a r X i v :q u a n t -p h /0412019v 1 2 D e c 2004Europhysics Letters PREPRINT Quantum diffusion in the quasiperiodic kicked rotor
Hans Lignier 1,Jean Claude Garreau 1,Pascal Szriftgir 1and Dominique Delande 2
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Laboratoire de Physique des Lars,Atomes et Mol´e cules,UMR CNRS 8523,Univer-sit´e des Sciences et Technologies de Lille,F-59655Villeneuve d’Ascq Cedex,France 2Laboratoire Kastler Brosl,Tour 12,Etage 1,4Place Jussieu,F-75005Paris,France PACS.05.45.Mt –Quantum chaos;miclassical methods.PACS.32.80.Lg –Mechanical effects of light on atoms,molecules,and ions.PACS.32.80.Pj –Optical cooling of atoms;trapping.Abstract.–We study the mechanisms responsible for quantum diffusion in the quasiperiodic kicked rotor.We report experimental measurements of the diffusion constant on the atomic version of the system and develop a theoretical approach (bad on the Floquet theorem)explaining the obrvations,especially the “sub-Fourier”character of the resonances obrved in the vicinity of exact he ability of the system to distinguish two neighboring driving frequencies in a time shorter than the inver of the difference of the two frequencies.Quantum chaos is the study of quantum systems who classical limit
is chaotic.A major challenge of quantum chaos is to understand the mechanisms that make quantum chaos dif-ferent from classical chaos.An important difference between classical and quantum systems is the existence in the latter of interferences between various paths.At long times,a large num-ber of complicated trajectories interfere.One could expect the contributions of the various paths to have uncorrelated phas,so that the interference terms vanish in the average after some time,implying that quantum and classical transport should be identical.This simple expectation is however too naive,becau phas of the various paths are actually correlated;
this is for example the ca for the kicked rotor.
The quantum kicked rotor has been extensively studied experimentally in recent years
[1–4].In its atomic version it consists of a cloud of lar-cooled atoms expod to short puls of a far detuned,standing lar wave,corresponding to the Hamiltonian (for the external motion of the atoms)H 0=p 2澳大利亚专业排名雅思
T cos θ n
δ(t −nT ).(1)where θis the 2π−periodic position of the rotor,p the conjugate momentum,T the period a
nd K is proportional to the strength of the kicks.In the classical limit,this system is chaotic for K 1[5],and the motion is an ergodic diffusion in momentum space for K >5.Dynamical localization (DL)is the suppression of such a diffusion in the quantum system by subtle quantum interference effects [6].
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新会计制度Number of kicks
050
100
150(a)
T cos θ
n [δ(t −nT )+δ(t −nrT −λT )](2)
where λis the initial pha between the two kick quences.If r is rational,the system is strictly time-periodic and DL takes place,but it is rapidly destroyed around any rational number.One way of characterizing this nsitivity to the time-periodicity is to measure the average kinetic energy p 2 of the system as a function of time (or number of kicks),which allows to deduce its diffusion constant.In this paper,we prent new experimental results and a theoretical interpretation of the physical mechanism responsible for the destruction of DL in the vicinity of rational r .
Our experimental tup is described in [8].Cold cesium atoms are produced in a magneto-optical trap,the trap is turned off,and a ries of short puls of a far-detuned (13.5GHz ∼2600Γ)standing wave (around 65mW in each direction)is applied.At the end of the pul ries,puls of counter-propagating pha-coherent beams perform velocity-lective Raman stimulated transitions between the hyperfine ground state sublevels F g =4and F g =3.A resonant probe beam is then ud to measure the fraction of transfered atoms,which corresponds to the population in a velocity class.Repeated measurements allow to reconstruct the atomic momentum distribution P (p ).In the periodic ca,one obrves,for t >t ℓ,where t ℓis the so-called localization time ,two manifestations of DL:i )P (p )is frozen in a characteristic exponential shape P (p )∼exp(−|p |/ℓ),where ℓis the localization
length ,and ii )the average kinetic energy of the system tends to a constant value,or,equivalently,the diffusion constant D =lim t →∞ p 2 /t vanishes.Experimental measurements of p 2 and the corresponding diffusion constant are displayed in Fig.1.In the periodic ca r =1(solid line),after an initial linear increa,the kinetic energy saturates to a constant value –corresponding to D (r =1)≈0in Fig.1(b).Due
H.Lignier et al.:Quantum diffusion in the quasiperiodic kicked rotor 30.980.991 1.01 1.02
高一英语必修1单词Ratio of the periods r
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P
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p
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Fig.2–Sub-Fourier resonances.The measured atomic population in the zero-momentum class,after 2
0(upper solid curve)and 100(lower solid curve)double kicks is plotted vs.the ratio r of the periods.The period of the first quence is T =27.8µs,and K ≈10.At r =1,dynamical localization is responsible for the large number of zero-momentum atoms.Away from the exact resonance,localization is progressively destroyed.Note the narrowness (5times smaller than Fourier limit)and the triangular shape of the resonance line.Dashed lines are fits to the experimental lines using Eq.(7).
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to residual spontaneous emission by the kicked atoms [9],D (r =1)is not strictly zero.In the quasiperiodic cas,there is a residual diffusion due to the quasiperiodicity,and Fig.1(b)shows that D (r )∝|r −1|.
Another way to characterize the obrved residual diffusion is to measure the population P 0of the zero-momentum class (1),as a function of r .Fig.2shows the experimentally measured P 0after 20and 100double kicks.This “localization resonance”displays a sharp peak at r =1,indicating the prence of DL,and decreas rapidly on both sides,evidencing the residual diffusion.The plot prents two surprising features:(i )The resonance is very narrow:after N kicks,it could be argued that the two quasi-periods can be distinguished only if they differ by 1/N (in relative value).This would predict a width of the order of ∆r =1/N =0.01for 100kicks,whereas we experimentally obrve a width five times smaller,0.0018(2).(ii )The “sub-Fourier”resonance is not smooth,but has a marked cusp at
the maximum.The understanding of the underlying mechanism behind the quantum behavior of the system shall also allow us to explain the features.
Our analysis of the residual diffusion takes the periodic ca as the reference system,becau,being periodic,it can be analyzed using the Floquet theorem .A Floquet state (FS)|ϕk is defined as an eigenstate of the unitary evolution operator U (T )of H 0over one period T :U (T )|ϕk =exp (−iǫk )|ϕk where ǫk is called the eigenpha.The temporal evolution of any state |ψ after n periods is |ψ(nT ) = k c k e −
inǫk |ϕk with c k = ϕk |ψ(0) .The
evolution of any quantity can be calculated using the basis of FS,for example:
p 2(nT ) = k,k ′
c k c ∗k ′e −in (ǫ
k −ǫk ′) ϕk ′|p 2|ϕk .(3)
The FS of the chaotic kicked rotor are well-known:they are on the average exponentially
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Fig.3–Eigenphasǫk of the evolution operator over one period for the doubly-kicked rotor,corre-sponding to the experimental conditions,versus the phaλ.Thefinite duration of the kicks(800ns) is taken into account.We have only plotted initially populated Floquet states having a significant weight:| ψ(0)|ϕk(λ0) |2>10−4(thin line),or>10−2(thick line).States appear(disappear)as their weights go above(below)the threshold.Note that Floquet states rapidly change when avoided crossings with other states are encountered.
localized in momentum space around a most probable momentum p k,with a characteristic localization lengthℓ[11,12].Such a localization–from which DL originates–is far from obvious and is cloly related to the Anderson localization in time-independent disordered
one-dimensional systems[13].
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The initial state is suppod to be localized in momentum space around zero-momentum, with a width much smaller than the widthℓof the FS(which is the ca in the experiment).
Hence,only FS with roughly|p k| ℓwill play a significant role in the dynamics;we shall call such states initially populated FS.Eq.(3)is a coherent sum over FS.However,as times goes
on,non-diagonal interference terms accumulate larger and larger phas.In a typical chaotic system,the phas will be uncorrelated at long enough times,leaving an incoherent sum over FS:
p2 ≈ k|c k|2 ϕk|p2|ϕk (4) This equation is valid once DL is ,for t>tℓ).How long does it take for the phas n(ǫk−ǫk′)to be scrambled?This can be simply estimated from the level spacing between initially populated FS,and turns out to be roughly tℓ=ℓT,while p2 saturates to
a value∝ℓ2.
A key point is to realize that,if r is clo enough to1,the quence of kicks is very sim-
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ilar to a periodic doubly-kicked rotor for r=1,except that the phaλbetween the two quences slowly drifts along the quence.A small part of the kick quence around the n th kick will em“instantaneously”periodic,with a pha differenceλ(n)=λ0+n(r−1), whereλ0is the initial pha between the two kick quences.The evolution operator of the quasiperiodic system from time(n−1)T to nT is thus given by the evolution operator U(λ)=exp(−ip2λT/ )exp(−iK cosθ/ T)exp(−ip2(1−λ)T/ )exp(−iK cosθ/ T)of the pe-riodic doubly-kicked rotor.The total evolution operator can thus be written as a product of the“instantaneous”evolution operators N n=1U(λ0+n(r−1)).For small enough|r−1|,the adiabatic approximation[14]applies:if the system is initially in a FS of U(λ0),it remains
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H.Lignier et al.:Quantum diffusion in the quasiperiodic kicked rotor5 in the corresponding FS of the“instantaneous”evolution operator asλchanges.This is il-lustrated in Fig.3,which shows the Floquet spectrum of the periodic doubly-kicked rotor, obtained by numerically diagonalizing U(λ).Whenλis varied the eigenergies evolve along complicated“spaghetti”,characteristic of quantum-chaotic systems,with a large number of avoided crossings(AC).
In the experiment,the initial state is a linear combination of FS.The adiabatic approxi-mation implies that it remains a linear combination of the“instantaneous”FS with the same weights(the phas evolve,but the squared moduli remain constant).As discusd above,the coherences between FS vanish after tℓ,which implies that Eq.(4)remains valid clo to the resonance,provided one us the“instantaneous”Floquet eigenbasis:
p2(λ) ≈ k|c k|2 ϕk(λ)|p2|ϕk(λ) .(5) The adiabatic approximation implies that there are no population exchanges among Floquet eigenstates.This means that the weights|c k|2are constant all along the evolution;they are evaluated at the initial time,corresponding to the initial wavefunction|ψ(0) and toλ=λ0: |c k|2=| ψ(0)|ϕk(λ0) |2,where|ϕk(λ0) are the“instantaneous”eigenstates corresponding to the initial time.The evolution of the average kinetic energy p2 is thus entirely due to the evolution of the“instantaneous”Floquet spectrum with the parameterλ,which evolves adiabatically from its initial v
alueλ0to thefinal valueλ=λ0+N(r−1)corresponding to the end of the kick quence.Thus,two distinct types of localization properties come into Eq.(5):tho of the initial Floquet spectrum,prent in the constants|c k|2,and tho of the “instantaneous”Floquet spectra,prent in the“instantaneous”eigenstates|ϕk(λ) .The|c k|2 coefficient gives important weights to the FS reprented in the initial distribution(that we shall call”initially populated”FS),which,becau the initial momentum distribution is very sharp compared to the localization length,are mostly localized around zero momentum.This limits the range of the sum in Eq.(5)to eigenstates|ϕk(λ0) centered at momenta|p k(λ0)|<ℓ. Asλmoves away fromλ0,the center p k(λ)of|ϕk(λ) moves away from zero momentum.As the eigenstates keep the same weight in the sum,this enlarges the momentum distribution. This is the mechanism at the origin of the quantum diffusion responsible for the growth of the average kinetic energy in the quasi-periodic ca.One thus expects p2(λ) to have a minimum atλ=λ0and to rapidly increa as the kicks are applied,that is,asλevolves. Sinceλ−λ0is proportional to r−1, p2 shall also prent a sharp minimum at r=1,which implies that the population in the zero-momentum class shall prent a sharp maximum at r=1,as experimentally obrved in Fig.2
What determines the linewidth of the resonance?A FS may considerably change becau of its AC with other FS,as tiny AC may be crosd diabatically.On the average,the typical variationλ−λ0of the
pha between kick ries for which|ϕk(λ) los the localization property of|ϕk(λ0) ,is the distance∆λc to the next AC.We immediately deduce that the full width∆f(in frequency)of the sub-Fourier line is such that:
2∆λc=(NT)∆f.(6) If the classical dynamics is regular,the Floquet eigenstates evolve smoothly with the parameter λ;a change inλof the order of one is thus required to significantly modify the Floquet states: 2∆λc≈1,which corresponds to∆f=1/(NT),that is,the Fourier limit.In a classically chaotic system,however,the level dynamics displays plenty of AC,e Fig.3,∆λc is then much smaller than unity,leading to sub-Fourier resonances.Eq.(6)also predicts the linewidth to be inverly proportional to the temporal length of the kick quence beyond the localization
6EUROPHYSICS LETTERS he sub-Fourier character is independent of N),as numerically obrved[10].The critical value∆λc depends on the detailed dynamics of the system.It can be roughly estimated by visual inspection of the quasi-energy level dynamics,Fig.3,to be of the order of0.05and a“factor10”sub-Fourier line,about twice the experimentally obrved factor(the additional experimental broadening is due to the transver profile of the lar mode leading to spatial inhomogeneities in K).
From Eq.(5)and the previous analysis,it is expected that p2 significantly increas from its minimum value atλ=λ0.AC between FS localized around the same momentum are rather large(this is what determines∆λc)whereas AC between states localized a distance L≫ℓapart in momentum space are typically much smaller and scale like exp(−L/ℓ),as a conquence of the exponential localization of the FS.There is thus a very broad distribution of AC widths,with a large number of tiny AC.A tiny AC typically extends over a smallλinterval and thus tends to produce small values of∆λc.The increa of p2 withλthus depends on the number of small AC encountered.In the prence of exponential localization,the AC density scales with size C as1/C for C→0,and p2(λ) − p2(λ0) shall behave like|λ−λ0| (3),producing the cusp experimentally obrved in the resonance line,Fig.2a.The large number of extremely small AC is responsible for the singularity of the sub-Fourier resonance line.Another conquence is the diffusive behavior obrved in the vicinity of the resonance, e Fig.1(b).Indeed, p2(λ) − p2(λ0) increas linearly with|λ−λ0|,itlf a linear function of time and of|r−1|.Thus,our model correctly predicts two non-trivial properties: p2 increas linearly with time and the corresponding diffusion constant is proportional to|r−1|. This is distinct from the prediction of Random Matrix Theory,which implies an increa of p2 initially quadratic inλ−λ0[15].
There will always be some degree of nonadiabaticity.Whatever small|r−1|is,tiny enough AC will be cr
osd diabatically.This puts a lower bound on the size of the AC effectively participating in the quantum dynamics and produces a rounding of the top of the sub-Fourier resonance line,too small to be en in the experiment after100kicks,but easily visible after 20kicks,Fig.2.
Our approach concentrates on the immediate vicinity of the resonance.What happens in the wings of the sub-Fourier line?Eq.(5)indicates that this depends on the residual correlation between|ϕk(λ0) and|ϕk(λ) for|λ−λ0|>∆λc.A quantitative answer to this question is difficult.However,it ems clear that the quantum diffusion constant does not exceed the classical one,which corresponds to vanishingly small interference terms.Random Matrix Theory tells us that this type of parametric correlation usually decays algebraically withλ−λ0.We thus propo the following ansatz:
|r−1|
p2(nT) = p2 DL+D cl
back怎么读(3)This basically comes from the fact that the probability to encounter a“bad”AC which expels the atoms from the zero-momentum region is proportional to the length of the interval|λ−λ0|.
