Floquet-state

Lecture3:PeriodicdifferentialequationsandFloquetstatesLearningoutcomes:Bytheendofthislecture,thestudentsshouldbeableto•recognisetime-periodicdifferentialequations,•understandFloquet’stheoremanditsunderlyingassumptions,•expandFloquetstatesasadiscreteFourierseries.I.INTRODUCTORYEXAMPLEANDANANALOGYConsideranatominteractingwithamonochromaticlaserfield,sothatitsHamiltonianisgivenbyˆH(t)=ˆH0−ˆd·E0cos(ωt).Here,ˆH0isthetime-independentHamiltonianoftheatom,ˆdisitselectricdipolemomentandE0istheamplitudeoftheelectriclaserfield.TheHamiltonianisperiodicintimewithperiodT=2π/ω.DoesthismeanthatthesolutionstotheSchr¨odingerequationareperiodicintimeaswell?Theanswerisobviouslyno:Forinstance,aresonantlaserfieldcanioniseanatom,withtherespectivemotionoftheelectronbeingfarfromperiodic.However,itwillbeshowninthislecturethatitispossibletofindabasissystemofquasi-periodiceigenstates.Tobeginwith,letusrecallthemorecommonlyknowcaseofspace-periodicsystemsarisinginsolid-statephysics.AnelectroninsideacrystalobeystheSchr¨odingerequationˆHψ(r)=~22m∇2+U(r)ψ(r)=Eψ(r)wherethepotentialofthelatticeionsisperiodic,U(r+R)=U(r).ItfollowsthattheeigenstatesoftheelectronaregivenbyBlochstatesψnk(r)=eik·runk(r)withperiodicfunctionsunk(r+R)=unk(r).Here,nisthediscretebandindexandkisthecrystalmomentum,whichcanbechosentolieinthefirstBrillouinzone.Equivalently,wecansaythatψnk(r+R)=eik·Rψnk(r).1II.FLOQUET’STHEOREMANDTHEFLOQUETSTATESReturningtoouroriginaltime-dependentproblem,weconsiderasystemwhoseHamil-tonianˆH(t)isperiodicwithperiodT,ˆH(t+T)=ˆH(t)∀t.ThesolutiontotheSchr¨odingerequationi~∂∂t|ψ(t)i=ˆH(t)|ψ(t)i,canformallybegivenas|ψ(t)i=ˆU(t,t0)|ψ(t0)iwithatime-evolutionoperatorˆU(t,t0)=ˆTexp−i~Ztt0dt′ˆH(t′)(ˆT:timeorderingoperator).ThisoperatorevolvessolutionsforwardsorbackwardsintimeandithasthecompositionpropertyˆU(t,t′)=ˆU(t,t′′)ˆU(t′′,t0)∀t,t′,t′′.ForaperiodicHamiltonian,wefurtherhaveˆU(t+T,t′+T)=ˆU(t,t′)∀t,t′.WewanttoexploittheperiodicityoftheHamiltoniantofindquasi-periodicsolutionstotheSchr¨odingerequation.Tothatend,weintroducetheFloquetHamiltonianˆH(t)=ˆH(t)−i~∂∂twhichisalsotime-periodicˆH(t+T)=ˆH(t+T)−i~∂∂(t+T)=ˆH(t).ItfollowsthattheFloquetHamiltoniancommuteswiththeperiod-shiftoperatorˆU(t+T,t):ˆU(t+T,t)ˆH(t)=ˆU(t+T,t)ˆH(t)ˆU−1(t+T,t)ˆU(t+T,t)=ˆH(t+T)ˆU(t+T,t)=ˆH(T)ˆU(t+T,t).WecanhencefindasetofsimultaneouseigenstatesforˆH(t)andˆU(t+T,t).2Letusinvestigatetheeigenvaluesoftheperiod-shiftoperatorˆU(t+T,t).AsˆU(t+T,t)isunitary,theeigenvalueequationhastobeoftheformˆU(t+T,t)|ψ(t)i=eiθ|ψ(t)i.TofindthedependenceofthephaseθonT,weusetocompositionalpropertyandexploittheperiodicitytofindˆU(t+nT,t)=ˆU[t+nT,t+(n−1)T]ˆU[t+(n−1)T,t+(n−2)T]···ˆU(t+T,t)=ˆU(t+T,t)nforn∈N.Combiningthisresultwiththeeigenvalueequation,wefindeiθ(nT)|ψ(t)=ˆU(t+nT,t)|ψ(t)i=ˆU(t+T,t)n|ψ(t)i=einθ(T)
n|ψ(t)i=eiθ(T)
n|ψ(t)i.showingthatθmustbeproportionaltoT.IntroducingtheyetundeterminedFloquetexponentε,wemayhencewriteθ=−(ε/~)T,sothatthesimultaneouseigenstatesofˆH(t)andˆU(t+T,t)obeyˆU(t+T,t)|ψ(t)i=e−i~εT|ψ(t)i.Theeigenstates|ψ(t)ioftheperiod-shiftoperatorhenceacquireaphase−i~εTduringtheintervalofoneperiodt=T.Wesplitofftheassociatedtime-dependencebywriting|ψ(t)i=e−i~ε(t−t0)|φ(t)i.With|φ(t)ibeinganeigenstate,theFloquetstate|ψ(t)iistime-periodic:|φ(t+T)i=ei~ε(t+T−t0)|ψ(t+T)i=ei~ε(t+T−t0)ˆU(t+T,t)|ψ(t)i=ei~ε(t+T−t0)e−i~εT|ψ(t)i=|φ(t)i.InsertingtheFloquetsolutionintotheSchr¨odingerequation,wefind0=ˆH(t)|ψ(t)i=ˆH(t)−i~∂∂te−i~εt|φ(t)i=e−i~εtˆH(t)−ε|φ(t)i,sotheFloquetstatesareeigenstatesoftheFloquet-typeSchr¨odingerequationˆH(t)|φ(t)i=ε|φ(t)i.Letussummarisetheseresults:3Theorem(Floquettheorem).Thebasicsolutionstothetime-dependentSchr¨odingerequa-tionwithtime-periodicHamiltonianˆH(t)=ˆH(t+T)canbegivenintheform|ψα(t)i=e−i~εα(t−t0)|φα(t)iwithFloquetexponentsεαandtime-periodicFloquetstates|φα(t)i=|φα(t+T)i,whicharesolutionstotheFloquet-typeSchr¨odingerequationˆH|φα(t)i=εα|φα(t)iwithˆH(t)=ˆH(t)−i~∂/∂t.Alternatively,εαand|φα(t)iareknownasquasi-eigenenergiesandquasi-eigenstates,respectively.Remarks:ForagivenFloquetexponentεαandstate|φα(t)ithereisaninfinitesetofequivalentexponentsandstates(ω=2π/T)εα′=εα+n~ωn∈Z,|φα′(t)i=einω(t−t0)|φα(t)iwhicharealsosolutionstotheFloquet-typeSchr¨odingerequation,butcorrespondtothesametotalsolution:|ψα′(t)i=e−i~εα′(t−t0)|φα′(t)i=e−i~εα(t−t0)|φα(t)i=|ψα(t)i.Toavoidredundanciesitisnecessarytorestricttherangeofallowedεαquasi-eigenenergiestoaBrillouinzoneofwidth~ω.Anarbitrarysolutiontothetime-dependentSchr¨odingerequationcanthenbeexpandedintermsofthenon-redundantFloquetstates:|ψ(t)i=Xαcαe−i~εα(t−t0)|φα(t)iwherecα=hφ(t0)|ψ(t0)i.WiththeFloquetstatesbeingtime-periodic,wecanapplyadiscreteFouriertransforma-tiontowrite|φα(t)i=∞Xn=−∞e−inω(t−t0)|φα,ni.Usingtherelation1TZt0+Tt0dtei(m−n)ωt=δmn,4theFouriercoefficientsarefoundtobe|φα,ni=1TZt0+Tt0dteinω(t−t0)|φα(t)i.Anarbitrarysolutiontothetime-periodicSchr¨odingerequationcanthenbegivenas|ψ(t)i=Xα∞Xn=−∞cαe−i(εα/~+nω)(t−t0)|φα,ni.FURTHERREADING•TheoreticalFemtosecondPhysics:AtomsandMoleculesinstronglaserfields,F.Gross-mann,Chaps.227,pp.45-48,159-227(Springer,Be