Table Of ContentAnomalous transport and quantum-classical correspondence
Bala Sundaram1 and G. M. Zaslavsky2
1 Department of Mathematics, CSI-CUNY, Staten Island, NY 10314
2 Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 and
Department of Physics, New York University, 2-4 Washington Place, New York, New York 10003
(28 February1998)
We present evidence that anomalous transport in the classical standard map results in strong
enhancement of fluctuations in the localization length of quasienergy states in the corresponding
quantum dynamics. This generic effect occurs even far from the semiclassical limit and reflects
the interplay of local and global quantum suppression mechanisms of classically chaotic dynamics.
Possible experimental scenarios are also discussed.
9
9 PACS numbers: PACS numbers: 03.65.Sq, 05.45.+b
9
1
n It is generally accepted that quantum mechanics sup- InthelimitoflargeK,itiswell-knownthattheclassi-
a presses chaotic classical motion. Numerous studies have calstandardmapdynamicsisdiffusiveintheactionvari-
J identifiedmechanismsforsuppressionwhich,forourpur- able p with diffusion constant < p2 > /t = K2/2≡ D .
ql
9
poses,fallintotwobroadclasses. Oneclassisexemplified Whatis alsoestablishedisthatawayfromthis limit, the
1
by ‘dynamical localization’ where quantum eigenstates diffusion constant exhibits peaks with changing K and
1 arelocalizedinanactionvariablelikemomentumdespite variesintime[8]. Recentresults[6,7]withimprovedpre-
v thedeterministicdiffusionexhibitedinthelimitingclassi- cision and detail show that there are values of K where
2 caldynamics[1]. Thiseffectisanalogoustotheinsulator the effective diffusion constant (over some time) can be
2
phase or Anderson localization in tight-binding models different from D by many orders of magnitude. Fig-
0 ql
[2]and,inone-dimension,thelocalizedwavefunctionhas ure 1 shows these windows of anomalous diffusion which
1
0 a characteristic exponential form. An associated scale is are not very narrowand exhibit sub-structure in each of
9 the localization length ξ which is related to the classi- the peaks.
9 cal diffusion constant,animportantfactfor the workre-
/ ∗
n portedhere. Therealsoexistsademarcatingtimescalet
y beyond which quantum and classical dynamics deviate.
d Despite a few counterexamples, dynamical localization
-
provides a ‘global’ mechanism for quantum suppression.
o
a The second class can be motivated by the fast devi-
h ation of wavepacket dynamics from the classical motion
c
even in the semiclassical limit. This effect is contained
:
v in the logarithmic break time τh¯ = ln(I/¯h)/Γ for the
i
X breakdownofquantum-classicalcorrespondence[3],rela-
tive to a characteristic Lyapunov exponent Γ and action
r
a I. However, the quantum dynamics retains features of
the classical dynamics for times beyond this estimate.
FIG.1. Diffusion coefficient D for the standard map, nor-
Thephenomenonof‘scarring’[4,5]referstoquantumco- 2
malized to Dql = K /2, as a function of K. The deviations
herences associated with local, unstable and marginally
fromthequasilinearpredictionspanawidewindowofK and,
stable, classical invariant structures. Though there are
asseenfromtheinset,canbelargeatselectvaluesofK,even
still many open questions, this effect has been numeri- far from thebifurcation point K =2π.
cally and experimentally observed in a wide variety of
strongly coupled quantum systems. It was shown that there is no diffusion at all and that
In this letter, we present evidence for another exam- the real random walk process corresponds to a L´evy-like
ple of ‘classical persistence’ related to anomalous diffu- wanderingcharacterizedbyatransportexponentµgiven
sion and accelerator modes in chaotic dynamics. Re- by < p2 >≈ tµ, where µ > 1. µ varies with K and has
cently, there has been considerable activity in the area the ‘normal’ value µ = 1 only at special values of K.
of anomalous classical transport [6,7,8]. What is of par- Superdiffusion occurs for µ > 1 leading to anomalous
ticular relevance to the present work is that even in an growth in momentum. It was explicitly shown [6] that
established paradigm like the standard map, described thiscouldresultfromanislandhierarchywithapeculiar
by qn+1 = qn +pn+1 and pn+1 = pn +Ksinqn , some topological structure near the ‘accelerator’ modes that
surprising new results were reported. appearatspecialvaluesofK. Nearthesevalues,classical
1
trajectoriessticktotheboundariesoftheseislandswhich, The computation uses fast-Fourier transforms to evolve
inturn,leadstoflightsofarbitrarylength. Thenetresult a plane wave initial state under repeated application of
is strong intermittency and superdiffusion. the single-kick evolution operator
Anothersignatureofanomaloustransportresultsfrom U =exp(−ip2/2h¯)exp(−iKcos(q)/¯h). (1)
the fact that a classical trajectory originating in a re-
gion of phase space recurs infinitely often. From the Theexpectationvaluesarethencomputedfromthetime-
set of recurrence times {τj}, j = 0,1,···, Poincar´e cy- evolved wavefunction. The value Kc = 6.908745··· was
cles {tj} = {τj+1 − τj} and an associated probability found [6] to be a criticalvalue in a wide parametricwin-
distribution function R(t) can be constructed. In the dow dominated by anomalous transport. We illustrate
∗
case of ‘perfect’ mixing and normal diffusion, R(t) is this regime by considering K = 6.905 and contrast it
strictly Poissonianwhereasit exhibits power-likeasymp- with K = 11, where the effective diffusion coefficient is
totics for anomalous kinetics. This classical characteris- actually below the quasilinear valueDql (seen in Fig. 1).
tic of anomalous transport is illustrated in Fig. 2, and The quantization scale h¯ =2πa, where several irrational
is important in the corresponding quantum dynamics as values of a in the range 0.04 − 1 were considered. In
well. the quasilinear limit, the quantum diffusion is expected
∗ 2
to saturate at t = αD /¯h where α was numerically
ql
shown to be 1/2 [1,11]. ξ measured in angular momen-
∗
tum quanta is ≈t .
FIG.2. Distribution of Poincar´e cycles. The inset clearly
displays thepower-law tail of thedistribution.
We raise several questions, arising from the classical
anomalous diffusion, in the corresponding quantum dy- FIG.3. Variation of <p2 > (normalized to the saturation
∗
namics of the standard map or, equivalently, the ‘delta- value Dqlt ) with the number of kicks t. Two representative
kickedrotor’,H =p2/2+KcosqP δ(t−n).The scale values of a, one large (Q,a≈.61803399) and the other small
n (SC,a ≈ 0.042340526), are shown for K = 11 (dashed lines)
length ξ for dynamical localizationis related to the clas- ∗
andK =6.905(solidlines). Notethatthelargeravaluewas
sical diffusion constant and even follows the variations
time-averaged to remove fluctuations that mask the trends.
in D with K [11]. Thus, we anticipate that the strong Theinsetshows alog-logplot (atlonger times,≈5000 kicks)
enhancement in D resulting from anomalous diffusion fora=0.042340526, K∗ =6.905(uppercurve)andK =11.0
should also be reflected in ξ. The nature of dynamical (lower curve). The corresponding slopes are 0.21 (solid line)
localization in these special windows is a related ques- and 0.06 (dashed line) respectively.
tion. Further, as small local structures in the classical
phase space lead to the superdiffusion, these issues di- Figure. 3 shows <p2 >/(D t∗) as a function of time
ql
rectlypertaintothe interplayoflocalandglobalaspects for two representative values of a. For large a, both val-
of quantum suppression. This idea is reinforced by the uesofK resultinstrongsaturationwithK =11havinga
∗
enhanced diffusion that ought to be visible in the quan- higher saturationvalue than K =6.905. However,with
∗
tumdynamicsevenfarfromthesemiclassicallimit,where decreasing a the enhanced diffusion for K = 6.905 be-
scarring is significant. Other authors have considered comes evident and there is a slowdown in growth in lieu
quantum dynamics in the presence of anomalous trans- of saturation (over the time considered). By contrast,
port [9,10] though in a different regime. We concentrate K = 11 appears to saturate at a value which is about
exclusively on the situation where the quantizationscale anorder of magnitude lower. This difference is consider-
2π¯h is much larger than the size of the islands. Thus, ably larger than earlier expectations based on quasilin-
any coherencesassociatedwith the islands wouldbe due ear analysis. The inset in Fig. 2 displays the existence
to ‘scarred’ quantum states. of power-law behavior in < p2 > for longer times at K∗
Consider the evolution of < p2 > with time t (mea- as compared with saturation at K = 11. Note that the
sured as the number of kicks) in the quantum dynamics slopepointstosub-diffusiveratherthantosuper-diffusive
fortwovalues ofthe classicalstochasticityparameterK. growthforwhichwehavenoclearexplanationatpresent.
2
A stronger signature appears on considering the vari- theneighborhoodofthecross-overtimefromexponential
ation of localization length ξ with K. It was shown [11] to power-law behavior shown in Fig. 1.
that ξ tracks the variations in the classical diffusion
coefficient D when plotted with respect to K =
q
Ksin(h¯/2)/(h¯/2). It is clear that this can be an im-
portantcorrectionforlargervaluesofh¯. Figure 4 shows
thevariationinξ,normalizedtothequasilinearestimate,
with respect to K . ξ is obtained by fitting an exponen-
q
tial to the time-dependent wavefunction for long times.
The peaks in the inset coincide with the classical accel-
erator modes at K = 2π and 4π and ξ computed at
different times reflect the same features. The more de-
tailedscandisplaysareasonablecorrespondencewiththe
results shown in Fig. 1, despite being far from the semi-
classical limit. It should be noted that the highest value
occurs at K ≈K defined earlier.
q c
FIG. 5. Lineshapes after 6000 kicks starting from
a plane wave initial condition. (a) corresponds to
Kq ≈Kc =6.90845··· while (b) corresponds to Kq =10.69.
a≈0.131267463 and thenoise in thelineshapes was reduced
byaveraging over 600 kicks.
Arguments for deviations of the localization length
from the simple estimates proceed from the long-
time averaged momentum distribution < f >=
n
Pm|ψm(n0)|2|ψm(n)|2, starting from a plane wave n =
n0 initialcondition. ψm(n) is the plane waverepresenta-
FIG. 4. Oscillations in the localization length ξ nor- tionoftheeigenfunctionwithquasienergyω . Theanal-
m
malized to the quasilinear estimate as a function of Kq. ysis of localization assumes that all quasienergy states
a ≈ 0.131267463 which makes the corrections to the classi-
are exponentially localized and that the fluctuations in
cal stochasticity parameter K significant. The inset shows
ξ are small. If this were true in the case of anomalous
peaks associated with the accelerator modes while the main
transport,theoscillationsinFig.4wouldbedirectlycor-
figure zooms in on the region considered in Fig.1. The inset
relatedwiththenumberofquasienergystatesparticipat-
also shows ξ computed after 3000 kicks (points) as well as
ing in the dynamics. To address this view, we computed
9000 (line) kicks. Note that the wavefunction averaged over
600 kicksis used to determine ξ. that participation ratio PR from the time averaged au-
tocorrelationfunction
We now consider the nature of the localized wave-
1
function when the classical transport is predominantly PR= lim |<ψ(T)|ψ(0)>|2 . (2)
T→∞T
anomalous. Figure 5 shows the evolved momentum dis-
tribution averagedover600kicks. Panel(a) corresponds As seen from Fig. 6, the only correlation with Fig. 4 oc-
to K ata peak in Fig.4 while (b) sits in the valley. Lo- cursinthevalleywherethelineshapeisexponentiallylo-
q
calizationdoes occur thoughthe characteristiclineshape calized. At the peaks, these results strongly support the
is strongly non-exponential for K in the neighborhood idea that anomalous transport results in strong fluctua-
q
of the peaks. The contrast with the clearly exponential tions in ξ among the individual quasienergy states. We
lineshapeshownin(b)isstriking. Itshouldbenotedthat have verified this by constructing the ‘entire’ spectrum
the central region in (a) also displays curvature and an of the quantum map U, in the momentum representa-
exponential fit results in a ξ which is considerablylarger tion, under conditions necessary for dynamical localiza-
than the quasilinear prediction. However, in case (b), tion. The average value of ξ does not reflect the large
the computed ξ is well-approximated by the quasilinear changes in D though dispersion ∆ξ exhibits large fluc-
estimate. Moving away from the peak in K decreases tuations. This view is being explored further at present.
q
the prominence of the shoulders in the lineshape though Aswearguedearlier,asimilarscenariomayapplytothe
thesearestillclearlyvisible. Wealsonotethattheshoul- interplay of scarring with dynamical localization.
dersdevelopoveratime(whichdependsonh¯)whichisin
3
be especially relevantin the weak noise limit and will be
considered in detail elsewhere.
Inconclusion,wehavepresentedevidenceoftheroleof
anomalous diffusion in enhancing quantum fluctuations.
Note that the standardmap is not ideal for isolating the
effects of anomalous transport. The kicked harmonic
oscillator, which results in the classical web map, is a
more interesting candidate where super diffusion domi-
natesthe classicaldynamics [6]. The absence oflocaliza-
tion in that system [14] and a recent suggestion for an
experimental realization in the context of ion traps [15]
FIG.6. Participation ratio as a function of Kq. Time av- makes this an attractive direction to pursue.
erages of the autocorrelation function over 1000(points) and The work of BS was supported by the National Sci-
6000(line) kicks are shown. a≈0.131267463. ence Foundationanda grantfromthe City Universityof
New York PSC-CUNY Research Award Program. GMZ
Theoretically, the classical flights of arbitrary length was supportedby U.S. Department of Navy Grants Nos.
result(forthe standardmap)fromahierarchyofbound- N00014-93-1-0218 and N00014-97-1-0426. The authors
ary layer islands [6]. The island area S , associated pe- are grateful to Mark Edelman for assistance with the
n
riodTn andLyapunovexponentΓn scaleas: Sn =λnSS0, Poincar´erecurrence plot.
Tn = λnTT0 and Γn = λnΓΓ0 with the generation n of the
island chain. λS < 1, λT > 1 and λΓ ≈ 1/λT. It is
straightforward to construct a semiclassical break-time
appropriate to anomalous transport using these scaling
rules.
Quantum flights are possible as long as S ≥ ¯h lead-
n
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4
