Pseudoclassical theory for fidelity of nearly resonant quantum rotors
Martina Abb,1Italo Guarneri,2,3and Sandro Wimberger1
1Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany
2Center for Nonlinear and Complex Systems, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy
3INFN, Sezione di Pavia, Via Bassi 6, 27100 Pavia, Italy 共Received 3 July 2009; published 29 September 2009兲
Using a semiclassical ansatz we analytically predict for the fidelity of␦-kicked rotors the occurrence of revivals and the disappearance of intermediate revival peaks arising from the breaking of a symmetry in the initial conditions. A numerical verification of the predicted effects is given and experimental ramifications are discussed.
DOI:10.1103/PhysRevE.80.035206 PACS number共s兲: 05.45.Mt, 37.10.Vz, 03.75.Dg Besides entanglement in multipartite systems, it is the
evolution of phases and the superposition principle which distinguishes a quantum from a classical system. Phase evo- lutions can be monitored in many ways, e.g., by correlation functions关1兴. A quantity which has gained interest in the last decade is fidelity 关2兴 defined as the overlap of two wave functions subjected to slightly different temporal evolutions.
The temporal evolution of this quantum fidelity crucially de- pends on evolving relative phases. For many-particle sys- tems, fidelity can be viewed as a Hilbert space measure to study quantum phase transitions 关3兴 and the regular-to- chaotic transition in complex quantum systems 关4兴. For single-particle evolutions fidelity was measured in electro- magnetic wave 关5兴 and matter wave 关6兴 billiards and with two different methods for periodically kicked cold atoms 关7,8兴.
The latter system is a realization of the quantum kicked rotor共QKR兲, the standard model for low-dimensional quan- tum chaos and the occurrence of dynamical localization关9兴.
Great interest in the QKR has reemerged in the study of its quantum resonant motion 关10–16兴 and related accelerator modes关17–20兴. These two regimes are far from the classical limit of the QKR and, therefore, governed by distinct quan- tum effects. Nevertheless, close to quantum resonance the system can be described 共pseudo兲classically with a new Planck’s constant, which is the detuning from the exact reso- nant value of the kicking period关10,18,21兴. For the quantum resonances, the underlying pseudoclassical model is com- pletely integrable and corresponds in good approximation to the dynamics of a classical pendulum关10,21兴.
In this Rapid Communication we apply well-known semi- classical methods to describe the behavior of fidelity close to the lowest-order quantum resonances of the QKR. We extend previous analytical results at exact resonance 关11兴 to a broader parameter regime, recently measured in experiments performed by Wu and co-workers 关8兴. The behavior of clas- sical 关22兴 and quantum fidelity 关23,24兴, in the case when classical motion is integrable, has mainly been addressed nu- merically so far, while our approach is both numerical and analytical. Also, the recurrences of fidelity found in关23兴 for the near-integrable regime of the kicked rotor are just pre- dicted for perturbative variations around small kicking strengths. Our results are more general, allowing, e.g., for strong changes of the fidelity parameter as long as the mo- tion remains nearly resonant. As expected, in the nearly reso-
nant regime, the temporal behavior of fidelity follows the behavior at exact resonance the longer, the smaller the de- tuning from resonance. Indeed, we show that the exactly resonant result predicted in 关11兴 by quantum calculations is retrieved by pseudoclassical analysis. At large times, how- ever, the exactly resonant fidelity and the nearly resonant one differ, as the latter displays recurrent revivals while the former steadily decays. Such revivals are approximately pe- riodic. Their period depends on the detuning from resonance and diverges as exact resonance is approached so this note- worthy phenomenon is unrelated to quantum resonant dy- namics. On the other hand, it is quite unexpected on classical grounds because the system is chaotic in the proper classical limit. Revivals of fidelity are thus a quantum effect and yet are explained by a 共pseudo兲classical analysis that relates them to periodic motion inside pseudoclassical resonant is- lands. Experimental possibilities to verify our predictions are discussed at the end of the Rapid Communication.
The dynamics of kicked atoms moving along a line in position space is described, in dimensionless units, by the Hamiltonian关10,25兴
H共t兲 =
2p2+ k cos共x兲
兺
t⬘=−⬁
+⬁
␦
共t − t⬘
兲, 共1兲where x is the position coordinate and p is its conjugate momentum. We use units in which ប=1 so the parameter
plays the role of an effective Planck’s constant; t is a con- tinuous time variable and t⬘
is an integer which counts the number of kicks. The evolution of the atomic wave function
共x兲 from immediately after one kick to immediately after the next is ruled by the one-period Floquet operator Uˆk= exp关−ik cos共xˆ兲兴exp共−i
pˆ2/2兲. Fidelity of the quantum evolution of a state
with respect to a change in the param- eter k from a value k1 to a value k2is the function of time t which for all integer t is defined byF共k1,k2,t兲 = 兩具Uˆk1 t
兩Uˆk2t
典兩2. 共2兲Periodicity in space of the kicking potential enforces conservation of quasimomentum

, which is just the frac- tional part of p thanks to ប=1. The atomic wave function decomposes into Bloch waves关10,18兴, which are eigenfunc- tions of quasimomentum,
共x兲=兰01d
eix冑
共
兲⌿共
兲, where PHYSICAL REVIEW E 80, 035206共R兲 共2009兲RAPID COMMUNICATIONS
1539-3755/2009/80共3兲/035206共4兲 035206-1 ©2009 The American Physical Society
= x mod共2
兲 and the factor
共
兲 is introduced in order to normalize ⌿ 共it weights the initial population in the Brillouin zone of width one in our units兲. The dynamics at any fixed value of
is formally that of a rotor on a circle parameterized by the angle coordinate
and described by the wave function ⌿. The Floquet propagator for the rotor is given by Uˆ,k= exp关−ik cos共
ˆ 兲兴exp关−i
共Nˆ+
兲2/2兴, where N=−idd. Fidelity Eq.共2兲 may then be written asF共k1,k2,t兲 =
冏 冕01d
共
兲具Uˆ,kt 1⌿兩Uˆ,kt 2⌿典冏
2 共3兲
so it results from averaging the scalar product under the integral sign over

with the weight
共
兲. Note that the rotor’s fidelity is the squared modulus of this quantity so the fidelity 关Eq. 共2兲兴 of atomic evolution does not coincide with the
average of the rotors’ fidelities, cf. 关11兴. Whenever
= 2
ᐉ 共ᐉ integer兲, the evolution is explicitly solvable 关10兴 and in particular the rotor’s fidelity is determined by关11兴兩具Uˆ,kt 1⌿兩Uˆ,kt 2⌿典兩2= J02共兩Wt兩
␦
k兲, 共4兲 where J0 is the Bessel function of first kind and order 0,␦
k = k2− k1, and 兩Wt兩=兩sin关
tᐉ共
−12兲兴csc关
ᐉ共
−12兲兴兩. If 2
− 1 is an integer then a so-called QKR resonance occurs and Eq. 共4兲 decays in time proportional to t−1. When
is close to a resonant value:
= 2
ᐉ+⑀
, the quantum rotor dy- namics may be viewed as the formal quantization of the pseudoclassical dynamics, defined by the map关10,18,21兴:It+1= It+ k˜ sin共
t+1兲,
t+1=
t+ It+
ᐉ +
mod共2
兲, 共5兲 using⑀
as the Planck’s constant, I =⑀
N, and k˜=⑀
k. It is thus possible to investigate the quantum fidelity in the limit of small⑀
by means of standard methods of semiclassical ap- proximation. In the limit⑀
→0, the physical parameter k is fixed so the pseudoclassical parameter k˜ →0. As a conse- quence, for sufficiently small⑀
, the pseudoclassical dynam- ics关Eq. 共5兲兴 is in the quasi-integrable regime even in cases when the classical kicked rotor dynamics is fully chaotic. It is dominated by the resonant islands at Ires=共2m+ᐉ兲
−
, with m as an integer. As we consider initial atomic states with a narrow distribution of momenta near p = 0, we may restrict ourselves to a portion of the pseudoclassical phase space that includes the one island which is located astride I = 0. We assume ᐉ=1 for simplicity. The pseudoclassical dynamics inside the resonant island is ruled, in continuous time, by the pendulum Hamiltonian 关26兴 H共
, I , k˜兲=12共I+
¯ 兲2+ k˜ cos共
兲, where
¯ ⬅
共
−12兲. We choose⌿共
兲=共2
兲−1/2 soUˆ,kt ⌿共
兲 ⬃ 1冑2
兺
s冏 ⬘ 冏
⬘=s⬘
−1/2
ei/⑀⌽s共,t兲−i共/2兲s, 共6兲
where
⑀
⬎0 is assumed with no limitation of generality. The sum is over all trajectories 共labeled by the index s兲 which start with I = 0 at time t = 0 and reach position
at time t. ⬘=
s⬘
are their initial positions, and the function whose de-
rivative is taken in the prefactor yields
at time t as a func- tion of position ⬘at time 0, given that the initial momentum
I⬘
= 0. Finally, the function⌽s共
, t兲=S共
,
s⬘
, t兲 is the action
of the sth trajectory and
sis the Morse-Maslov index关27兴.
We restrict ourselves to librational motion inside the stable island. The frequency of this motion decreases from
=冑
˜k at the island center to
= 0 at the separatrix. For times less than the minimal half period
/冑
˜, there is a single trajec-k tory in Eq.共6兲. Furthermore,
⌽s共
,t兲
t =冉
S共
, ⬘ ⬘,t兲
⬘共
,t兲
t +
S共
, ⬘,t兲
t冊
⬘=s⬘=
冉
S共
,t ⬘,t兲冊
⬘=
s⬘= − H共
s⬘
,0兲 = −
¯2 2+ k˜ cos共
s⬘
兲. 共7兲For t fixed and
⑀
→0 we use ⬘共
, t兲⬃
−
¯ t in
this equation so ⌽共
, t兲⬃−12
¯2t + k˜兰0tdt⬘
cos共
−
¯ t兲
= −12

¯2t + 2˜¯ksin共¯2t兲cos共
−¯t2兲. Replacing all this in Eq. 共3兲, we find for the rotor’s fidelity in the limit when⑀
→0 at constant t:兩具Uˆ,k
1 t ⌿兩Uˆ,k
2
t ⌿典兩2⬃
冏
21 冕
02d
eiB共,t兲cos关−共¯t/2兲兴冏
2= J02关B共

,t兲兴, 共8兲 where B共
, t兲=2␦k¯sin共¯t2兲. Since B共
, t兲⬇兩Wt兩 in Eq. 共4兲 for
= 2
ᐉ and
⬇0.5, we see that the pseudoclassical approxi- mation along with the pendulum approximation well repro- duce the exact quantum calculation关Eq. 共4兲兴 when⑀
→0 at fixed t. In the final step of integrating over quasimomenta to find the fidelity for atoms 共as distinct from fidelity for ro- tors兲, the pseudoclassical approximation plays no role since the particle’s dynamics, unlike the rotor’s, does not turn pseudoclassical in the limit⑀
→0 关18兴. Replacing Eq. 共4兲 into Eq. 共3兲 and computing the integral with a uniform distribu- tion of
in关0,1兲 shows that the complete fidelity 关Eq. 共3兲兴 saturates to a nonzero value in the course of time 关11兴.Next we address the asymptotic regime where
⑀
→0 and t⑀
1/2⬃const. To this end, the exact solution of the pen- dulum dynamics is needed in order to compute actions; how- ever, some major features of fidelity are accessible by ex- ploiting the harmonic approximation of the pendulum Hamiltonian. We replace the pendulum by the quadratic Hamiltonian H共I,
兲=1/2共I+
¯ 兲2+
2/2
2, where
=冑
˜k and a shift of
by
is understood. Except at exact multiples of the period, there is one harmonic oscillator trajectory in the sum in Eq. 共6兲; moreover, Maslov indices do not depend on the trajectory. Straightforward calculations yield ⬘共
, t兲=sec共
t兲关
−
¯
−1sin共
t兲兴 and
⌽共
, t兲=
¯
关sec共
t兲−1兴−共
−1
¯2+
2兲tan共
t兲/2, and soABB, GUARNERI, AND WIMBERGER PHYSICAL REVIEW E 80, 035206共R兲 共2009兲
RAPID COMMUNICATIONS
035206-2
具Uˆ,kt 1⌿兩Uˆ,kt 2⌿典 ⬃ ei共t兲
2
冑兩cos共
1t兲cos共
2t兲兩冕
−
d
⫻ei/2⑀兵A共t兲2+C共t兲¯2−2¯B共t兲其, 共9兲
where A共t兲=
2tan共
2t兲−
1tan共
1t兲,B共t兲= sec共
2t兲−sec共
1t兲 and C共t兲=
2−1tan共
2t兲−
1−1tan共
1t兲.共t兲 is a phase factor accumulated by the Maslov indices and it just depends on time, rendering it irrelevant for our present purposes. We next insert Eq.共9兲 into Eq. 共3兲 and choose for
共
兲 a uniform distribution in some interval 关12− b ,12+ b兲, with 0ⱕbⱕ1/2. It is necessary to assume that b is smaller than the halfwidth of the pseudoclassical resonant island be- cause the harmonic approximation we have used is valid only inside that island. ThenF共k1,k2,t兲 ⬃ 1
16
2b2
2兩cos共
1t兲cos共
2t兲兩⫻
冏 冕−
d
e−i/2⑀⌳1共,⑀,t兲冕
−bb d
¯ e−i/2⑀⌳2共¯,,⑀,t兲冏
2,共10兲
where ⌳1共
,⑀
, t兲=关A共t兲−B2共t兲C共t兲−1兴
2 and ⌳2共
¯ ,
,⑀
, t兲=关

¯冑C共t兲−B共t兲C共t兲−1/2
兴2. As ⌳2⬃⑀
−1/2 in the limit when⑀
→0 and t冑⑀
⬃const, the limits in the
¯ integral in Eq. 共10兲 may be taken to⫾⬁:冕
−bb
d

¯ e−i/2⑀⌳2共¯,,⑀,t兲⬃ 共2
兲1/2⑀
1/2C共t兲−1/2e−i/4.Due to this approximation, Eq. 共11兲 below is valid in the asymptotic regime where
⑀
is small compared to b2. The remaining
integral is dealt with similarly because the pref- actor of
2 in⌳1is⬃⑀
−1/2. Thus finallyF共k1,k2,t兲 ⬃
⑀
216
2b2兩C共t兲A共t兲 − B共t兲2兩兩cos共
1t兲cos共
2t兲兩=
⑀
2
1
28
2b2兩4
1
2−
+2 cos共
−t兲 −
−2cos共
+t兲兩, 共11兲 where
⫾=
1⫾
2. Singularities of this expression are arti- facts of the approximations used in evaluating the integrals in Eq. 共10兲, which indeed break down when the divisor in Eq. 共11兲 is small compared to⑀
. However, they account for the periodic “revivals” that are observed in the fidelity at large times, with the beating period T12= 2
/兩
−兩 关Fig.2共a兲兴.With a quite narrow distribution of

, however, fidelity is at long times dominated by the “resonant” rotors 共
= 0 or
= 1/2, respectively兲, and then revivals occur with the period T12/2 共Fig.1兲. Indeed, with the purely resonant
, Eq. 共10兲 yieldsF共k1,k2,t兲 ⬅ Fres共k1,k2,t兲
⬃
⑀
2
1
兩
2cos共
1t兲sin共
2t兲 −
1cos共
2t兲sin共
1t兲兩, 共12兲 which has singularities in time with the mentioned periodic- ity of T12/2. This behavior of resonant rotors has a simple qualitative explanation. As the initial state of the rotor corre- sponds to momentum I = 0, at that value of quasimomentum 共
=12兲 the stationary-phase trajectories of the two harmonic oscillators, which were started at I = 0, exactly return to I = 0 whenever time is a multiple of the half period T12/2 and0 200 400 600 800 1000
t
0 0.2 0.4 0.6 0.8
F(t)
0 200 400 600 800
t 0
0.2 0.4 0.6 0.8 1
F(t)0.4
FIG. 1. 共Color online兲 Fidelity as predicted by Eq. 共12兲 because of the singularities of the analytical formula the curve is folded with normalized Gaussians with a standard deviation of t⬇6 kicks 共solid black line兲 and numerical data 共gray/green curve兲 for k1= 0.8, k2
= 0.6, and detuning⑀=0.01 from −⑀=2. In the inset, the non- smoothed result关Eq. 共12兲兴 is shown.
200 400 600 800 1000
0 0.1 0.2 0.3 0.4 0.5
F(t)
200 400 600 800 1000
t
0 0.1 0.2 0.3 0.4 0.5
F(t)
(a)
(b)
FIG. 2. 共Color online兲 Same as in Fig. 1 for an ensemble of 5000 equidistantly chosen rotors 共solid gray/green lines兲 with a width of 共a兲 ⌬=0.05 共or ⌬¯⬇0.31兲 around the resonant value covering half the width of the resonance island in the phase space induced by Eq. 共5兲 and 共b兲 ⌬=1 covering the full phase space, compared with the smoothed共see caption of Fig.1兲 version of Eq.
共11兲 共solid black lines兲. In 共a兲 the intermediate revival peaks ob- served in Fig.1disappear as predicted by Eq.共11兲. The dashed line in共a兲 reproduces the smoothed analytic formula from Fig.1. For distributed over the full Brillouin zone in共b兲, the revivals are barely visible since the average includes many nonresonant rotors per- forming rotational motion in phase space, which is not decribed by our theory valid just for the librational island motion.
PSEUDOCLASSICAL THEORY FOR FIDELITY OF NEARLY… PHYSICAL REVIEW E 80, 035206共R兲 共2009兲 RAPID COMMUNICATIONS
035206-3
so fully contribute to fidelity in spite of their angles being different by
in the case of odd multiples. At
⫽0 this symmetry is lost. Comparing numerical data 共obtained by repeated application of the Floquet operator to the initial wave function兲 with the analytical predictions we find excel- lent agreement. We observe the expected peak structure of the revivals in Fig. 1 and the loss of intermediate revival peaks at T12/2 in Fig. 2共a兲. The time scale on which the revivals occur is proportional to⑀
−1/2and of crucial impact to experimental measurements: conservation of coherence has been shown for up to 150 kicks 共see 关19兴兲 with cold atoms, making an observation of the revivals for reasonable⑀
ⱗ0.01 possible. Earlier realizations of the QKR were implemented using cold atoms关7,12,14兴 with broad distribu- tions in quasimomentum. Nowadays, much better control of quasimomentum is provided by using Bose-Einstein conden- sates 共see 关13,16兴兲, which allow for a restriction in
up to 0.2%共as achieved in 关16兴兲 of the Brillouin zone. This would allow us to verify our results by conveniently reducing the intervals in quasimomentum and thus retracing the revivals with period T12/2 to the exactly resonant and the revivals with period T12 to the near-resonant rotors. There exists an interesting second possibility to measure the transition from Eq.共12兲 to Eq. 共11兲 with just cold atoms since the
¯ we usescales with the kicking period, i.e.,

¯ =
共
− 1/2兲. Due to this scaling, the limit
→0 共automatically implying also⑀
→0, cf. 关14兴兲 permits a measurement of Eq. 共12兲 even with an ensemble of cold atoms whose quasimomenta occupy the full Brillouin zone. Also the momentum selective interfero- metric measurements of fidelity关8兴 allow us to select narrow intervals of quasimomenta and hence would permit to check our predictions experimentally.To summarize, we predict fidelity revivals in the QKR close to quantum resonance using a semiclassical ansatz. Our results are supported by numerical data showing the same characteristic revival peaks. Every second peak vanishes once the symmetry of the initial quasimomentum distribution on the resonance island is broken. This makes for a surpris- ing transition that could be measured with both cold and ultracold atoms owing to the scaling of

¯ or the use of mo- mentum selective methods as described in the previous para- graph.Support by the Excellence Initiative through the Global Networks Mobility Measures and the Heidelberg Graduate School of Fundamental Physics共DFG Grant No. GSC 129/1兲 and by a Short Visit Grant共DAAD兲 is acknowledged.
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