek]to \ je posljedica relativno velikog raspona vrijednosti izraza(64). Bolje \ slaganje dobili bismo da smo proveli rekurzijski postupak opisan u \ posljednjem odjeljku cjeline 2.2, no popravka u odnosu na rezonantni slu\ \[CHacek]aj je i ovako vrlo zna\[CHacek]ajna. Desni grafovi u prvom retku \ prikazuju potankosti oko trenutka ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["T", FontFamily->"Courier", FontSlant->"Plain"], "prijenosa"], FontFamily->"Courier", FontWeight->"Plain"], TraditionalForm]]], StyleBox[" odre\:0111enog prema (95). Teorijska nepopravljena odnosno \ popravljena vrijednost vremena prijenosa iznose 17720 a.u. odnosno 18041 \ a.u., dok iz grafa mo\:017eemo o\[CHacek]itati da je vrijeme prijenosa \ otprilike 18150 a.u.. Mo\:017eemo dakle zaklju\[CHacek]iti da je u ovom slu\ \[CHacek]aju popravka radi curenja valjana, iako ne i savr\[SHacek]ena. \ Posljednji graf prikazuje dinamiku jedine znatno pobu\:0111ene razine \ smetnje, (3,7,0).", FontWeight->"Plain", FontSlant->"Italic"] }], "Commentary", FontWeight->"Bold", FontSlant->"Plain"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ POGLAVLJE 3 Vi\[SHacek]estruko rezonantni lan\[CHacek]asti prijenos\ \>", "Section 1", CellTags->"POGLAVLJE 3"], Cell[TextData[{ "U ovom, posljednjem poglavlju rada prikazat \[CAcute]emo metodu laserskog \ upravljanja koju \[CAcute]emo nazvati ", StyleBox["lan\[CHacek]astim", FontSlant->"Italic"], " ", StyleBox["vi\[SHacek]estruko rezonantnim (LaVoR) prijeenosom", FontSlant->"Italic"], ". Prvi dio rasprave posvetit \[CAcute]emo ispitivanju samog principa ove \ vrste prenosa u idealiziranom kvantnom sustavu gra\:0111enom od samih stanja \ koja tvore odabranu putanju populacije. Potom \[CAcute]emo pogledati kako \ moramo promijeniti na\[SHacek]e zaklju\[CHacek]ke kada u obzir uzmemo da \ svako od stanja du\:017e putanje mo\:017ee imati na se vezan \[CHacek]itav \ niz razina smetnje. Na poslijetku \[CAcute]emo se okoristit rezultatima \ prethodnog poglavlja: opisat \[CAcute]emo algoritam za oblikovanje pogodnog \ pogona za postizanje LaVoR prenosa u stvarnim sustavima, te prikazati \ upotrebu istoga na primjeru dinamike numeri\[CHacek]kog modela \ ro-vibracijskih stupnjeva molekule HF. " }], "Text"], Cell[CellGroupData[{ Cell["\<\ 3.1 Jednad\:017eba dinamike lan\[CHacek]astih sustava\ \>", "Subsection", CellTags->"3.1"], Cell["\<\ Op\[CAcute]u jednad\:017ebu dinamike vi\[SHacek]erazinskih kvantnikh sustava \ pod utjecajem vanjske smenje iznijeli smo na samom po\[CHacek]etku ovog rada \ (jed. (7)). U prikazanoj jednad\:017ebi svaka od razina sustava vezana je sa \ svim ostlim razinama sustava. Naravno, ve\[CAcute]ina stvarnih sustava nije \ ovakva, ve\[CAcute] unutar sustava postoje odre\:0111ena 'izborna pravila' \ kojima su ograni\[CHacek]ena vezanja pod utjecajem odre\:0111enog oblika \ vanjske sile. \ \>", "Text"], Cell[CellGroupData[{ Cell["Putanja i smetnje", "Subsubsection"], Cell[TextData[{ "U nastavku \[CAcute]emo se ograni\[CHacek]iti na ", StyleBox["lan\[CHacek]aste", FontSlant->"Italic"], " ", StyleBox["vi\[SHacek]estruko rezonantne", FontSlant->"Italic"], " sustave. Ovakvi su sustavi gra\:0111eni od dvije skupine svojstvenih \ razina. Prvu skupinu sa\[CHacek]injavaju ", StyleBox["razine putanje", FontSlant->"Italic"], " me\:0111u kojima mo\:017eemo ostvariti prijenos populacije pomo\[CAcute]u \ niza jednostavnih rezonantnih Rabijevih prenosa. Svaka od razina u ovoj \ skupini vezana je sa najvi\[SHacek]e dvije druge razine iz skupine i to tako \ da je putanja prijenosa populacije izme\:0111u svakih dviju razina jednozna\ \[CHacek]no odre\:0111ena. Dakle, izme\:0111u razina prve skupine mo\:017eemo \ posti\[CAcute]i 'lan\[CHacek]asti' prijenos populacije. U drugu skupinu \ razina spadaju ", StyleBox["razine smetnje", FontSlant->"Italic"], ". Svaka od razina smetnje vezana je na jednu jedinu razinu putanje. \ Populacija koja tokom prenosa dospije izvan putanje zavr\[SHacek]i razasuta \ po razinama smetnje pa kona\[CHacek]na populacija razina smetnje ima ulogu \ odljeva za lan\[CHacek]asti prijenos. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Vi\[SHacek]estruko rezonantni pogon", "Subsubsection"], Cell[TextData[{ "Prijenos populacije kroz lan\[CHacek]astu putanju mo\:017eemo ostvariti \ korak po korak, nizom jednostavnih Rabijevih prenosa. Vrijeme potrebno za \ takav prijenos jednako je zbroju pojedina\[CHacek]nih vremena u svakom od \ koraka. Mi bismo \:017eeljeli smanjiti to vrijeme na na\[CHacek]in da sustav \ podvrgnemo istovremenom djelovanju vi\[SHacek]e rezonantnih pogona, svaki za \ po jedan prijelaz du\:017e putanje. Na ovaj se na\[CHacek]in nadamo skratiti \ ukupno vrijeme prijenosa, te uz to dobiti jednako potpun prijenos populacije. \ Uzmimo da se putanja sastoji od N razina. Tada za prijenos moramo \ upotrijebiti ", StyleBox["N-1", FontFamily->"Courier"], " rezonantnu frekvenciju. ", StyleBox["N", FontFamily->"Courier"], " svojstvenih razina ", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\_i\)], FontFamily->"Courier"], " du\:017e putanje ozna\[CHacek]it \[CAcute]emo indeksom ", StyleBox["\[Alpha]", FontFamily->"Courier"], " prema redoslijedu njihovog naseljavanja; ", StyleBox["\[Alpha]=1", FontFamily->"Courier"], " za po\[CHacek]etnu razinu putanje ", Cell[BoxData[ FormBox[ StyleBox[\(\[CurlyPhi]\_\(i[\[Alpha] = 1]\)\), FontFamily->"Courier"], TraditionalForm]]], " a ", StyleBox["\[Alpha]=N", FontFamily->"Courier"], " za kona\[CHacek]nu razinu ", Cell[BoxData[ FormBox[ StyleBox[\(\[CurlyPhi]\_\(i[\[Alpha] = N]\)\), FontFamily->"Courier"], TraditionalForm]]], ". Sli\[CHacek]no \[CAcute]emo napraviti i sa odgovaraju\[CAcute]im \ koeficijentima razvoja: ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["A", FontFamily->"Courier", FontSlant->"Plain"], "1"], FontFamily->"Courier"], StyleBox["[", FontFamily->"Courier"], StyleBox["t", FontFamily->"Courier", FontSlant->"Plain"], StyleBox["]", FontFamily->"Courier"]}], TraditionalForm]]], " je koeficijent pridru\:017een po\[CHacek]etnoj razini putanje, dok je ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["A", FontSlant->"Plain"], "N"], FontFamily->"Courier"], "[", StyleBox["t", FontFamily->"Courier", FontSlant->"Plain"], StyleBox["]", FontFamily->"Courier"]}], TraditionalForm]]], " pridru\:017een kona\[CHacek]noj. Nadalje, sa ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["E", FontFamily->"Courier", FontSlant->"Plain"], "12"], FontFamily->"Courier"], TraditionalForm]]], " \[CAcute]emo ozna\[CHacek]iti amplitudu sastavnice pulsa kojime pogonimo \ prvi prelaz u lancu izme\:0111u razina sa indeksima ", StyleBox["1", "InlineInput"], " i ", StyleBox["2", "InlineInput"], ", a sa ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], RowBox[{ StyleBox["N", FontSlant->"Plain"], "-", RowBox[{"2", " ", StyleBox["N", FontSlant->"Plain"]}], "-", "1"}]], FontFamily->"Courier"], TraditionalForm]]], " onaj kojime pogonimo posljednji prelaz. Bez smanjenja op\[CAcute]enitosti \ pretpostaviti da je ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], \(\[Alpha]\ \[Alpha] + 1\)], FontFamily->"Courier"], ">", "0"}], TraditionalForm]]], " Op\[CAcute]eniti vi\[SHacek]erezonantni pogonski puls prikazat \ \[CAcute]emo u obliku" }], "Text"], Cell[BoxData[ \(V[x, t, \[Omega]\_\(\[Alpha]\ \[Alpha] + 1\); \[Alpha] = 1, ... , N - 1] = \(D\&^[ x]\[CenterDot]\(\[Sum]\+\(\[Alpha] = 1\)\%\(N - 1\)\(\(E\& \ \[Rule] \)\_\(\[Alpha]\ \[Alpha] + 1\)\) Cos[\(\[Omega]\_\(\[Alpha]\ \[Alpha] + 1\)\) t]\) = \(1\/2\) \(\[Sum]\+\(\[Alpha] = 1\)\%\(N - \ 1\)\((D\&^[x]\[CenterDot] E\&^\_\(\[Alpha]\ \[Alpha] + 1\))\) \(E\_\(\[Alpha]\ \ \[Alpha] + 1\)\) \((\[ExponentialE]\^\(\(\[Omega]\_\(\[Alpha]\ \ \[Alpha] + 1\)\) t\) + \[ExponentialE]\^\(\(-\[Omega]\_\(\[Alpha]\ \[Alpha] + \ 1\)\) t\))\)\)\)\)], "NumberedEquation"], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ OverscriptBox[ StyleBox["D", FontSlant->"Plain"], "^"], "[", "x", "]"}], FontFamily->"Courier"], TraditionalForm]]], " je prostorni dipolni operator a ", Cell[BoxData[ FormBox[ StyleBox[\(\[Omega]\_\(\[Alpha]\ \[Alpha] + 1\)\), FontFamily->"Courier"], TraditionalForm]]], " je frekvencija pojedine sastavnice pogona koja je op\[CAcute]enito \ bliska, ali ne nu\:017eno i to\[CHacek]no jednaka, rezonantnoj frekvenciji ", Cell[BoxData[ StyleBox[\(\(\[Omega]\^0\)\_\(i[\[Alpha]] i[\[Alpha] + 1]\)\), FontFamily->"Courier"]]], " prelaza izme\:0111u svojstvenih razina ", Cell[BoxData[ FormBox[ StyleBox[\(\[CurlyPhi]\_\(i(\[Alpha])\)\), FontFamily->"Courier"], TraditionalForm]]], " i ", Cell[BoxData[ FormBox[ StyleBox[\(\[CurlyPhi]\_\(i(\[Alpha] + 1)\)\), FontFamily->"Courier"], TraditionalForm]]], ". Zadatak nam je ugoditi amplitude i frekvencije pogonskog pulsa tako da \ dobijemo \[SHacek]to je mogu\[CAcute]e \[CHacek]i\[SHacek]\[CAcute]i i br\ \:017ei prenos populacije iz po\[CHacek]etne u kona\[CHacek]nu razinu." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3.2 Potpuna rezonancija i vi\[SHacek]estruko pribli\:017eenje rotiraju\ \[CAcute]eg vala\ \>", "Subsection", CellTags->"3.2"], Cell["\<\ U prvome slu\[CHacek]aju kojega \[CAcute]emo razmotriti, problem \[CAcute]emo \ pojednostavnit time da frekvencije sastavnica pogona ugodimo to\[CHacek]no na \ odgovaraju\[CAcute]e rezonantne frekvencije pojedinih prijelaza du\:017e \ lanca. Time nam kao slobodni parametri ostaju samo amplitude pojedinih \ sastavnica. Za po\[CHacek]etak \[CAcute]emo tako\:0111er pretpostaviti da u \ sustavu nema niti jedne razine smetnje.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Dinami\[CHacek]ka jednad\:017eba i zahtjevi na rje\[SHacek]enje\ \>", "Subsubsection"], Cell[TextData[{ "Poop\[CAcute]avanjem argumenata rasprave o RWA iznesene u posljednjem \ odjeljku cjeline 1.2 na ovaj slu\[CHacek]aju ", StyleBox["potpune rezonancije", FontSlant->"Italic"], " uo\[CHacek]avamo da u matrici hamiltonijana mo\:017eemo zanemariti sve \ osim samih rezonantnih elemenata \[CHacek]ime se dinami\[CHacek]ka jednad\ \:017eba svodi na" }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/dt\) \(A\& \[Rule] \) \((t)\)\), "=", RowBox[{\(-\[ImaginaryI]\), \(1\/2\), RowBox[{"(", GridBox[{ {"0", \(\(E\_12\) D\_12\), "0", ".", ".", "0"}, {\(\(E\_12\) D\_12\), "0", \(\(E\_23\) D\_23\), "0", ".", "."}, {"0", \(\(E\_23\) D\_23\), "0", ".", "0", "."}, {".", "0", ".", "0", \(\(E\_\(N - 2\ N - 1\)\) D\_\(N - 2\ N - 1\)\), "0"}, {".", ".", "0", \(\(E\_\(N - 2\ N - 1\)\) D\_\(N - 2\ N - 1\)\), "0", \(\(E\_\(N - 1\ N\)\) D\_\(N - 1\ N\)\)}, {"0", ".", ".", "0", \(\(E\_\(N - 1\ N\)\) D\_\(N - 1\ N\)\), "0"} }], ")"}], \(A\& \[Rule] \), \((t)\)}]}]], "NumberedEquation"], Cell["gdje je ", "Text"], Cell[BoxData[ \(D\_\(\[Alpha]\ \[Alpha] + 1\) \[Congruent] E\&^\_\(\[Alpha]\ \[Alpha] + 1\)\[CenterDot]\(\[Integral]\((\[CurlyPhi]\ \_\(i[\[Alpha]]\)[x]\ D\&^[x]\ \[CurlyPhi]\_\(i[\[Alpha] + 1]\)[ x]\ dx)\)\)\)], "NumberedEquation"], Cell[TextData[{ "projekcija dipolnog elektri\[CHacek]nog momenta u smjeru pojedine \ sastavnice pulsa. U ovom je slu\[CHacek]aju matrica hamiltonijana odmah \ nezavisna o vremenu. Na\[SHacek] je problem stoga formalno rije\[SHacek]en \ \[CHacek]im zadamo jakosti vezanja ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["I", FontSlant->"Plain"], \(\[Alpha]\ \[Alpha] + 1\)], "InlineInput"], "\[Congruent]", RowBox[{ SubscriptBox[ StyleBox["E", FontSlant->"Plain"], \(\[Alpha]\ \[Alpha] + 1\)], SubscriptBox[ StyleBox["D", FontSlant->"Plain"], \(\[Alpha]\ \[Alpha] + 1\)]}]}], TraditionalForm]]], ". Iskazano preko svojstvenih vrijednosti ", Cell[BoxData[ FormBox[ StyleBox[\(\[Mu]\_\[Alpha]\), FontFamily->"Courier"], TraditionalForm]]], " i svojstvenih vektora ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[ SubscriptBox[ OverscriptBox[ StyleBox["a", FontSlant->"Plain"], "\[Rule]"], StyleBox["k", FontSlant->"Plain"]], FontFamily->"Courier"], StyleBox[";", FontFamily->"Courier"], StyleBox[" ", FontFamily->"Courier"], RowBox[{ StyleBox["k", FontFamily->"Courier", FontSlant->"Plain"], StyleBox["=", FontFamily->"Courier"], StyleBox["1", FontFamily->"Courier"]}]}], StyleBox[",", FontFamily->"Courier"], StyleBox["...", FontFamily->"Courier"], StyleBox[",", FontFamily->"Courier"], StyleBox["N", FontFamily->"Courier", FontSlant->"Plain"]}], TraditionalForm]]], " matrice hamiltonijana u gornjoj jednad\:017ebi, rje\[SHacek]enje jednad\ \:017ebe glasi" }], "Text"], Cell[BoxData[ \(\(A\& \[Rule] \)[ t] = \[Sum]\+\(\[Alpha] = 1\)\%N\( b\_\[Alpha]\) \(a\& \[Rule] \)\_\[Alpha]\ \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Mu]\_\[Alpha]\ t\)\)], \ "NumberedEquation"], Cell[TextData[{ "Koeficjente ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["b", FontSlant->"Plain"], "\[Alpha]"], FontFamily->"Courier"], TraditionalForm]]], " treba odrediti na temelju po\[CHacek]etnih uvjeta." }], "Text"], Cell["\<\ Sustav koji promatramo je takav da je u po\[CHacek]etnom trenutku sva \ populacija skupljena u samo po\[CHacek]etnoj razini putanje, dok u kona\ \[CHacek]nom trenutku \:017eelimo posti\[CAcute]i potpuni prijenos populacije \ u kona\[CHacek]nu razinu. Ove po\[CHacek]etne i kona\[CHacek]ne uvjete mo\ \:017eemo zapisati kao\ \>", "Text"], Cell[BoxData[{ RowBox[{\(\(A\& \[Rule] \)[t = 0]\), "\[Congruent]", RowBox[{"(", GridBox[{ {GridBox[{ {"1"}, {"0"} }]}, {"."}, {"."}, {"0"} }], ")"}]}], "\[IndentingNewLine]", RowBox[{\(\(A\& \[Rule] \)[t = T]\), "\[Congruent]", RowBox[{"(", GridBox[{ {GridBox[{ {"0"}, {"0"} }]}, {"."}, {"."}, {"1"} }], ")"}]}]}], "NumberedEquation"], Cell[TextData[{ "T je vrijeme prijenosa populacije koje za sada ne znamo. \ Periodi\[CHacek]ku dinamiku opisanu uvjetima (103) mo\:017eemo ostvariti ako \ je razvoj naseljenosti pojedine razine du\:017e putanje simetri\[CHacek]an s \ obzirom na obrat u vremenu oko to\[CHacek]aka", StyleBox[" t=0", FontFamily->"Courier"], " i ", StyleBox["t=T", FontFamily->"Courier"], ". Ovo \[CAcute]e biti zadovoljeno ako je matrica potpuno rezonantnog \ hamiltonijana simetri\[CHacek]na s obzirom na traspoziciju oko sporedne \ dijagonale. Jakosti su vezanja tada me\:0111usobno povezane relacijama" }], "Text"], Cell[BoxData[ RowBox[{\(I\_12 = I\_\(N - 1\ N\)\), ";", " ", \(I\_23 = I\_\(N - 2\ N - 1\)\), ";", StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[\(\(itd\)\(.\)\), FontFamily->"Times New Roman"]}]], "NumberedEquation"], Cell[TextData[{ "Zbog poznatih svojstava Fourierovih razvoja (rje\[SHacek]enje (102) jest \ Fourierov razvoj), cjelokupna dinamika mo\:017ee biti periodi\[CHacek]ka samo \ ako su svojstvene vrijednosti ", Cell[BoxData[ FormBox[ StyleBox[\(\[Mu]\_\[Alpha]\), FontFamily->"Courier"], TraditionalForm]]], " cjelobrojni vi\[SHacek]ekratnici neke temeljne frekvencije ", Cell[BoxData[ FormBox[ StyleBox[\(\[Mu]\_0\), FontFamily->"Courier"], TraditionalForm]]], " koja je s vremenom prijenosa T povezana preko" }], "Text"], Cell[BoxData[ \(\[Mu]\_0 = \[Pi]\/T\)], "NumberedEquation"], Cell[TextData[{ "Ako to ne bi bilo zadovoljeno, dinamika bi bila neperiodi\[CHacek]ka, a \ dinamika kakvu mi zahtjevamo nu\:017eno jest periodi\[CHacek]ka, ba\[SHacek] \ kao \[SHacek]to je to slu\[CHacek]aj sa jednostavnom Rabijevom dinamikom u \ sustavu sa dvije razine. Dinami\[CHacek]ku \[CAcute]emo jednad\:017ebu (100) \ daljnje pojednostavniti time da iz matrice hamiltonijanae izvu\[CHacek]emo \ element ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["I", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "12"], "InlineInput", FontFamily->"Courier"], StyleBox["\[Congruent]", FontFamily->"Courier"], StyleBox[ RowBox[{ SubscriptBox[ StyleBox["E", FontFamily->"Courier", FontSlant->"Plain"], "12"], SubscriptBox[ StyleBox["D", FontFamily->"Courier", FontSlant->"Plain"], "12"]}], FontFamily->"Courier"]}], TraditionalForm]]], " (pogodan jer se pojavljuje u matrici i za najmanju mogu\[CAcute]u \ vrijednost ", StyleBox["N", FontFamily->"Courier"], ", ", StyleBox["N=2", FontFamily->"Courier"], "), te ga ugradimo u novu vremensku veli\[CHacek]inu, ", Cell[BoxData[ \(\[Tau] \[Congruent] t\ | I\_12 | \)]], " " }], "Text"], Cell[BoxData[ RowBox[{\(\(d\/d\[Tau]\) \(a\& \[Rule] \)[\[Tau]]\), "=", RowBox[{\(-\[ImaginaryI]\), RowBox[{"(", GridBox[{ {"0", "1", "0", ".", ".", "0"}, {"1", "0", \(C\_23\), "0", ".", "."}, {"0", \(C\_23\), "0", ".", "0", "."}, {".", "0", ".", "0", \(C\_23\), "0"}, {".", ".", "0", \(C\_23\), "0", "1"}, {"0", ".", ".", "0", "1", "0"} }], ")"}], \(\(a\& \[Rule] \)[\[Tau]]\)}]}]], "NumberedEquation"], Cell["pri \[CHacek]emu je ", "Text"], Cell[BoxData[ \(C\_\(\[Alpha]\ \[Alpha] + 1\) \[Congruent] I\_\(\[Alpha]\ \[Alpha] + 1\)\/I\_12\)], "NumberedEquation"], Cell[TextData[{ "dok je ", Cell[BoxData[ \(\(a\& \[Rule] \)[\[Tau]]\)]], " vektor stanja izra\:017een preko nove vremenske veli\[CHacek]ine. Na \ temelju karakteristi\[CHacek]nog tridijagonalog oblika matrice hamiltonijana \ mo\:017eemo zaklju\[CHacek]iti da \[CAcute]e karakteristi\[CHacek]ni polinom \ uklju\[CHacek]ivati samo parne ili samo neparne potencije od ", StyleBox["\[Mu]", FontFamily->"Courier"], ", ovisno o tome je li ", StyleBox["N", FontFamily->"Courier"], " paran ili neparan. Budu\[CAcute]i je matrica simetri\[CHacek]na, \ svojstvene vrijednosti su realne. Za parne \[CAcute]emo ", StyleBox["N", FontFamily->"Courier"], " ", StyleBox["(N=2k)", FontFamily->"Courier"], " dakle dobiti niz parova svojstvenih vrijednosti ", Cell[BoxData[ FormBox[ StyleBox[\(\(\[PlusMinus]\[Mu]\_i\); \ i = 1, 2, ... , k\), FontFamily->"Courier"], TraditionalForm]]], ". U slu\[CHacek]aju neparnog ", StyleBox["N (N=2k+1", FontFamily->"Courier"], "), uz k takvih parova svojstvenih vrijednosti, dobit \[CAcute]emo jo\ \[SHacek] jednu dodatnu svojstvenu vrijednost ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox["\[Mu]", StyleBox["N", FontSlant->"Plain"]], "=", "0"}], FontFamily->"Courier"], TraditionalForm]]], ". Dalje uoo\[CHacek]imo da 'vezanja' ", Cell[BoxData[ FormBox[ StyleBox[\(C\_\(\[Alpha]\ \[Alpha] + 1\)\), FontFamily->"Courier"], TraditionalForm]]], " u karakteristi\[CHacek]ni polinom ulaze samo kroz kvadrate svojih \ vrijednosti, ", Cell[BoxData[ FormBox[ StyleBox[\(\((C\_\(\[Alpha]\ \[Alpha] + 1\))\)\^2\), FontFamily->"Courier"], TraditionalForm]]], ". Zbog svega navedenog, mo\:017eemo zaklju\[CHacek]iti da cjelokupna \ dinamika ne\[CAcute]e ovisiti o predznacima jakosti vezanja ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["I", FontFamily->"Courier", FontSlant->"Plain"], \(\[Alpha]\ \[Alpha] + 1\)], FontFamily->"Courier"], TraditionalForm]]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Op\[CAcute]e rje\[SHacek]enje LaVoR prijenosa", "Subsubsection"], Cell["\<\ Iz (106) uo\[CHacek]imo da je problem sklapanja vi\[SHacek]estruko \ rezonantnog pulsa za lanac sa odre\:0111enim brojem karika u slu\[CHacek]aju \ potpune rezonancije i bez uklju\[CHacek]ivanja razina smetnje \[CHacek]isto \ matemati\[CHacek]ke prirode. Ovo \[CAcute]e biti istina i u slu\[CHacek]aju \ kada u obzir uzmemo i razine smetnje du\:017e putanje. Uvr\[SHacek]tavanjem i \ numeri\[CHacek]kom simulacijom dinamike mo\:017ee se provjeriti da je \ hamiltnoijan LaVoR prenosa koja zadovoljava sve prethodno navedene uvjete \ oblika\ \>", "Text"], Cell[BoxData[ RowBox[{\(\(H\&^\) \((N)\)\), "=", RowBox[{\(1\/\@\(N - 1\)\), RowBox[{"(", GridBox[{ {"0", \(\(\@1\) \@\(N - 1\)\), "0", "0", "0", ".", ".", "."}, {\(\(\@1\) \@\(N - 1\)\), "0", \(\(\@2\) \@\(N - 2\)\), "0", ".", ".", ".", "."}, {"0", \(\(\@2\) \@\(N - 2\)\), "0", "...", "0", ".", ".", "."}, {".", "0", "...", "0", \(\(\@k\) \@\(N - k\)\), "0", ".", "."}, {".", ".", "0", \(\(\@k\) \@\(N - k\)\), "0", "...", "0", "."}, {".", ".", ".", "0", "...", "0", \(\(\@\(N - 2\)\) \@2\), "0"}, {".", ".", ".", ".", "0", \(\(\@\(N - 2\)\) \@2\), "0", \(\(\@\(N - 1\)\) \@1\)}, {"0", "0", "0", "0", "0", "0", \(\(\@\(N - 1\)\) \@1\), "0"} }], ")"}]}]}]], "NumberedEquation"], Cell["pri \[CHacek]emu je", "Text"], Cell[BoxData[ RowBox[{"k", "=", RowBox[{"{", GridBox[{ { RowBox[{\(N\/2\), RowBox[{"(", RowBox[{ StyleBox["za", FontFamily->"Times New Roman"], " ", "N", " ", StyleBox["paran", FontFamily->"Times New Roman"]}], ")"}]}]}, { RowBox[{\(\(N - 1\)\/2\), " ", RowBox[{"(", RowBox[{ StyleBox["za", FontFamily->"Times New Roman"], " ", "N", " ", StyleBox["neparan", FontFamily->"Times New Roman"]}], ")"}]}]} }]}]}]], "NumberedEquation"], Cell["\<\ Ovaj rezultat navodimo bez dokaza jer smo do njega do\[SHacek]li indukcijom. \ Za ovakav hamiltonijan, ukupno vrijeme LaVoR prijenosa iznosi\ \>", "Text"], Cell[BoxData[ \(\(\[Tau]\_LaVoR\) \((N)\) = \(\@\(N - 1\)\) \[Pi]\/2\)], "NumberedEquation"], Cell[TextData[{ "Ovaj je rezultat izuzetno zanimljiv jer nam govori da LaVoR prijenosom za \ veliki broj razina mo\:017eemo posti\[CAcute]i bitno br\:017ei prijenos \ populacije nego \[SHacek]to bismo to mogli posti\[CAcute]i nizom \ obi\[CHacek]nih dvorazinskih Rabijevih prenosa (vrijeme kojega raste \ razmjerno broju koraka ", StyleBox["N-1", FontFamily->"Courier"], "). Naravno, ovo je u potpunosti istina samo u idealiziranom \ slu\[CHacek]aju, dok \[CAcute]e u stvarnim slu\[CHacek]ajevima odnos dvaju \ vremena biti slo\:017eeniji. Skup svojstvenih vrijednosti hamiltonijana (108) \ za pojedini ", StyleBox["N", FontFamily->"Courier"], " je " }], "Text"], Cell[BoxData[ \({\[Mu]\_i; i = 1, ... , N} = {\(\[PlusMinus]k\)\ \[Mu]\_0; \ \(k > 0\ &\)\ k = N, N - 2, ... }\)], "NumberedEquation"], Cell[TextData[{ "Na ", StyleBox["Slici 12", FontWeight->"Bold"], " prikazano je nekoliko primjera numeri\[CHacek]kih rje\[SHacek]enja jednad\ \:017ebe (106) za razli\[CHacek]ite N sa odgovaraju\[CAcute]im VRL \ hamiltonijanom." }], "Text"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.214361 0 0.618034 [ [.23817 -0.0125 -3 -9 ] [.23817 -0.0125 3 0 ] [.45253 -0.0125 -3 -9 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5e-005 L .96108 4e-005 L .963 2e-005 L .96417 2e-005 L .96528 1e-005 L .96628 1e-005 L .96737 0 L .96842 0 L .96959 0 L .9708 0 L .97195 0 L .97301 0 L .97399 0 L .97506 0 L .97619 0 L Mfstroke .2 0 1 r .02381 0 m .02499 0 L .02605 0 L .02729 0 L .02846 0 L .02954 0 L .03053 0 L .03163 0 L .03279 0 L .03395 0 L .0352 0 L .03638 0 L .03746 0 L .03873 0 L .03944 0 L .04011 0 L .04132 0 L .04262 0 L .04396 0 L .04519 0 L .04654 0 L .04797 0 L .04922 0 L .05038 0 L .05165 0 L .05302 0 L .05423 1e-005 L .05555 1e-005 L .05795 1e-005 L .06028 2e-005 L .06244 3e-005 L .06486 4e-005 L .06745 5e-005 L .06991 7e-005 L .07216 9e-005 L .07465 .00013 L .07738 .00017 L .08023 .00023 L .08288 .0003 L .0857 .00039 L .08837 .0005 L .09337 .00077 L .09875 .00117 L .10458 .00178 L .10947 .00247 L .11404 .00329 L .11948 .00451 L .12446 .00592 L .1346 .0098 L .14019 .01261 L Mistroke .1455 .01579 L .15511 .02293 L .16534 .03268 L .18374 .05608 L .22538 .13077 L .24629 .16962 L .25545 .18424 L .2655 .19749 L .27019 .20247 L .27522 .20685 L .27996 .21003 L .28206 .21112 L .28428 .21206 L .28549 .21248 L .28682 .21286 L .28804 .21314 L .28916 .21334 L .29048 .2135 L .29117 .21355 L .2919 .21358 L .29313 .21358 L .29382 .21354 L .29446 .21349 L .29572 .21334 L .29691 .21312 L .29797 .21287 L .29913 .21255 L .30172 .21158 L .3041 .21043 L .30879 .2074 L .31382 .20308 L .3229 .19265 L .3337 .17637 L .34355 .15865 L .38337 .07776 L .40156 .0464 L .41193 .03219 L .42167 .02152 L .43117 .01358 L .44142 .00751 L .44675 .00525 L .45237 .00344 L .45716 .00229 L .46242 .00138 L .46495 .00105 L .46769 .00077 L .47011 .00056 L .47267 .0004 L .47498 .00028 L .47752 .00018 L Mistroke .47984 .00012 L .482 8e-005 L .48329 6e-005 L .4847 4e-005 L .48604 3e-005 L .48727 2e-005 L .4885 1e-005 L .48961 1e-005 L .49089 1e-005 L .49209 0 L .49341 0 L .49408 0 L .49481 0 L .4961 0 L .49731 0 L .49804 0 L .49873 0 L .50003 0 L .50075 0 L .50143 0 L .50216 0 L .50294 0 L .50408 0 L .50533 0 L .50663 0 L .50784 0 L 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.06132 0 L .06244 0 L .0649 0 L .06627 0 L .06752 0 L .06995 0 L .07225 0 L .07355 0 L .07492 0 L .07739 0 L .08005 0 L .08297 1e-005 L .08578 1e-005 L .08835 1e-005 L .09132 2e-005 L .09413 2e-005 L Mistroke .09673 3e-005 L .09913 4e-005 L .10197 5e-005 L .10458 7e-005 L .10967 .00011 L .11235 .00013 L .11517 .00017 L .11993 .00025 L .12514 .00037 L .1298 .00052 L .13478 .00073 L .13948 .00098 L .14376 .00128 L .14858 .0017 L .1537 .00227 L .16288 .00367 L .16798 .00472 L .17277 .0059 L .18366 .00949 L .19352 .01403 L .20388 .02043 L .21273 .02743 L .2224 .03688 L .24205 .06237 L .26301 .09869 L .30456 .1859 L .31469 .20587 L .3254 .22436 L .33455 .23717 L .3398 .24302 L .3446 .24727 L .34696 .24896 L .34953 .25049 L .35195 .25162 L .3542 .2524 L .35543 .2527 L .35658 .25292 L .35784 .25307 L .35855 .25312 L .3592 .25314 L .36043 .25312 L .36113 .25307 L .36178 .25299 L .3631 .25277 L .36384 .25261 L .36451 .25243 L .3669 .2516 L .3691 .25055 L .37151 .24908 L .37408 .24716 L Mistroke 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.68838 .01096 L .69712 .01553 L .70667 .02207 L .71696 .03123 L .72631 .04168 L .74727 .07302 L .78729 .15749 L .80564 .19783 L .81606 .2176 L .82115 .2259 L .8258 .23252 L .83039 .23806 L .8354 .24283 L .83814 .24485 L .84069 .24632 L .84312 .24737 L .84435 .24776 L .84499 .24793 L .8457 .24808 L .84639 .2482 L .84703 .24828 L .8477 .24834 L .84842 .24837 L .84973 .24834 L .85092 .24821 L .85164 .24809 L .85231 .24795 L .85382 .24751 L .85515 .24701 L .85656 .24635 L .85906 .24484 L .86141 .24305 L .86676 .23761 L .87172 .23089 L .87714 .22181 L .88701 .20097 L Mistroke .90554 .15039 L .92686 .08521 L .93614 .05894 L .94619 .03447 L .95424 .01891 L .95777 .01343 L .96164 .00844 L .9653 .00475 L .96872 .00225 L .9706 .00126 L .97232 .0006 L .97327 .00034 L .97431 .00014 L .97529 3e-005 L .97619 0 L Mfstroke 0 1 1 r .02381 0 m .02499 0 L .02605 0 L .02729 0 L .02846 0 L .02954 0 L .03053 0 L .03163 0 L .03279 1e-005 L .03395 1e-005 L .0352 2e-005 L .03638 3e-005 L .03746 4e-005 L .03873 5e-005 L .04011 8e-005 L .04262 .00013 L .04396 .00018 L .04523 .00023 L .04759 .00034 L .05014 .00051 L .0529 .00075 L .05532 .00103 L .05787 .00139 L .06244 .00227 L .06746 .00362 L .07283 .00561 L .07791 .0081 L .08253 .01094 L .09385 .02055 L .10458 .03342 L .12449 .06683 L .14291 .10565 L .16316 .14946 L .17308 .16834 L .18218 .18281 L .19103 .19356 L .19548 .19756 L .20034 .20078 L .20291 .20197 L .20525 .20276 L .20654 .20307 L .20727 .20321 L .20794 .20331 L .20918 .20343 L .21049 .20346 L .21161 .20343 L .21281 .20331 L .21394 .20313 L .21498 .20291 L .2162 .20258 L Mistroke .21752 .20213 L .21993 .20108 L .22249 .19965 L .22526 .19774 L .23027 .19336 L .23964 .18229 L .26062 .14729 L .28021 .10882 L .30168 .06912 L .31156 .05355 L .32214 .03934 L .33206 .02845 L .34122 .02041 L .35142 .01354 L .35669 .01076 L .3625 .00823 L .37227 .00503 L .37762 .00376 L .38263 .00281 L .38734 .00211 L .39249 .00152 L .39786 .00105 L .40285 .00073 L .40755 .00051 L .41181 .00036 L .41418 .00029 L .41673 .00023 L .42137 .00015 L .42374 .00012 L .42628 9e-005 L .43086 6e-005 L .43366 4e-005 L .43631 3e-005 L .43869 2e-005 L .44129 2e-005 L .4438 1e-005 L .44659 1e-005 L .44798 1e-005 L .44949 1e-005 L .45079 0 L .45222 0 L .45479 0 L .45619 0 L .4575 0 L .45984 0 L .46115 0 L .46234 0 L .46345 0 L .46465 0 L .46596 0 L Mistroke .46718 0 L .46855 0 L .46986 0 L .47103 0 L .47231 0 L .47352 0 L .47462 0 L .47588 0 L .47708 0 L .47819 0 L .4792 0 L .48032 0 L .4815 0 L .4828 0 L .48398 0 L .48463 0 L .48534 0 L .48663 0 L .48784 0 L .48912 0 L .49021 0 L .49142 0 L .4921 0 L .49283 0 L .49412 0 L .49484 0 L .49551 0 L .49624 0 L .49703 0 L .49771 0 L .49845 0 L .49914 0 L .49979 0 L .50107 0 L .50229 0 L .50346 0 L .50474 0 L .5059 0 L .50697 0 L .50823 0 L .50941 0 L .51006 0 L .51077 0 L .512 0 L .5131 0 L .5143 0 L .51556 0 L .51673 0 L .51783 0 L .51882 0 L Mistroke .51998 0 L .52107 0 L .52226 0 L .52357 0 L .52423 0 L .52494 0 L .52622 0 L .52749 0 L .5287 0 L .52978 0 L .53097 0 L .53238 0 L .53367 0 L .53515 0 L .53654 0 L .53784 0 L .53903 0 L .54041 0 L .54172 0 L .54289 0 L .54417 0 L .54641 0 L .54768 0 L .54903 0 L .55145 1e-005 L .55367 1e-005 L .55576 1e-005 L .55806 1e-005 L .56053 2e-005 L .56304 3e-005 L .56578 4e-005 L .56864 5e-005 L .57131 7e-005 L .57609 .00012 L .57876 .00015 L .58119 .00019 L .58604 .0003 L .58848 .00037 L .59117 .00046 L .59592 .00066 L .60037 .00092 L .6056 .00133 L .61039 .00183 L .61591 .00258 L .62116 .00351 L .63077 .0059 L .63613 .0077 L .641 .00969 L .65047 .01466 L .65939 .02089 L Mistroke .66918 .02968 L .6795 .04138 L .69798 .06855 L .736 .14081 L .74688 .16102 L .75717 .17776 L .76641 .18985 L .77163 .19515 L .77646 .19896 L .77869 .20035 L .7811 .20156 L .78337 .20244 L .78437 .20274 L .78545 .20301 L .78664 .20323 L .78775 .20337 L .78908 .20346 L .79028 .20345 L .79153 .20336 L .79224 .20328 L .79289 .20318 L .79409 .20293 L .79535 .20258 L .79774 .2017 L .80029 .20042 L .80491 .19725 L .8104 .19208 L .81541 .1861 L .83446 .15422 L .85521 .10981 L .87456 .06869 L .88485 .04973 L .89595 .03271 L .90649 .02021 L .91111 .01586 L .91617 .01186 L .92064 .00893 L .92542 .0064 L .93047 .00431 L .93517 .00285 L .93772 .00223 L .94043 .00168 L .94528 .00095 L .94798 .00067 L .95051 .00046 L .95328 .00029 L .95483 .00022 L .95626 .00017 L .95748 .00013 L .95879 .0001 L Mistroke .95992 8e-005 L .96114 6e-005 L .96259 4e-005 L .96394 2e-005 L .96526 2e-005 L .96648 1e-005 L .96764 1e-005 L .96873 0 L .96992 0 L .9712 0 L .97184 0 L .97254 0 L .97381 0 L .97505 0 L .97619 0 L Mfstroke .2 0 1 r .02381 0 m .02499 0 L .02605 0 L .02729 0 L .02846 0 L .02954 0 L .03053 0 L .03163 0 L .03279 0 L .03395 0 L .0352 0 L .03638 0 L .03746 0 L .03873 0 L .03944 0 L .04011 0 L .04132 0 L .04262 0 L .04396 0 L .04519 0 L .04654 0 L .04797 0 L .04922 0 L .05038 1e-005 L .05165 1e-005 L .05302 1e-005 L .05423 1e-005 L .05555 2e-005 L .05795 2e-005 L .06028 4e-005 L .06244 5e-005 L .06486 7e-005 L .06745 .0001 L .06991 .00014 L .07216 .00018 L .07465 .00025 L .07738 .00033 L .08023 .00045 L .08288 .00058 L .0857 .00076 L .08837 .00096 L .09337 .00146 L .09875 .0022 L .10458 .00332 L .10947 .00456 L .11404 .00601 L .11948 .00816 L .12446 .01058 L .1346 .01709 L .1455 .02676 L Mistroke .15537 .03811 L .16452 .05087 L .18526 .08659 L .20409 .12347 L .22444 .16049 L .23562 .17639 L .24052 .1819 L .24589 .18676 L .24835 .18855 L .25098 .19014 L .25325 .19125 L .25571 .19217 L .25716 .19257 L .2578 .19271 L .2585 .19284 L .25978 .19302 L .26048 .19308 L .26114 .19312 L .26238 .19313 L .26352 .19307 L .26476 .19293 L .26608 .19269 L .26732 .1924 L .26866 .19199 L .27109 .19101 L .27346 .18979 L .27565 .18842 L .28061 .18449 L .28586 .17914 L .29461 .16777 L .30424 .15237 L .34463 .07531 L .36324 .0451 L .37306 .03248 L .3835 .02179 L .38874 .01745 L .39452 .01341 L .40472 .00798 L .40971 .00603 L .41509 .00435 L .42017 .00312 L .42483 .00224 L .42956 .00156 L .43462 .00103 L .43939 .00067 L .44373 .00044 L .44639 .00033 L .44884 .00025 L .45152 .00019 L .45437 .00013 L Mistroke .45684 9e-005 L .45916 7e-005 L .4617 5e-005 L .46313 4e-005 L .46443 3e-005 L .46697 2e-005 L .46836 2e-005 L .46967 1e-005 L .47201 1e-005 L .47333 1e-005 L .47454 0 L .47586 0 L .4771 0 L .47845 0 L .47991 0 L .48064 0 L .48143 0 L .48213 0 L .48287 0 L .48354 0 L .48428 0 L .48558 0 L .48684 0 L .48748 0 L .48818 0 L .48942 0 L .49057 0 L .49127 0 L .49193 0 L .49317 0 L .49386 0 L .4945 0 L .4952 0 L .49594 0 L .4971 0 L .49834 0 L .49951 0 L .50058 0 L .50183 0 L .50253 0 L .50319 0 L .50439 0 L .50567 0 L .50692 0 L .50763 0 L .50829 0 L .50899 0 L .50962 0 L .51036 0 L .51105 0 L Mistroke .51228 0 L .51342 0 L .51467 0 L .5153 0 L .51599 0 L .51673 0 L .51744 0 L .51876 0 L .5195 0 L .52019 0 L .52175 0 L .52311 0 L .52435 0 L .52579 0 L .52714 1e-005 L .52959 1e-005 L .53083 1e-005 L .53219 2e-005 L .53459 3e-005 L .53683 4e-005 L .53948 6e-005 L .5419 8e-005 L .54468 .00012 L .54732 .00016 L .54969 .00022 L .55191 .00028 L .55455 .00037 L .55697 .00047 L .5619 .00076 L .5645 .00096 L .56722 .00121 L .57178 .00174 L .57602 .00239 L .58067 .0033 L .58564 .00455 L .59097 .00627 L .59588 .00825 L .60698 .01438 L .61676 .02203 L .62713 .0327 L .64669 .06029 L .665 .0934 L .68357 .12986 L .70381 .16544 L .70913 .17298 L .71479 .17986 L .71969 .18475 L .72498 .18884 L .72781 .19048 L .73041 .19165 L Mistroke .7329 .19245 L .73415 .19274 L .73486 .19287 L .73552 .19296 L .73677 .19309 L .73791 .19313 L .73914 .1931 L .73983 .19306 L .74047 .19299 L .74153 .19283 L .7427 .1926 L .74393 .19227 L .74508 .19191 L .74752 .1909 L .74979 .18971 L .75493 .1861 L .75994 .18142 L .76529 .17526 L .7838 .1464 L .80375 .10799 L .82534 .0673 L .83547 .05088 L .84618 .03617 L .85621 .02505 L .86536 .01712 L .87482 .01097 L .88003 .00837 L .88497 .00636 L .89374 .00371 L .8987 .00265 L .90329 .00189 L .90791 .00132 L .9129 .00086 L .91519 .0007 L .9176 .00055 L .92192 .00036 L .92438 .00027 L .92701 .0002 L .92949 .00015 L .93175 .00011 L .9344 8e-005 L .93728 5e-005 L .94 3e-005 L .94251 2e-005 L .9446 2e-005 L .9465 1e-005 L .94755 1e-005 L .9487 1e-005 L .9498 1e-005 L .95079 0 L Mistroke .95274 0 L .9538 0 L .95481 0 L .95568 0 L .9566 0 L .95761 0 L .95854 0 L .95964 0 L .96064 0 L .9618 0 L .96289 0 L .9639 0 L .96484 0 L .96587 0 L .96696 0 L .96805 0 L .96922 0 L .97033 0 L .97135 0 L .97247 0 L .97371 0 L .97499 0 L .97619 0 L Mfstroke 1 0 .6 r .02381 0 m .02499 0 L .02605 0 L .02729 0 L .02846 0 L .02954 0 L .03053 0 L .03163 0 L .03279 0 L .03395 0 L .0352 0 L .03638 0 L .03746 0 L .03873 0 L .03944 0 L .04011 0 L .04132 0 L .04262 0 L .04327 0 L .04396 0 L .04519 0 L .04588 0 L .04652 0 L .04726 0 L .04797 0 L .04925 0 L .05041 0 L .05166 0 L .05302 0 L .05414 0 L .05539 0 L .05669 0 L .05791 0 L .06009 0 L .06132 0 L .06244 0 L .0649 0 L .06627 0 L .06752 0 L .06995 0 L .07225 0 L .07492 1e-005 L .07739 1e-005 L .0801 1e-005 L .08297 2e-005 L .08553 2e-005 L .08829 3e-005 L .09089 5e-005 L .09326 6e-005 L .09604 8e-005 L Mistroke .09907 .00011 L .10193 .00014 L .10458 .00019 L .10967 .00029 L .11254 .00037 L .11517 .00046 L .12002 .00068 L .12514 .00099 L .12974 .00136 L .13402 .00181 L .13873 .00243 L .14376 .00327 L .14906 .00441 L .15396 .00571 L .15923 .00744 L .16504 .00979 L .17481 .01499 L .18517 .02249 L .19529 .03208 L .20456 .04296 L .22543 .07476 L .246 .11356 L .26511 .1509 L .28357 .18146 L .28888 .1883 L .2939 .19371 L .2984 .19762 L .30327 .20077 L .30563 .20187 L .30694 .20236 L .30817 .20274 L .30932 .20302 L .31057 .20325 L .31171 .20339 L .31276 .20345 L .31399 .20346 L .31516 .20338 L .31643 .20322 L .31715 .20309 L .31781 .20295 L .32043 .20216 L .32174 .20162 L .32318 .20093 L .32845 .19747 L .3314 .19491 L .33409 .19221 L .34424 .179 L .3634 .14379 L .38419 .09841 L .40365 .05862 L Mistroke .41397 .0411 L .42507 .02594 L .43052 .01998 L .4357 .01519 L .44073 .01132 L .44541 .00837 L .45018 .00596 L .45451 .00423 L .45932 .00276 L .46176 .00218 L .46445 .00165 L .46933 .00093 L .47199 .00065 L .47448 .00045 L .47681 .00031 L .47893 .00021 L .48011 .00017 L .48139 .00013 L .48371 8e-005 L .48499 6e-005 L .4862 4e-005 L .48753 3e-005 L .48821 2e-005 L .48895 2e-005 L .49037 1e-005 L .49114 1e-005 L .49186 0 L .49315 0 L .49387 0 L .49453 0 L .49581 0 L .497 0 L .4983 0 L .49895 0 L .49967 0 L .50088 0 L .50218 0 L .50341 0 L .50453 0 L .50583 0 L .50704 0 L .50814 0 L .50932 1e-005 L .51061 1e-005 L .51126 2e-005 L .51197 2e-005 L .51324 3e-005 L .51441 5e-005 L .5156 6e-005 L .51673 8e-005 L Mistroke .51884 .00014 L .52111 .00021 L .52357 .00033 L .52606 .00049 L .5288 .00072 L .53167 .00105 L .53435 .00144 L .53915 .00239 L .54431 .00383 L .54955 .00584 L .55441 .00827 L .55975 .01166 L .56544 .01617 L .57597 .0272 L .58552 .04033 L .60476 .07496 L .62558 .11998 L .64512 .16077 L .65547 .17855 L .66117 .1866 L .66657 .1929 L .67168 .19757 L .67457 .19963 L .67725 .20114 L .67949 .20212 L .68078 .20257 L .68197 .2029 L .68321 .20316 L .68386 .20327 L .68455 .20336 L .68571 .20344 L .68698 .20346 L .68827 .20339 L .68948 .20324 L .69057 .20304 L .69176 .20276 L .69441 .20185 L .69686 .2007 L .70132 .19783 L .70604 .19376 L .71548 .18277 L .72561 .16738 L .74392 .13354 L .76366 .09473 L .78518 .05739 L .79517 .04331 L .80581 .03092 L .81577 .02171 L .82493 .01515 L .83524 .00967 L Mistroke .84066 .00748 L .84637 .00561 L .856 .00332 L .86146 .0024 L .86655 .00174 L .87157 .00125 L .87703 .00085 L .88218 .00057 L .88694 .00039 L .89122 .00027 L .89594 .00018 L .89832 .00014 L .90088 .00011 L .9055 7e-005 L .90791 5e-005 L .91047 4e-005 L .9129 3e-005 L .91509 2e-005 L .91768 2e-005 L .92049 1e-005 L .92313 1e-005 L .92557 1e-005 L .9281 0 L .93046 0 L .93182 0 L .93305 0 L .9345 0 L .93583 0 L .9371 0 L .9385 0 L .93991 0 L .9414 0 L .94271 0 L .94412 0 L .94546 0 L .94668 0 L .94763 0 L .94853 0 L .94934 0 L .95021 0 L .95115 0 L .95201 0 L .95295 0 L .95397 0 L .95486 0 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00<000000=WIf@3IfMT0g`3IfMT000P0fMWI1`0000090=WIf@8000002`3IfMT200000>40fMWI003o 0=WIf@T0fMWI003o0=WIf@T0fMWI003o0=WIf@T0fMWI0000\ \>"], ImageRangeCache->{{{506.75, 717.375}, {365.313, 213.563}} -> {-28.9482, 1.76271, 0.0549299, 0.00898125}}]]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]]]]], "Text", TextAlignment->Left], Cell[TextData[{ StyleBox["Slika 12 -", FontWeight->"Bold", FontSlant->"Plain"], " Dinamika populacija LaVoR sustava u pribli\:017eenju potpune rezonancije \ i bez razina smetnje. Grafovi redom odgovaraju slu\[CHacek]ajevnima \ N=3,4,5,6,7 i 10 \[SHacek]to se lako mo\:017ee razabrati iz broja populacija. \ Jasno se uo\[CHacek]ava potpuni prijenos izme\:0111u po\[CHacek]etne i kona\ \[CHacek]ne razine. Takoo\:0111er, budu\[CAcute]i da je vremenski raspon \ svakog grafa ", Cell[BoxData[ \(TraditionalForm\`2\ \(\(\[Tau]\_VLP\)(N)\)\)]], " mo\:017ee se provjeriti ispravnost (barem za ove slu\[CHacek]ajeve) \ relacije (110). " }], "Commentary"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3.3 Utjecaj linija smetnje na pogodne zna\[CHacek]ajke pogona\ \>", "Subsection", CellTags->"3.3"], Cell["\<\ Kao posljenji korak istra\:017eivanja iznesenog u ovome radu ostalo nam je \ sve dosada\[SHacek]nje rezultate zatvoriti u zaokru\:017eenu cjelinu. Stoga \ \[CAcute]emo sada razmotriti ograni\[CHacek]enja koja na mehanizam LaVoR \ prijenosa name\[CAcute]u fizikalna svojstva stvarnih kvantnih sustava. Tu \ prije svega mislimo na razine smetnje koje u stvarnim sustavima gotovo \ neizostavno postoje. Temeljni oslonac u slijede\[CAcute]im razmatranjima bit \ \[CAcute]e nam otkri\[CAcute]a do kojih smo do\[SHacek]li u prethodnom \ poglavlju.\ \>", "Text"], Cell[CellGroupData[{ Cell["Optimizacija zna\[CHacek]ajki pogona", "Subsubsection"], Cell[TextData[{ "U drugom smo poglavlju pokazali kako postojanje vi\[SHacek]estrukih \ vezanja svojstvenih razina unutar sustava efektivno uzrokuje promjenu spektra \ vlastitih frekvencija sustava, kao i promjenu efektivnih vezanja u sustavu \ (kroz promjenu ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[Gamma]\[Delta]"], "InlineInput", FontFamily->"Courier"], TraditionalForm]]], "). \[CapitalSHacek]tovi\[SHacek]e, ove promjene funkcije su jakosti \ vanjske sile koja djeluje na sustav. Dok je god jakost vanjske sile dovoljno \ mala, tako da preklapanje Rabijevih profila linija koje vezuju pojedinu \ razinu sa ostatkom sustava zanemarivo, Rabijev profil pojedine linije izvrsno \ opisuje dinamiku prijenosa populacije izme\:0111u odgovaraju\[CAcute]ih dviju \ razina. Me\:0111utim, kada jakost vanjske sile postane prevelika, nastalu je \ dinamiku vrlo te\[SHacek]ko analiti\[CHacek]ki opisati pa stoga nadzor i \ upravljanje sustavom postaju prakti\[CHacek]no neizvedivi." }], "Text"], Cell["\<\ Kao izvrstan pokazatelj apsolutne jakosti vanjskog pogona mo\:017ee nam poslu\ \:017eiti curenje koje - dok je malo - znamo analiti\[CHacek]ki opisati. \ Stoga, da bismo zadr\:017eali mogu\[CAcute]nost nadzora i upravljanja \ prijenosom populacije u mnogorazinskom sustavu, prva stvar koju moramo \ odrediti jest gornja granica jakosti vanjskog pogona iznad koje curenje \ postaje 'preveliko'. Na temelju rezultata (97) i (98) znamo da je granica \ iznad koje curenje mo\:017eemo smatrati 'velikim' reda veli\[CHacek]ine 1%. \ Naime, do te vrijednosti nemamo problema sa prili\[CHacek]no nezgodnom i \ analiti\[CHacek]ki te\[SHacek]ko 'uhvatljivom' promjenom efektivne \ vrijednosti projekcije dipolnog elektri\[CHacek]nog momenta. Svakom pojedinom \ prelazu u sustavu pridru\:017eena je ovakva grani\[CHacek]na vrijednost, a \ iskazuje se maksimalnom vrijedno\[SHacek]\[CAcute]u jakosti vezanja.\ \>", "Text"], Cell[TextData[{ "Ve\[CAcute] smo u prvom poglavlju uo\[CHacek]ili vezu izme\:0111u jakosti \ vezanja i perioda Rabijevog prijenosa za pojedini prelaz unutar sustava (vidi \ (27)). Prema tome, nije te\[SHacek]ko zaklju\[CHacek]iti da je svakoj \ putanji kroz sustav jednozna\[CHacek]no pridru\:017eena neka minimalna \ vrijednost vremena prenosa koja nam jo\[SHacek] uvijek omogu\[CAcute]ava \ dobro upravljanje dinamikom sustava. Svakoj karici du\:017e putanje pridru\ \:017eena je neka maksimalna vrijednost jakosti vezanja, a najmanja od tih \ jakosti vezanja du\:017e neke putanje ugrubo odre\:0111uje minimalno vrijeme \ LaVoR prijenosa kroz doti\[CHacek]ni lanac (vidi (110)). Kako je projekcija \ dipolnog integrala odre\:0111ena sustavom, ovo nam zapravo daje maksimalnu \ jakost vanjske sile kojom smijemo pogoniti doti\[CHacek]nu kariku u lancu. \ Kada smo jednom za odabrani lanac odredili 'usko grlo' po pitanju vremena \ prijenosa, ostaje nam uskladiti preostale (nekriti\[CHacek]ne) komponente \ pogonskog pulsa. To \[CAcute]emo na\[CHacek]initi tako da odaberemo takve \ jakosti preostalih sastavnica pogonskog pulsa uz koje \[CAcute]e matrica \ hamiltonijana poprimiti to\[CHacek]no oblik (108) uz odgovaraju\[CAcute]i ", StyleBox["N", "InlineInput", FontWeight->"Plain"], "." }], "Text"], Cell["\<\ Posljednji korak koji moramo napraviti je prilagoditi frekvencije pojedinih \ sastavnica pogonskog pulsa na novu optimalnu frekvenciju za dani prijelaz \ odre\:0111enu izrazom (96). Time \[CAcute]emo sve prelaze du\:017e lanca \ poslo\:017eiti na maksimume njihovih 'pomaknutih' Rabijevih profila, te \ \[CAcute]emo posti\[CAcute]i prijenos \[CHacek]ija je potpunost ograni\ \[CHacek]ana jedino ukupnim curenjem du\:017e odabrane putanje.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Algoritam optimizacije", "Subsubsection"], Cell["\<\ Prika\:017eimo sada kako bismo operativno proveli postupak optimizacije \ pogonskog pulsa.\ \>", "Text"], Cell[TextData[{ StyleBox["Odabir putanje i odre\:0111ivanje razina smetnje", FontSlant->"Italic"], "\nPrvi je korak odabrati putanju kojom populaciju \:017eelimo provesti \ kroz sustav. Kada smo jednom odabrali razine putanje, na temelju poznate \ unutra\[SHacek]nje gra\:0111e sustava (vlastitih frekvencija ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox[\(\[Omega]\^0\), FontWeight->"Plain"], RowBox[{\(\[Alpha]\ \[Alpha]\), "+", StyleBox["1", FontWeight->"Plain"]}]], "InlineInput"], TraditionalForm]]], ", projekcija dipolnog elektri\[CHacek]nog momenta ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["D", FontWeight->"Plain", FontSlant->"Plain"], RowBox[{\(\[Alpha]\ \[Alpha]\), "+", StyleBox["1", FontWeight->"Plain"]}]], "InlineInput"], TraditionalForm]]], " za pojedine prijelaze, izbornih pravila, ...) moramo sakupiti podatke o \ svim razinama sustava koje \[CAcute]e \[CHacek]initi 'smetnju' u bilo kojoj \ od karika lanca prenosa." }], "Text"], Cell[TextData[{ StyleBox["Odre\:0111ivanje 'uskog grla' putanje i minimalnog vremena LaVoR \ prenosa", FontSlant->"Italic"], "\nZa svaku pojedinu kariku u lancu prenosa moramo odrediti maksimalnu \ vrijednost jakosti ", Cell[BoxData[ \(E\_\(\[Alpha]\ \[Alpha] + 1\)\)]], " sastavnice pogonske sile kojom taj prelaz mo\:017eemo pogoniti. \ Maksimalno curenje, kako smo to ranije pokazali, odre\:0111eno je sa" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox[ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "max"], \((\[Alpha] \[LeftRightArrow] \[Alpha] + 1)\)}], "=", \(E\_\(\[Alpha]\ \[Alpha] + 1\)\^2\ \[Lambda]\_\(\[Alpha]\ \ \[Alpha] + 1\)\^2\)}]], "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ \(\(\(\ \)\(\[Lambda]\_\(\[Alpha]\ \[Alpha] + 1\)\)\)\)]], " definiran analogno kao u (93)" }], "Text"], Cell[BoxData[{\(\[Lambda]\_\(\[Alpha]\ \[Alpha] + 1\)\^2 \[Congruent] \ \[Sum]\+\(i = 1\)\%n\((\[Lambda]\_i\^\[Alpha])\)\^2 + \[Sum]\+\(j = 1\)\%m\((\ \[Lambda]\_j\^\(\[Alpha] + 1\))\)\^2\), "\[IndentingNewLine]", \(\[Lambda]\_i\ \^\[Alpha] = \(1\/2\) \(\(\(|\)\(D\_\(\[Alpha], i\)\)\(|\)\)\/\(\(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\) \ - \[Omega]\_\(\[Alpha], i\)\)\) \((1 + 1\/\(\(|\)\(1 - \(2\/\(1 - \[Omega]\_\(\[Alpha], \ i\)\/\(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\)\)\) \ \[Omega]\/\(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\)\)\(|\)\))\)\), "\ \[IndentingNewLine]", RowBox[{\(\[Lambda]\_j\^\[Beta]\), "\[LongDash]", " ", StyleBox["analogno", FontFamily->"Times New Roman"]}]}], "NumberedEquation"], Cell[TextData[{ "Pri izvrijednjavanju ovog izraza kao frekvenciju pogona ", StyleBox["\[Omega]", FontFamily->"Courier"], " upotrijebit \[CAcute]emo rezonantnu frekvenciju ", Cell[BoxData[ FormBox[ StyleBox[\(\(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\)\), FontFamily->"Courier"], TraditionalForm]]], ". Sve dok je curenje malo \[LongDash] \[SHacek]to \[CAcute]e redovito biti \ slu\[CHacek]aj \[LongDash] ovo \[CAcute]e nam pribli\:017eenje poslu\:017eiti \ kao izvrstan po\[CHacek]etni korak za cijeli izlo\:017eeni postupak. Kada \ provedemo cjelokupni postupak optimizacije mo\:017eemo radi sigurnosti \ provjeriti jesu li uistinu sve optimizirane frekvencije toliko blizu \ rezonancijskima da se izraz (113) ne razlikuje bitno izme\:0111u dvaju slu\ \[CHacek]ajeva. Ako ne, postupak mo\:017eemo provesti ponovo, ali sada sa tim \ optimiziranim frekvencijama kao ulazom. Nakon nekoliko koraka dolazimo do po \ volji preciznog rje\[SHacek]enja. Uz postavljanje gornje granice dozvoljenog \ maksimalnog curenja, obratom izraza (112) nalazimo tra\:017eenu jakost" }], "Text"], Cell[BoxData[ RowBox[{\(\(E\^max\)\_\(\[Alpha]\ \[Alpha] + 1\)\), "=", SqrtBox[ RowBox[{" ", FractionBox[\(\[Lambda]\_\(\[Alpha]\ \[Alpha] + 1\)\^2\), RowBox[{ SuperscriptBox[ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "max"], \((\[Alpha] \[LeftRightArrow] \[Alpha] + 1)\)}]]}]]}]], "NumberedEquation"], Cell["Sada za svaki od prijelaza odredimo vrijednost izraza", "Text"], Cell[BoxData[ \(\(I\^max\)\_\(\[Alpha]\ \[Alpha] + 1\)\/\(\(\@\[Alpha]\) \@\(N - \ \[Alpha]\)\) \[Congruent] \(\(\(E\^max\)\_\(\[Alpha]\ \[Alpha] + 1\)\) D\_\(\ \[Alpha]\ \[Alpha] + 1\)\)\/\(\(\@\[Alpha]\) \@\(N - \[Alpha]\)\)\)], \ "NumberedEquation"], Cell[TextData[{ "te utvrdimo za koju od karika on ima najmanju vrijednost (neka to bude ona \ ozna\[CHacek]ena indeksom \[Beta] \[Beta]+1)", StyleBox[".", "InlineInput", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], "Ta nam najmanja vrijednost odre\:0111uje minimalno vrijeme LaVoR prijenosa \ koje mo\:017eemo posti\[CAcute]i za odabranu putanju, a iznosi" }], "Text"], Cell[BoxData[ \(\(T\^min\)\_LaVoR = \[Pi]\ \(\(\@\[Beta]\) \@\(N - \[Beta]\)\)\/\(\(\(E\ \^max\)\_\(\[Beta]\ \[Beta] + 1\)\) D\_\(\[Beta]\ \[Beta] + 1\)'\)\)], \ "NumberedEquation"], Cell[TextData[{ "gdje je ", Cell[BoxData[ \(D\_\(\[Beta]\ \[Beta] + 1\)'\)]], " sada ", StyleBox["popravljena", FontSlant->"Italic"], " vrijednost projekcije dipolnog momenta odre\:0111ena prema" }], "Text"], Cell[TextData[{ Cell[BoxData[ RowBox[{ StyleBox[\(D\_\(\[Beta]\ \[Beta] + 1\)'\), "InlineInput"], StyleBox["\[Congruent]", "InlineInput"], FractionBox[ StyleBox[\(D\_\(\[Beta]\ \[Beta] + 1\)\), "InlineInput"], StyleBox[\(\@\(1 + P\_s\)\), "InlineInput"]]}]]], " " }], "NumberedEquation", FontFamily->"Courier"], Cell[TextData[{ "a ", Cell[BoxData[ \(InputForm\`P\_s\)]], " je odre\:0111en sa" }], "Text"], Cell[BoxData[ RowBox[{ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "\[Congruent]", \(\(\((\(E\^max\)\_\(\[Beta]\ \[Beta] + 1\)\/2)\)\^2\) \ \((\[Sum]\+\(i = 1\)\%n\(\((D\_\(\[Beta], i\))\)\^2\/\((\(\[Omega]\^0\)\_\(\ \[Beta]\ \[Beta] + 1\) - \[Omega]\_\(\[Beta], i\))\)\^2\) \((1 + 1\/\(1 - \(2\ \/\(1 - \[Omega]\_\(\[Beta], i\)\/\(\[Omega]\^0\)\_\(\[Beta]\ \[Beta] + 1\)\)\ \) \[Omega]\/\(\[Omega]\^0\)\_\(\[Beta]\ \[Beta] + 1\)\))\)\^2 + \[Sum]\+\(j \ = 1\)\%m\(\((D\_\(\[Beta] + 1, j\))\)\^2\/\((\(\[Omega]\^0\)\_\(\[Beta]\ \ \[Beta] + 1\) - \[Omega]\_\(\[Beta] + 1, j\))\)\^2\) \((1 + 1\/\(1 - \(2\/\(1 \ - \[Omega]\_\(\[Beta] + 1, j\)\/\(\[Omega]\^0\)\_\(\[Beta]\ \[Beta] + 1\)\)\) \ \[Omega]\/\(\[Omega]\^0\)\_\(\[Beta]\ \[Beta] + 1\)\))\)\^2)\)\)}]], \ "NumberedEquation"], Cell[TextData[{ "pri \[CHacek]emu je ", StyleBox["N", "InlineInput", FontWeight->"Plain"], " broj razina putanje. Dok je curenje malo, razlika izme\:0111u ", Cell[BoxData[ \(\((D\_\(\[Beta]\ \[Beta] + 1\))\)'\)]], " i ", Cell[BoxData[ \(D\_\(\[Beta]\ \[Beta] + 1\)\)]], " bit \[CAcute]e gotovo zanemarivo mala. Osim toga, uvo\:0111enje \ popravljene vrijednosti u (116) umjesto nepopravljene uz istu vrijednost \ jakosti elektri\[CHacek]nog polja imat \[CAcute]e za posljedicu ", StyleBox["produljavanje", FontSlant->"Italic"], " vremena prijenosa, tako da \[CAcute]e efektivno vezanje biti ", StyleBox["slabije", FontSlant->"Italic"], " nego ono kojega smo odredili kao maksimalno dozvoljeno." }], "Text"], Cell[TextData[{ StyleBox["Optimizacija sastavnica pogonskog pulsa", FontSlant->"Italic"], "\nOptimalne vrijednosti jakosti preostalih sastavnica vanjskog pogona odre\ \:0111ujemo iz izraza" }], "Text"], Cell[BoxData[ \(E\_\(\[Alpha]\ \[Alpha] + 1\) = \[Pi] \(\(\(\@ \[Alpha]\) \@\(N - \ \[Alpha]\)\)\/\(T\^min\)\_VLP\) 1\/D\_\(\[Alpha]\ \[Alpha] + 1\)'\)], "NumberedEquation"], Cell[TextData[{ "Naravno, popravljena vrijednost projekcije dipolnog momenta ovisi o \ primjenjenom elektri\[CHacek]nom polju pa \[CAcute]emo to\[CHacek]nu \ vrijednost na\[CAcute]i iteriraju\[CAcute]i ovaj izraz, s time da u nultom \ koraku iteracije uzmemo nepopravljenu vrijednost ", Cell[BoxData[ \(D\_\(\[Alpha]\ \[Alpha] + 1\)' = D\_\(\[Alpha]\ \[Alpha] + 1\)\)]], ". Kao posljednji korak algoritma odre\:0111ujemo optimalne frekvancije \ pojedinih sastavnica pogona. Ovaj je postupak tako\:0111er iterativan, s time \ da u nultom koraku vrijednost frekvencije postavljamo na rezonantnu \ vrijednost ", Cell[BoxData[ \(TraditionalForm\`\[Omega] = \(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\ \)\)]], ", " }], "Text"], Cell[BoxData[ \(\[Omega]\_\(\[Alpha]\ \[Alpha] + 1\)' = \(\[Omega]\^0\)\_\(\[Alpha]\ \ \[Alpha] + 1\) - \(s\_\(\[Alpha]\ \[Alpha] + 1\)\) \(\((E\_\(\[Alpha]\ \[Alpha] + 1\)\/2)\)\^2\) \((\[Sum]\ \+\(i = 1\)\%n\( s\_\(\[Alpha], i\)\) \(\((D\_\(\[Alpha], \ i\))\)\^2\/\(\(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\) - \[Omega]\_\(\ \[Alpha], i\)\)\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\[Alpha], i\)\/\(\ \[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\)\)\) \[Omega]\/\(\[Omega]\^0\)\_\(\ \[Alpha]\ \[Alpha] + 1\)\))\) - \[Sum]\+\(j = 1\)\%m\( s\_\(\[Alpha] + 1, j\)\) \(\((D\_\(\[Alpha] + 1, \ j\))\)\^2\/\(\(\[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\) - \[Omega]\_\(\ \[Alpha] + 1, j\)\)\) \((1 + 1\/\(1 - \(2\/\(1 - \[Omega]\_\(\[Alpha] + 1, j\)\/\(\ \[Omega]\^0\)\_\(\[Alpha]\ \[Alpha] + 1\)\)\) \[Omega]\/\(\[Omega]\^0\)\_\(\ \[Alpha]\ \[Alpha] + 1\)\))\))\)\)], "NumberedEquation"], Cell[TextData[{ "dok u slijede\[CAcute]im koracima uvr\[SHacek]tavamo optimiziranu \ vrijednost ", Cell[BoxData[ FormBox[ StyleBox[\(\[Omega] = \[Omega]\_\(\[Alpha]\ \[Alpha] + 1\)'\), FontFamily->"Courier"], TraditionalForm]]], " odre\:0111enu prethodnim korakom. Naj\[CHacek]e\[SHacek]\[CAcute]e \ \[CAcute]e rezultat dobiven u drugoj iteraciji biti sasvim zadovoljavaju\ \[CAcute]e to\[CHacek]nosti." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Numeri\[CHacek]ki model ro-vibratora HF\ \>", "Subsubsection"], Cell[TextData[{ StyleBox["Rezultate dobivene u prethodnim odljeljcima provjerit \ \[CAcute]emo na numeri\[CHacek]kom modelu molekule HF u osnovnom elektronskom \ stanju. Dinamika tog sustava uklju\[CHacek]uje 10 vibracijskih i 31 \ rotacijsku razina. Model je izra\:0111en na temelju ", FontSlant->"Plain"], "ab initio", StyleBox[" potencijala izlo\:017eenom u radu [10] te operatoru dipolnog \ elektri\[CHacek]nog momenta izlo\:017eenom u radu [11]. Pri ra\[CHacek]unanju \ dinamike u obzir nisu uzeti relaksacijski procesi, tako da treba biti oprezan \ sa zaklju\[CHacek]cima vezanim za dinamike \[CHacek]iji je vremenski raspon \ jako duga. Tako\:0111er je zanemaren i utjecaj ekstremno jakih pogonskih \ polja koja bi za posljedicu mogla imati zna\[CHacek]ajno izobli\[CHacek]enje \ me\:0111unuklearnog potencijala. Linearno polariziran vanjski pogon ograni\ \[CHacek]ava prenos populacija na razine sa istom vrijedno\[SHacek]\[CAcute]u \ magnetskog kvantnog broja. ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["Na ", FontSlant->"Plain"], StyleBox["Slici 13", FontWeight->"Bold", FontSlant->"Plain"], StyleBox[" prikazano je numeri\[CHacek]ko rje\[SHacek]enje dinamike \ ro-vibratora HF za lan\[CHacek]asti prenos kroz niz vlastitih razina", FontSlant->"Plain"], StyleBox[" ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["(1,0,0)\[Rule](2,1,0)\[Rule](1,2,0)\[Rule](2,3,0)\[Rule](1,4,0)", FontFamily->"Courier", FontSlant->"Plain"], StyleBox[". Po\[CHacek]etno je sva populacija usredoto\[CHacek]ena u \ osnovnoj razini sustava. Parametri vanjske smetrnje numeri\[CHacek]ki su \ optimizirani upravo opisanim postupkom uz maksimalno dopu\[SHacek]teno \ curenje ", FontSlant->"Plain"], Cell[BoxData[ RowBox[{ SuperscriptBox[ SubscriptBox["P", StyleBox["s", FontSlant->"Italic"]], "max"], \((\[Alpha] \[LeftRightArrow] \[Alpha] + 1)\)}]], FontSlant->"Plain"], " ", StyleBox["od 1%. Slaganje sa teorijskim rezultatima je vrlo dobro.", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[Cell[GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg`00000B0=WIfCD00000603IfMTl00000?l0fMWIh03IfMU90000 02X0fMWI00D000000=WIf@3IfMT0fMWI000000020=WIf@8000000P3oool0 0080fMWI0P00000_0=WIfOl00000o`00003o000002@00000603IfMT00`00 0000fMWI0=WIf@040=WIf@8000000P3oool00080fMWI0P00000o0=WIfOl0 00005`00000E0=WIf@030000003IfMT0fMWI02P0fMWI00<000000=WIf@3I fMT09`3IfMT00`000000fMWI0=WIf@0W0=WIf@030000003IfMT0fMWI02P0 fMWI00<000000=WIf@3IfMT09`3IfMT00`000000fMWI0=WIf@0:0=WIf@H0 0000103IfMWZ000002T0fMWI00<000000=WIf@3IfMT0103IfMT200000080 oooo00020=WIf@800000@03IfMWo000002<00000203IfMT00`000000fMWI 0=WIf@0X0=WIf@030000003IfMT0fMWI02L0fMWI00<000000=WIf@3IfMT0 9`3IfMT00`000000fMWI0=WIf@0X0=WIf@030000003IfMT0fMWI0240fMWI o`000008000002T0fMWI00<000000=WIf@3IfMT0103IfMT200000080oooo 00020=WIf@800000@P3IfMT600000080fMWI5`00000F0=WIfCL00000403I 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FontSlant->"Plain"], StyleBox[".", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" Zbog postojajnja velikog broja razina smetnji na svakoj od \ razina putanje dinamika nije '\[CHacek]ista' kao u idealiziranim primjerima \ na ", FontWeight->"Plain"], StyleBox["Slici 12", FontSlant->"Plain"], StyleBox[", ve\[CAcute] se uo\[CHacek]avaju dodatne sitne brze \ vanrezonancijske oscilacije svake pojedine razine. Ipak ukupna dinamika se \ izvrsno poklapa sa idealiziranim slu\[CHacek]ajem. Teorijski \ o\[CHacek]ekivano vrijeme prenosa iznosi 880878 a.u., dok je eksperimentalna \ vrijednost o\[CHacek]itana iz ovog grafa oko 882000 a.u., \[SHacek]to je \ sasvim prihvatljiva to\[CHacek]nost s obzirom na slo\:017eenost cjelokupne \ dinamike. Ukupno curenje iznosi oko 2%.", FontWeight->"Plain"] }], "Commentary", TextAlignment->Left, FontWeight->"Bold", FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell["Zavr\[SHacek]ne primjedbe", "Subsubsection"], Cell["\<\ Formalno je ovim problem optimizacije LaVoR pogona rije\[SHacek]en. No kako \ je (108) vrlo idealiziran oblik potpune matrice dinamike LaVoR sustava, treba \ provjeriti sa kakvim se problemima mo\:017eemo susresti ako tu \ \[CHacek]injenicu uva\:017eimo. \ \>", "Text"], Cell[TextData[{ "Kao prvo, elementi su matrice u (108) 'slijepi' na sve sastavnice pogona \ osim one koja povezuje njih same sa njihovim prvim susjedima du\:017e \ putanje. U stvarnom sustavu to naravno ne\[CAcute]e biti tako; svaki od \ elemenata matrice osje\[CAcute]at \[CAcute]e ", StyleBox["sve", FontSlant->"Italic"], " sastavnice pulsa, ali ve\[CAcute]ina njih biti \[CAcute]e daleko od \ rezonancija kojima je pojedina razina povezana sa 'smetnjama' iz ostatka \ sustava. No mo\:017ee se dogoditi da je npr. frekvencija sastavnice koja \ povezuje razine ", StyleBox["\[Gamma]", FontFamily->"Courier"], " i ", StyleBox["\[Gamma]+1", FontFamily->"Courier"], " vrlo blizu rezonancije s frekvencijom linije smetnje na razini ", StyleBox["\[Delta]", FontFamily->"Courier"], " udaljenoj za nekoliko mjesta du\:017e putanje od razina ", StyleBox["\[Gamma]", FontFamily->"Courier"], " i ", StyleBox["\[Gamma]+1", FontFamily->"Courier"], ". Tada bi ta sastavnica mogla uzrokovati ne\:017eeljeno (i u gore opisanom \ algoritmu sasvim neo\[CHacek]ekivano) jako curenje iz razine ", StyleBox["\[Delta]", FontFamily->"Courier"], ", \[CHacek]ime bi cijeli na\[SHacek] optimizacijski postupak postao \ nevaljan. Da bi se ovo izbjeglo trebalo bi istra\:017eiti graf Rabijevih \ profila svih mogu\[CAcute]ih linija kojima su razine putanje vezane sa \ ostatkom sustava (me\:0111usobno i sa smetnjama), te iz njega unaprijed \ utvrditi postoji li mogu\[CAcute]nost ovakvog problema." }], "Text"], Cell[TextData[{ "Nadalje, iako je poklapanje teorije i numerike vrlo dobro, ono nije i savr\ \[SHacek]eno. Naime, potankim promatranjem ", StyleBox["Slike 13", FontWeight->"Bold"], " oko trenutka ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ StyleBox["T", FontSlant->"Plain"], "LaVoR"], FontFamily->"Courier"], " "}], TraditionalForm]]], " mo\:017ee se uo\[CHacek]iti kako dinamika populacija me\:0111ustanja \ prenosa nije savr\[SHacek]eno simetri\[CHacek]na s obzirom na obrat u vremenu \ oko te to\[CHacek]ke, \[SHacek]to je bio jedan od zahtjeva na dinamiku \ navedenih u prvom odjeljku cjeline 3.2. Naime, populacije me\:0111ustanja u \ tom trenutku niti su jednake nuli, niti im je prva derivacija jednaka nuli. \ Ovo mo\:017eemo poku\[SHacek]ati objasniti \[CHacek]injenicom da na\[SHacek]i \ 'eksperimentalni' kona\[CHacek]ni uvjeti ionako nikada ne mogu biti potpuno \ jednaki onim idealiziranima zbog toga \[SHacek]to curenje du\:017e lanca nu\ \:017eno smanjuje kona\[CHacek]nu amplitudu prijenosa. Tada bismo mogli i o\ \[CHacek]ekivati da \[CAcute]e se primje\[CAcute]eno odstupanje i javiti. \ Problem je to izra\:017eeniji \[SHacek]to vi\[SHacek]e razina imamo uklju\ \[CHacek]enih u lanac, odnosno \[SHacek]to je ve\[CAcute]i ", StyleBox["N", FontFamily->"Courier"], ". No, novu i stro\:017eu analizu ovoga ostavit \[CAcute]emo za neki drugi \ rad jer iznesene rezultate smatramo dovoljno zaokru\:017eenima i \ \[CHacek]vrstima." }], "Text"], Cell[TextData[{ "Na poslijetku, opisani je postupak mogu\[CAcute]e upotrijebiti za odre\ \:0111ivanje najbr\:017ee putanje kroz sustav za prijenos populacije izme\ \:0111u dviju proizvoljno odabranih razina sustava. Ovo bismo postigli tako \ da za svaku mogu\[CAcute]u putanju koja vezuje dvije odabrane razine odredimo \ vrijeme prijenosa, te usporedimo na\:0111ena minimalna vremena. Pri \ opse\:017enijoj razradi ove metode ", StyleBox["komparativne analize prijenosa usporednim lanicma", FontSlant->"Italic"], " (KAPULa) svakako bi trebalo u obzir uzeti i vremena relaksacije pojedinih \ svojstvenih razina, jer ona za razli\[CHacek]ite putanje izme\:0111u istih \ dvaju krajnjih razina mogu biti bitno razli\[CHacek]ita." }], "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["ZAKLJU\[CapitalCHacek]AK", "Section 1", CellTags->"ZAKLJU\[CapitalCHacek]AK"], Cell["\<\ Ve\[CAcute]inu smo zaklju\[CHacek]aka ve\[CAcute] izlo\:017eili po pojedinim \ poglavljima rada, pa nam se stoga ostaje samo osvrnuti na uspje\[SHacek]nost \ savladavanja problema koje smo u uvodu naveli kao temeljne ciljeva ovog istra\ \:017eivanja.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Nadgradnja teorije Rabijevih oscilacija i \[Pi]-puls teorije\ \>", "Subsubsection"], Cell["\<\ Analiza provedena u prvim dvama poglavljima daje nadgradnju odavno poznatom \ mehanizmu Rabijevih oscilacija. Kroz prvo su mu poglavlje analiti\[CHacek]ki \ i u potpunosti odre\:0111ene granice valjanosti u sustavima sa samo dvije \ razine za koje je izvorna teorija i bila razvijena. U drugom smo poglavlju \ smo zna\[CHacek]ajan korak naprijed od izvorne teorije, analiti\[CHacek]ki ju \ pro\[SHacek]iriv\[SHacek]i i izvan dvorazinskih sustava, te utvrdiv\[SHacek]i \ potankosti utjecaja razina smetnje na promjenu polo\:017eaja i oblika \ Rabijevog profila pojedine linije prijelaza u sustavu.\ \>", "Text"], Cell["\<\ Navedeni rezultati omogu\[CAcute]avaju analiti\[CHacek]ku dogradnju \ \[Pi]-puls teorije, a rezultati koje smo razvili za slu\[CHacek]aj ravnog \ vala u cijelosti se mogu upotrijebiti i u slu\[CHacek]aju adijabatski \ moduliranih pogonskih pulseva. Tada rezonancijska frekvencija i konstanta \ vezanja postaju parametri \[CHacek]iji vremenski razvoj mo\:017eemo unaprijed \ analiti\[CHacek]ki odrediti u zavisnosti o obliku anvelope pulsa. Bitan \ doprinos predstavlja i mogu\[CAcute]nost analiti\[CHacek]kog odre\:0111ivanja \ makismalne dozvoljene vrijednosti jakosti pogona koja bi, zajedno sa \ prethodnim rezultatima, mogla olak\[SHacek]ati odabir po\[CHacek]etnih \ parametara za numeri\[CHacek]ke simulacije u laserskoj kontroli.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Potpuno upravljanje ro-vibracijskom dinamikom dvoatomne heteronuklearne \ molekule\ \>", "Subsubsection"], Cell["\<\ Ve\[CAcute]ina dosada\[SHacek]njih istra\:017eivanja upravljanja molekularnom \ dinamikom usredoto\[CHacek]ila su se na upravljanje elektronskim ili \ vibracijskim stupnjevima slobode. Rotacijski stupnjevi slobode se pri tome \ ili zanemaruju, ili uzimaju u obzir kao smetnja i to kroz statisti\[CHacek]ko \ usrednjenje. Na\[SHacek]i rezultati daju jasnu analiti\[CHacek]ku sliku o \ tome na koji se na\[CHacek]in mo\:017ee u\[CHacek]inkovito upravljati, kako \ pojedinim izdvojenim ro-vibracijskim prijelazom, tako i \[CHacek]itavom \ kaskadom vezanih ro-vibracijskih prijelaza (LaVoR pristup). Tako\:0111er \ jasno odre\:0111uje ograni\[CHacek]enja \[LongDash] odre\:0111enih unutra\ \:0111njom fizi\[CHacek]kom gra\:0111om sustava \[LongDash] s kojima se takvo \ upravljanje susre\[CAcute]e. Tu prvenstveno mislimo na minalno vrijeme u\ \[CHacek]inkovitog prijenosa koje se mo\:017ee posti\[CAcute]i izme\:0111u \ pojedinih ro-vibracijskih razina unutar sustava.\ \>", "Text"], Cell["\<\ Kao zanimljiv vid mogu\[CAcute]e sprege LaVoR metode i prijenosa izdvojenim \ prijelazima napomenut \[CAcute]emo KAPULa pristup. Njegovu bismo upotrebu mo\ \:017eda mogli o\[CHacek]ekivati u budu\[CAcute]im kvantnim \ ra\[CHacek]unalima za optimizaciju brzine prijenosa informacije unutar \ memorije ili procesora. Ipak, valja naglasiti da bi ovakva primjena svakako \ zahtjevala dodatnu op\[SHacek]irnu razradu cjelokupnog pristupa pri kojoj bi \ trebalo uzeti u obzir mno\[SHacek]tvo tehni\[CHacek]kih i pojmovnih problema \ koji su u ovome radu ili samo dotaknuti, ili sasvim zanemareni.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["POPIS LITERATURE", "Section 1", CellTags->"zaklju\[CHacek]ak"], Cell[TextData[{ "[1] J. Manz. ", StyleBox["Molecular wavepacket dynamics: theory for experiments 1926-1996", FontWeight->"Bold"], ".", StyleBox[" ", FontWeight->"Bold"], StyleBox["Femtochemistry and Femtobiology (ed. V Sundstrom), ", FontSlant->"Italic"], "World Sicentific, Singapore (1997.)\n", StyleBox["[2] H. Rabitz. ", FontFamily->"Times New Roman"], StyleBox["Shaped Laser Pulses as Reagents", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[". ", FontFamily->"Times New Roman"], StyleBox["Science", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" 299: pp 525-527 (2003)", FontFamily->"Times New Roman"], "\n[3] G. K. Paramonov and V. A. Savva. ", StyleBox["Resonance effects in molecule vibrational excitation by \ picosecond laser pulses", FontWeight->"Bold"], ". ", StyleBox["Phys. Letters A", FontSlant->"Italic"], ", 97A (8): pp 340-342 (1983.)\n[4] M. Holthaus and B. Just. ", StyleBox["Generalized \[Pi] pulses", FontWeight->"Bold"], ". ", StyleBox["Phys. Rev. A", FontSlant->"Italic"], ", 49(3): pp 1950-1960 (1994.)\n[5] J. H. Shirley. ", StyleBox["Solution of the schroedinger equation with a hamiltonian periodic \ in time", FontWeight->"Bold"], ". ", StyleBox["Phys. Rev.", FontSlant->"Italic"], ", 138 (4B):pp B979-B987 (1965.)\n[6] W. Demtroeder. ", StyleBox["Laser spectroscopy", FontWeight->"Bold"], ". p 57, Springer-Werlag, Berlin (1988.)\n[7] M. Kalu\:017ea, J. T. \ Muckerman, P. Gross, H. Rabitz. ", StyleBox["Optimaly controlled five-laser infrared multiphoton dissociation \ of HF", FontWeight->"Bold"], ", ", StyleBox["J. Chem. Phys.", FontSlant->"Italic"], ", 100 (6): pp 4211-4228 (1993.)\n[8] S. Chelkowski, A. Bandrauk, P. B. \ Corkum. ", StyleBox["Efficient molecular dissociation by chirped ultrashort infrared \ laser pulse", FontWeight->"Bold"], ", ", StyleBox["Phys. Rev. Lett.", FontSlant->"Italic"], ", 65 (19): pp 2355-2358 (1990.)\n[9] S. Chelkowski and A. Bandrauk. ", StyleBox["Control of molecular vibrational excitation and dissociation by \ chirped intense infrared pulses. Rotational effects", FontWeight->"Bold"], ", ", StyleBox["J. Chem. Phys.", FontSlant->"Italic"], ", 99 (6): pp 4279-4287 (1993.)\n[10] I.I.Rabi. ", StyleBox["Space quantization in a gyrating magnetic field", FontWeight->"Bold"], ". Phys. Rev. 51, pp 652-654 (1937.)\n[11] H. Mueller et. al.. ", StyleBox["Toward spectroscopic accuracy of ab initio calculations of \ vibrational frequencies and related quantities: a case study of the HF \ molecule", FontWeight->"Bold"], ", ", StyleBox["Theo. Chem. Acc.", FontSlant->"Italic"], ", 100 (19): pp 85-102 (1998.)\n[12] W. T. Zemke, W. C. Stwalley, S. R. \ Langhoff, G. L. Valderrama and M. J. Berry. ", StyleBox["Radiative transition probabilities for all vibrational levels in \ the ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ StyleBox["X", FontWeight->"Bold"], StyleBox["1", FontWeight->"Bold"]], StyleBox[\(\[CapitalSigma]\^*\), FontWeight->"Bold"]}], TraditionalForm]]], " ", StyleBox[" state of HF", FontWeight->"Bold"], ", ", StyleBox["J. Chem. Phys.", FontSlant->"Italic"], ", 95 (11): pp 7846-7853 (1991.)" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["SA\:017dETAK", "Section 1", CellTags->"POPIS LITERATURE"], Cell[TextData[StyleBox["Mogu\[CAcute]nost preciznog upravljanja strukturnim \ zna\[CHacek]ajkama diskretnih kvantnih sustava kao \[SHacek]to su atomi, \ molekule ili kvantne to\[CHacek]ke jest zanimljiva kao vrhunski alat u mnogim \ razli\[CHacek]itim poljima kao npr. lasersko upravljanje kemijskim \ reakcijama, atomska optika, metrologija i kvantno ra\[CHacek]unarstvo. \ Upotreba Rabijevih titranja spada me\:0111u najjednostavnije a istovremeno \ vrlo korisne pristupe takvom upravljanju. Rabijeva teorija pokazuje kako u \ dvorazinskim sustavima rezonantna pobuda vodi do periodi\[CHacek]kih i \ potpunih oscilacija populacije izme\:0111u dvaju kvantnih razina.", FontFamily->"Times-Roman"]], "Text", TextJustification->1, LineSpacing->{1.3, 3}], Cell[TextData[StyleBox["Ovaj rad ima trostruki cilj: prvo, u njemu su potanko \ istra\:017eena ograni\[CHacek]enja dvorazinske Rabijeve teorije te je \ razvijen jednostavan grafi\[CHacek]ki kriterij za ocjenu njezine valjanosti u \ sustavima s vi\[SHacek]e razina; potom, izvedeno je analiti\[CHacek]ko poop\ \[CAcute]enje Rabijeve teorije za vi\[SHacek]erazinske sustave; na \ poslijetku, dobiveno poop\[CAcute]enje primjenjeno je na razvoj i \ uga\:0111anje jedne specifi\[CHacek]ne vrste mehanizma za upravljanje \ populacijom mnogorazinskih kvantnih sustava.", FontFamily->"Times-Roman"]], "Text", TextJustification->1, LineSpacing->{1.3, 3}] }, Open ]], Cell[CellGroupData[{ Cell["SUMMARY", "Section 1", CellTags->"POPIS LITERATURE"], Cell[TextData[StyleBox["State selective preparation and manipulation of \ discrete-level quantum systems such as atoms, molecules, or quantum dots is \ the ultimate tool for many diverse fields such as laser control of chemical \ reactions, atom optics, high-precision metrology, and quantum computing. Rabi \ oscillations are one of the simplest, yet potentially quite useful mechanisms \ for achieving such manipulation. Rabi theory establishes that in the \ two-level systems resonant drive leads to the periodic and complete \ population oscillations between the two system levels.", FontFamily->"Times New Roman CE", FontSize->12, CharacterEncoding->"WindowsEastEurope"]], "Text", TextJustification->1, LineSpacing->{1.3, 3}, FontFamily->"Times New Roman CE", FontSize->12, CharacterEncoding->"WindowsEastEurope"], Cell[TextData[StyleBox["The scope of this paper is threefold: first, the \ limitations of the two level Rabi theory are thoroughly revised and a simple \ graphical criterium for its validity in many-level szstems is established; \ further, an analytic extension to the Rabi theory for the case of many-level \ quantum systems is developed; finally, the obtained analytic procedure is \ employed in producing and optimizing a very specific case of population \ manipulation of a many level quantum system.", FontFamily->"Times New Roman CE", FontSize->12]], "Text", TextJustification->1, LineSpacing->{1.3, 3}, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["POPIS RADOVA", "Section 1", CellTags->"zaklju\[CHacek]ak"], Cell[TextData[{ "[1] D. Bonacci: ", StyleBox["Rabi spectra - a simple tool for analyzing the limitations of RWA \ in modelling of the selective population transfer in many-level quantum \ systems", FontWeight->"Bold"], ".", StyleBox[" ", FontWeight->"Bold"], StyleBox["Unpublished manuscript (2003) (", FontVariations->{"CompatibilityType"->0}], "quant-ph/0309126", StyleBox[")", FontVariations->{"CompatibilityType"->0}], "\n\n[1] D. Bonacci: ", StyleBox["Analytic pulse design for selective population transfer in \ many-level quantum systems:\nMaximizing the amplitude of population \ oscillations", FontFamily->"Times-Bold", FontWeight->"Bold"], ".", StyleBox[" ", FontWeight->"Bold"], StyleBox["Phys.Rev.A", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], StyleBox["70", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" 043413 ", FontVariations->{"CompatibilityType"->0}], StyleBox["(2004) (", FontVariations->{"CompatibilityType"->0}], "quant-ph/0310034", StyleBox[")", FontVariations->{"CompatibilityType"->0}] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\:017dIVOTOPIS AUTORA", "Section 1", CellTags->"POPIS LITERATURE"], Cell["\<\ Ro\:0111en sam u Zagrebu, 01. lipnja 1973. Osnovnu \[SHacek]kolu i gimnaziju \ poha\:0111ao sam u Splitu. Maturirao sam u ljeto 1992. u Oxfordu. U jesen \ 1998. sam diplomirao fiziku na Prirodoslovno-matemati\[CHacek]kom fakultetu \ Sveu\[CHacek]ili\[SHacek]ta u Zagrebu, a 2000. na istom sveu\[CHacek]ili\ \[SHacek]tu upisao poslijediplomski studij iz atomske i molekulske fizike i \ astrofizike.\ \>", "Text"], Cell["\<\ Tijekom poslijediplomskog studija radio sam u nastavi kao mla\:0111i asistent \ na Fakultetu prirodoslovno-matemati\[CHacek]kih znanosti i odgojnih podru\ \[CHacek]ja Sveu\[CHacek]ili\[SHacek]ta u Splitu, te na Fizi\[CHacek]kom \ odsjeku Prirodoslovno-matemati\[CHacek]kog fakulteta \ Sveu\[CHacek]ili\[SHacek]ta u Zagrebu. U jesen 2000. sam nakratko radio kao \ profesor fizike i politehnike u Osnovnoj \[SHacek]koli 'Stro\:017eanac' \ pokraj Splita. Od prosinca 2000. zaposlen sam kao znanstveni novak pri Zavodu \ za fizi\[CHacek]ku kemiju Instituta 'Ru\:0111er Bo\[SHacek]kovi\[CAcute]'. U \ okviru svoje magistarske teme bavio sam se izu\[CHacek]avanjem metoda \ koherentnog upravljanja populacijom kvantnih sustava.\ \>", "Text"], Cell["\<\ Od 1998. intenzivno se bavim popularizacijom i posredni\[SHacek]tvom \ (komunikacijom) znanosti. Tijekom akademske godine 2003./'04. poha\:0111ao \ sam jednogodi\[SHacek]nji poslijediplomski studij iz posredni\[SHacek]tva \ znanosti (communicating science) na Sveu\[CHacek]ili\[SHacek]tu Glamorgan u \ Cardiff, Velika Britanija. Kao zavr\[SHacek]nu (magistarsku) radnju za taj \ studij izradio sam rad pod naslovom 'Computer Games As Interactive Labels For \ Science Centre Exhibits'.\ \>", "Text"] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 800}, {0, 555}}, ScreenStyleEnvironment->"Printout", PrintingStyleEnvironment->"Printout", WindowSize->{792, 528}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{68, 69}, PageHeaders->{{None, Inherited, None}, {None, Inherited, None}}, PageFooters->{{Inherited, Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}, { Inherited, Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{Inherited, False}, PageFooterLines->{True, True}, PrintingOptions->{"PrintingMargins"->{{2.1875, 2.1875}, {2.1875, 2.8125}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False}, IndexCreationOptions->{"Format"->"Text"}, Magnification->1, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle", PageBreakAbove->Automatic], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. 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"Mma"->"Mathematica", "MMA"->"Mathematica", Inherited}, CounterIncrements->"Subsubsection", FontFamily->"Times", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{31, 30}, {0, 12}}, PageBreakAbove->Automatic], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{18, 30}, {0, 12}}, PageBreakAbove->Automatic, FontFamily->"Times New Roman CE", FontSize->14, CharacterEncoding->"WindowsEastEurope"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{55, 10}, {6, 6}}, PageBreakAbove->Automatic, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, -0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", Inherited}, TextJustification->1, Hyphenation->True, LineSpacing->{1, 2}, FontFamily->"Times"], Cell[StyleData["Text", "Presentation"], CellMargins->{{20, 10}, {6, 6}}, PageBreakAbove->Automatic, TextAlignment->Left, TextJustification->0, LineSpacing->{1.3, 0}, FontSize->14], Cell[StyleData["Text", "Printout"], CellMargins->{{18, 4}, {4, 4}}, PageBreakAbove->Automatic, LineSpacing->{1, 3}, FontFamily->"Times New Roman CE", CharacterEncoding->"WindowsEastEurope"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Commentary"], CellMargins->{{55, 10}, {2, 6}}, PageBreakAbove->Automatic, TextJustification->1, Hyphenation->True, LineSpacing->{1, 2}, FontFamily->"Helvetica", FontSize->10, FontColor->RGBColor[0, 0, 0.4]], Cell[StyleData["Commentary", "Presentation"], CellMargins->{{60, 30}, {2, 6}}, PageBreakAbove->Automatic, TextJustification->1, LineSpacing->{1.3, 0}, FontSize->12], Cell[StyleData["Commentary", "Printout"], CellMargins->{{18, 30}, {3, 0}}, PageBreakAbove->Automatic, LineSpacing->{1, 3}, FontFamily->"Times New Roman CE", FontSize->10, CharacterEncoding->"WindowsEastEurope"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section", PageBreakAbove->Automatic], Cell["\<\ The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names.\ \>", "Text", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{1, 1}, {0, 1}}, CellMargins->{{55, 10}, {0, 0}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelPositioning->Automatic, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, FontSize->12, FontWeight->"Bold", Background->GrayLevel[0.966674]], Cell[StyleData["Input", "Presentation"], CellMargins->{{60, 10}, {0, 10}}, PageBreakAbove->Automatic, Background->GrayLevel[0.850004]], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 10}, {0, 10}}, PageBreakAbove->Automatic, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->10, Background->GrayLevel[0.850004]] }, Closed]], Cell[StyleData["InlineInput"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, GroupPageBreakWithin->False, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellFrame->{{1, 1}, {1, 0}}, CellMargins->{{55, 10}, {15, 0}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, FontSize->12, Background->GrayLevel[0.850004]], Cell[StyleData["Output", "Presentation"], CellMargins->{{60, Inherited}, {10, 0}}, PageBreakAbove->Automatic], Cell[StyleData["Output", "Printout"], CellMargins->{{55, Inherited}, {10, 0}}, PageBreakAbove->Automatic, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["InputOnly"], CellFrame->1, CellMargins->{{55, 10}, {15, 0}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelPositioning->Automatic, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, FontSize->12, FontWeight->"Bold", Background->GrayLevel[0.966674]], Cell[StyleData["InputOnly", "Presentation"], CellMargins->{{60, Inherited}, {10, 10}}, PageBreakAbove->Automatic], Cell[StyleData["InputOnly", "Printout"], CellMargins->{{55, Inherited}, {10, 10}}, PageBreakAbove->Automatic, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellFrame->{{1, 1}, {0, 0}}, CellDingbat->"\[LongDash]", CellMargins->{{55, 10}, {0, 0}}, CellGroupingRules->"OutputGrouping", PageBreakAbove->Automatic, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, FormatType->InputForm, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontSlant->"Oblique", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{60, Inherited}, {Inherited, Inherited}}, PageBreakAbove->Automatic, LineSpacing->{1, 0}], Cell[StyleData["Message", "Printout"], CellMargins->{{55, Inherited}, {0, 0}}, PageBreakAbove->Automatic, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{55, 26}, {1, 6}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, FormatType->InputForm, StyleMenuListing->None, Background->GrayLevel[0.700008]], Cell[StyleData["Print", "Presentation"], CellMargins->{{60, Inherited}, {10, 2}}, PageBreakAbove->Automatic], Cell[StyleData["Print", "Printout"], CellMargins->{{54, Inherited}, {2, 6}}, PageBreakAbove->Automatic, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellFrame->{{1, 1}, {0, 0}}, CellMargins->{{55, 10}, {0, 0}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, StyleMenuListing->None, Background->GrayLevel[0.850004]], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{60, Inherited}, {0, 0}}, PageBreakAbove->Automatic, ImageMargins->{{10, 10}, {10, 10}}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, Inherited}, {0, 0}}, PageBreakAbove->Automatic, ImageSize->{0.0625, 0.0625}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], PageBreakAbove->Automatic, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontSlant->"Oblique", FontColor->RGBColor[0.6, 0, 0.6]], Cell[StyleData["CellLabel", "Presentation"], CellMargins->{{18, Inherited}, {Inherited, Inherited}}, PageBreakAbove->Automatic], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, PageBreakAbove->Automatic, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Section", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{20, 30}, {45, 5}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakAbove->Automatic, PageBreakBelow->False, CellFrameMargins->{{0, 4}, {8, 4}}, LineSpacing->{1, 0}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Author", "Presentation"], CellMargins->{{20, 30}, {45, 10}}, PageBreakAbove->Automatic], Cell[StyleData["Author", "Printout"], CellMargins->{{18, 30}, {45, 5}}, PageBreakAbove->Automatic] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellFrame->False, CellMargins->{{18, 140}, {4, 30}}, PageBreakAbove->Automatic, Hyphenation->True, LineSpacing->{0.9, 0}, FontFamily->"Times", FontSize->12], Cell[StyleData["Abstract", "Presentation"], CellFrame->True, CellMargins->{{20, 10}, {Inherited, 30}}, PageBreakAbove->Automatic], Cell[StyleData["Abstract", "Printout"], PageBreakAbove->Automatic, LineSpacing->{1, 2}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Caption"], CellMargins->{{55, 10}, {5, 3}}, PageBreakAbove->Automatic, Hyphenation->True, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["Caption", "Presentation"], CellMargins->{{60, 65}, {6, 4}}, PageBreakAbove->Automatic, FontSize->10], Cell[StyleData["Caption", "Printout"], CellMargins->{{55, 55}, {5, 4}}, PageBreakAbove->Automatic, LineSpacing->{1, 2}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{24, 40}, {6, 6}}, PageBreakAbove->Automatic, TextJustification->1, Hyphenation->True, LineSpacing->{1, 0}, FontFamily->"Times"], Cell[StyleData["Reference", "Presentation"], CellMargins->{{20, 40}, {Inherited, 6}}, PageBreakAbove->Automatic, TextJustification->0, LineSpacing->{1, 4}, FontSize->12], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 4}, {4, 4}}, PageBreakAbove->Automatic, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PictureGroup"], CellFrame->{{1, 1}, {0, 0}}, CellMargins->{{55, Inherited}, {0, 0}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakAbove->Automatic, PageBreakWithin->False, ShowCellLabel->False, ImageMargins->{{35, Inherited}, {Inherited, 0}}, StyleMenuListing->None, Background->GrayLevel[0.850004]], Cell[StyleData["PictureGroup", "Presentation"], CellMargins->{{60, Inherited}, {0, 0}}, PageBreakAbove->Automatic, ImageMargins->{{10, 10}, {10, 10}}], Cell[StyleData["PictureGroup", "Printout"], CellMargins->{{55, Inherited}, {0, 0}}, PageBreakAbove->Automatic, ImageSize->{0.0625, 0.0625}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section", PageBreakAbove->Automatic], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], PageBreakAbove->Automatic, FontSize->16], Cell[StyleData["Hyperlink", "Printout"], PageBreakAbove->Automatic, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], PageBreakAbove->Automatic, FontSize->16], Cell[StyleData["MainBookLink", "Printout"], PageBreakAbove->Automatic, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], PageBreakAbove->Automatic, FontSize->16], Cell[StyleData["AddOnsLink", "Printout"], PageBreakAbove->Automatic, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], PageBreakAbove->Automatic, FontSize->16], Cell[StyleData["RefGuideLink", "Printout"], PageBreakAbove->Automatic, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], PageBreakAbove->Automatic, FontSize->16], Cell[StyleData["GettingStartedLink", "Printout"], PageBreakAbove->Automatic, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], PageBreakAbove->Automatic, FontSize->16], Cell[StyleData["OtherInformationLink", "Printout"], PageBreakAbove->Automatic, FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section", PageBreakAbove->Automatic], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text", PageBreakAbove->Automatic], Cell[StyleData["Paste"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], PageBreakAbove->Automatic, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, PageBreakAbove->Automatic, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, StyleMenuListing->None, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}, PageBreakAbove->Automatic, LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Printout"], CellMargins->{{18, 4}, {4, 4}}, PageBreakAbove->Automatic] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{55, 10}, {Inherited, 0}}, CellHorizontalScrolling->True, PageBreakAbove->Automatic, DefaultFormatType->DefaultInputFormatType, TextJustification->1, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}, PageBreakAbove->Automatic], Cell[StyleData["ChemicalFormula", "Printout"], CellMargins->{{18, 4}, {4, 4}}, PageBreakAbove->Automatic, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{55, 10}, {Inherited, 0}}, CellHorizontalScrolling->True, PageBreakAbove->Automatic, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}, PageBreakAbove->Automatic], Cell[StyleData["Program", "Printout"], CellMargins->{{18, 4}, {4, 4}}, PageBreakAbove->Automatic, LineSpacing->{1, 3}, FontSize->9.5] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section", PageBreakAbove->Automatic], Cell["\<\ The following styles are useful for numbered equations, figures, etc. They \ automatically give the cell a FrameLabel containing a reference to a \ particular counter, and also increment that counter.\ \>", "Text", PageBreakAbove->Automatic], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CellMargins->{{55, 85}, {Inherited, Inherited}}, PageBreakAbove->Automatic, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedEquation", "Presentation"], CellMargins->{{60, 10}, {Inherited, 6}}, PageBreakAbove->Automatic, LineSpacing->{1, 0}], Cell[StyleData["NumberedEquation", "Printout"], CellMargins->{{18, 4}, {4, 4}}, PageBreakAbove->Automatic, FontFamily->"Arial CE", FontSize->10, CharacterEncoding->"WindowsEastEurope"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellMargins->{{55, 95}, {Inherited, Inherited}}, PageBreakAbove->Automatic, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}], FontWeight -> "Bold"], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{60, 80}, {Inherited, 6}}, PageBreakAbove->Automatic, LineSpacing->{1, 0}], Cell[StyleData["NumberedFigure", "Printout"], CellMargins->{{18, 80}, {4, 4}}, PageBreakAbove->Automatic, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{55, 95}, {Inherited, Inherited}}, PageBreakAbove->Automatic, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}], FontWeight -> "Bold"], None}}, TextAlignment->Center, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{60, 80}, {Inherited, 6}}, PageBreakAbove->Automatic, LineSpacing->{1, 0}], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{18, 80}, {4, 4}}, PageBreakAbove->Automatic, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section", PageBreakAbove->Automatic], Cell[StyleData["Header"], PageBreakAbove->Automatic, StyleMenuListing->None, FontFamily->"Times New Roman CE", FontSize->10, FontSlant->"Italic", CharacterEncoding->"WindowsEastEurope"], Cell[StyleData["PageNumber"], PageBreakAbove->Automatic, StyleMenuListing->None, FontFamily->"Times New Roman CE", FontSize->10, FontWeight->"Bold", CharacterEncoding->"WindowsEastEurope"], Cell[StyleData["Footer"], PageBreakAbove->Automatic, TextAlignment->Center, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9] }, Open ]] }, Open ]] }] ] (******************************************************************* Cached data follows. 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