arXiv:quant-ph/0105086 v2 14 Dec 2001TheÆ-fu

nctio

n-kickedrotor:

Momentum

di�usio

nandthequantum-classic

alboundary

LA-UR-01-1909

Tanmoy

Bhatta

chary

a,1Salm

anHabib,1Kurt

Jacobs,1andKosu

keShizu

me2

1T-8,Theoretica

lDivisio

n,MSB285,LosAlamosNatio

nalLaboratory,LosAlamos,NewMexico

87545

2University

ofLibrary

andInform

atio

nScience,1-2

Kasuga,Tsukuba,Iba

raki305,Japan

(Dated

:Decem

ber

17,2001)

Weinvestigate

thequantum-classical

transition

inthedelta-k

ickedrotor

andtheattain

ment

oftheclassical

limit

interm

sof

measu

rement-in

duced

state-localization

.It

ispossib

leto

study

thetran

sitionby�xingtheenviron

mentally

induced

distu

rbance

atasuÆcien

tlysm

allvalu

e,and

exam

iningthedynam

icsas

thesystem

ismademore

macroscop

ic.When

thesystem

actionis

relativelysm

all,thedynam

icsisquantum

mech

anical

andwhen

thesystem

actionissuÆcien

tlylarge

there

isatran

sitionto

classicalbehavior.

Thedynam

icsoftherotor

intheregion

oftran

sition,

characterized

bythelate-tim

emom

entum

di�usion

coeÆ

cient,can

bestrik

ingly

di�eren

tfrom

both

thepurely

quantum

andclassical

results.

Rem

arkably,

theearly

timedi�usive

behavior

ofthe

quantum

system

,even

when

di�eren

tfrom

itsclassical

counterp

art,isstab

ilizedbythecon

tinuous

measu

rementprocess.

This

show

sthat

such

measu

rements

cansucceed

inextractin

gessen

tiallyquantum

e�ects.

Thetran

sitionregim

estu

died

inthispaper

isaccessib

lein

ongoin

gexperim

ents.

I.INTRODUCTION

Exploration

sof

thetran

sitionfro

mquantum

toclas-

sicalbehaviorin

nonlin

eardynam

icalsystem

scon

sti-tute

oneofthefro

ntier

areas

ofpresen

ttheoretical

re-sea

rch.Therecen

tlyrea

lizedpossib

ilityofcarry

ingout

controlled

experim

ents

inthisregim

e[1,

2,3]has

added

greatlyto

theim

petu

sfor

increa

singourunderstan

ding

ofthistra

nsition

,quite

asidefrom

theundoubted

lyfun-

dam

entalim

portan

ceofthesubject.

Thepointat

issueis

notso

much

thestatu

sofform

alsemiclassical

approx

ima-

tionsinthesen

seoftakingthemathem

aticallim

it�h!

0,butadescrip

tionandundersta

ndingoftheprocesses

that

takeplace

inactu

alexperim

ents

onthedynamical

be-

havior

ofobserved

quantum

system

s.Quantum

decoh

er-ence

andcondition

edevolution

arisingasacon

sequence

ofsystem

-enviro

nmentcou

plin

gsandtheact

ofobserva-

tionprov

ideanatu

ralpathway

totheclassica

llim

itas

has

been

dem

onstra

tedquantitatively

inRef.

[4](see

alsoreferen

ces[5])

andisrev

iewed

inthenextsection

.Com

parison

ofthedynam

icsofclo

sedquantum

and

classicalsystem

sin

explicit

exam

ples

hasshow

navari-

etyofbehaviors.

Attheoneextrem

e,there

exist

system

swhere

quantum

andclassical

averages

track

eachoth

erfor

longtim

es[6],

andattheotherextrem

e,there

aresys-

temsin

which

theaverag

esbrea

kaw

ayfro

meach

other

decisively

at�nite

times

[7].Since

even

thissecon

dclass

ofsystem

sdoattain

classicaldynam

icswhen

theaction

issuÆcien

tlylarge,they

undergo

aclear

transition

froma`quantum'to

a`cla

ssical'behavior

astheparam

etersof

thesystem

contro

llingits

actionare

varied

,andare

ofparticu

larinterest

toastu

dyofthequantum-classical

transition

.Thequantum

delta

-kick

edrotor

(QDKR)is

awell-stu

died

example

ofthisclass

ofsystem

anditis

ourpurpose

here

toinvestig

atethequantum-classical

transition

inthis

system

with

inthefra

mew

ork

based

onthetheory

ofcon

tinuousmeasurem

entpresen

tedin

Ref.

[4].Aswill

bedem

onstra

tedbelow

,thequantum-

classicaltra

nsition

regim

epossesses

somerem

arkablefea-

tures

which

are

well

with

inthereach

ofpresen

t-day

ex-

perim

ents.

Oneof

therem

arkable

features

oftheQDKR

which

distin

guish

esitfrom

itsclassical

counterp

artisthebe-

havior

ofits

latetim

emom

entum

di�usion

constan

t,Dp�

Lim

t!1dhp

2(t)i=dt.

Intheclassical

case,this

quantity

attainsacon

stantvalu

e,D

cl,whereas

forthe

QDKR,it

fallsto

zero.Thislatter

phenom

enon

isterm

eddynam

icallocalization

[8].In

addition

,theQDKR

alsodisp

laysanontriv

ialvariation

oftheearly

-timeintrin

sicmom

entum

di�usion

coeÆ

cient,D

init ,

asafunction

ofthesto

chasticity

param

eter[9],

andstron

gresistan

ceto

decoh

erence

[10,11,

12,13,

14,15].

Thislast

feature

re-lates

tothefact

that

itispossib

lefor

extern

alnoise

and

decoh

erence

tobreak

dynam

icallocalization

inthesen

sethat

thelate-tim

equantity

Dp 6=

0,neverth

elessDpre-

main

ssm

allandfar

fromits

classicalvalu

eunless

verystron

gnoise

strength

isem

ployed

.Finally,

theQDKR

possesses

theadded

attractionthat

itcan

bestu

died

via

atomic

optics

experim

ents

utilizin

glaser-co

oledatom

sandallow

ingsom

edegree

ofcon

trolof

thecou

plin

gto

anextern

alenviron

mentvia

spontan

eousem

issionpro-

cesses[2].

Thissystem

consists

ofnoninteractin

gatom

sin

amagn

eto-optical

trap(M

OT);

onturningon

ade-

tuned

standingligh

twave

with

wavenumberkL=2�=�,

theatom

sscatter

photon

sresu

ltinginan

atomicmom

en-

tum

kick

of2�hkLwith

everysuch

event.

Intern

aldegrees

offreed

omcan

beelim

inated

since

thelarge

detu

ningpre-

cludes

population

transfer

betw

eenatom

icstates.

The

resultin

gdynam

icsisrestricted

tothemom

entum

space

oftheatom

sandcan

bedescrib

edbysim

ple

e�ective

Ham

iltonian

s,theHam

iltonian

fortheQDKRbein

gone

oftheim

plem

entab

leexam

ples.

Ourmain

consid

erationisasystem

aticanaly

sisof

the

quantum-classical

transition

induced

via

position

mea-

surem

entfor

thenon-dissip

ativedelta-k

ickedrotor

us-

ingthelate-tim

emom

entum

di�usion

rateas

adiagn

os-tic

tool.

Even

though

quantum

dynam

icallocalization

typically

suppress

latetim

edi�usion

(Dp=

0at

latetim

es),we�ndthat

asm

allmeasu

rementstren

gth(i.e.,

weak

noise)

canstab

ilizean

intrin

sicallyquantum

early

2

time e�ect (the enhancement of the initial quantum dif-fusion rate predicted by Shepelyansky [16]) and producea nonzero �nal di�usion rate much larger than the clas-sical value. The predicted e�ect is large and should beobservable in present experiments studying the quantum-classical transition in the QDKR. We also comment onthe behavior of the di�usion coeÆcient near a quantumresonance and on the nontrivial nature of the approach tothe classical noise-dominated value for the di�usion rateas the noise induced by the measurement is increased.Our theoretical results have been con�rmed recently bya more detailed analysis relevant to speci�c experimentalrealizations [17].The plan of the paper is as follows. In Section II,

we provide a short review of recent work on continuousmeasurement and the quantum-classical transition, mov-ing on to the speci�c case of the QDKR in Section III.In Section IV, we explain how the late-time di�usion co-eÆcient provides a window to investigate the quantum-classical transition in this system and in Section V wedescribe the detailed nature of the transition. Section VIis a short conclusion.

II. CONTINUOUS MEASUREMENT AND THE

QUANTUM TO CLASSICAL TRANSITION

Macroscopic mechanical systems are observed to obeyclassical mechanics. However, the atoms which ulti-mately make up the macroscopic systems certainly obeyquantum mechanics. Since classical and quantum evolu-tion are di�erent, the question of how the observed classi-cal mechanics emerges from the underlying quantum me-chanics arises immediately. This emergence, referred toas the quantum to classical transition, is particularly cu-rious in the light of the fact that classical mechanical tra-jectories are governed by nonlinear dynamics that oftenexhibit chaos, whereas the very concept of a phase spacetrajectory for a closed quantum system is ill-de�ned, andany signatures of chaos are, at best, indirect.If quantummechanics is really the fundamental theory,

then one must be able to predict the emergence of (the of-ten chaotic) classical trajectories by describing a macro-scopic object with suÆcient realism, but fully quantummechanically. SuÆcient ingredients to perform such adescription have now been found [4, 5]. The solution hasinvolved the realization that all real systems are subjectto interaction with their environment. This interactiondoes at least two things. First, it subjects the systemto noise and damping [18, 19] (as a consequence all realclassical systems are subject to noise and damping - evenif very small), and second, the environment provides ameans by which information about the system can beextracted (e�ectively continuously if desired), providinga measurement of the system [20].Since observation of a system is essential in order that

the trajectories followed by that system can be obtainedand analyzed (this being just as true classically as quan-

tum mechanically), it may be expected that this processmust be included in the description of the macroscopicsystem in order to correctly predict the emergence of clas-sical trajectories. An example of an environment thatnaturally provides a measurement is that of the (quan-tum) electromagnetic �eld with which the system is sur-rounded. Monitoring this environment consists of focus-ing the light which is re ected from the system, allowingthe motion to be observed. If the environment is notbeing monitored, then the evolution is simply given byaveraging over all the possible motions of the system.(Classically this means an average over any uncertaintyin the initial conditions, and over the noise realizations.)It has now been established quantitatively that contin-

uous observation of the position of a single quantum me-chanical degree of freedom, is suÆcient to correctly pre-dict the emergence of classical motion when the action ofthe system is suÆciently large compared to �h. In particu-lar, inequalities have been derived involving the strengthof the environmental interaction and the Hamiltonian ofthe system, which, if satis�ed, will result in classical mo-tion [4] (see also [5]). These inequalities, therefore, re�nethe notion of what it means for a system to be macro-scopic.A detailed explanation of the reason that continu-

ous measurement induces the qantum-to-classical tran-sition may be found in Refs. [4, 21, 22]. To recapitulatethat analysis, the emergence of classical behavior arisesfrom simultaneous satisfaction of two counteracting con-straints. The �rst is that a suÆciently strong observationprocess is needed to maintain localization of the particlein phase space, and, thereby, produce classical motionof the centroid of the Wigner function [4] from Ehren-fest's theorem. On the other hand, even for a localizeddistribution, a strong measurement introduces noise intothe system, and this needs to be bounded to a micro-scopic scale. These constraints are satis�ed for an everincreasing range of measurement strengths as the systemparameters become large enough to make the quantumunit of action, i.e., �h, negligible. Thus, when systemsare suÆciently macroscopic, they exhibit classical motionwith a negligible amount of irreducible quantum noise, anoise that, in practice, is always swamped by classicalmeasurement uncertainties and tiny environmental dis-turbances.As mentioned above, these arguments can be codi�ed

into a set of inequalities. First, to maintain enough local-ization to guarantee that, at a typical point on the tra-jectory, one has for the force F (x), hF (x)i � F (hxi), asrequired in the classical limit, the measurement strength(de�ned precisely in the next section), k, must stop thespread of the wavefunction at the unstable points [23],@xF > 0:

8�k�����@

2xF

F

����rj@xF j2m

: (1)

Second, as noted already, a large measurementstrength introduces noise into the trajectory. Demand-

3

ing that averaged over a characteristic time period of thesystem, the change in position and momentumdue to thenoise are small compared to those induced by the classi-cal dynamics, it is suÆcient that, at a typical point onthe trajectory, the measurement satisfy

2 j@xF j�s

� �hk� j@xF j s4

; (2)

where s is the typical value of the action [24] of the sys-tem in units of �h and � ranges from zero to unity andcharacterizes the eÆciency of the measurement (for themeasurements considered in this paper, � = 1). Obvi-ously as s becomes large, this relationship is satis�ed foran ever larger range of k, and this de�nes the classicallimit.This understanding of the quantum-to-classical tran-

sition in terms of quantum measurement which hasemerged within the last ten years is, of course, com-pletely consistent with the mechanism usually referredto as decoherence, since averaging over the results of themeasurement process gives the same evolution as an in-teraction with an unobserved environment (in particular,the environment through which the system is being moni-tored) in which the environment is traced over. Thus, thetreatment in terms of measurement is actually a micro-scopic analysis of the process of decoherence. However,examining the measurement process allows us to obtainan understanding of why it is that decoherence causesclassical motion to emerge, and also allows us to realizethe trajectories themselves, something that is impossi-ble when the environment is merely traced over. Theknowledge of the dynamics of the individual quantumtrajectories provides new information not available froma traditional decoherence analysis and, as we show below,this more microscopic information can be helpful in un-derstanding phenomena even at the level of expectationvalues.With this understanding of the mechanismof the emer-

gence of classical motion, it becomes pertinent to ask thequestion, how does the dynamics of a particular systemchange as it is made more macroscopic? That is, whathappens to the dynamics as it passes through the transi-tion fromquantummotion (when its action is very small),to classical motion (when its action is suÆciently large).In the following sections we address this question for thedelta-kicked rotor.

III. QDKR UNDER CONTINUOUS

OBSERVATION

The Hamiltonian controlling the evolution of theQDKR is

~H(p0; q0; t0) =1

2

p02

m+ �0cos(2kLq

0)Xn

Æ(t0 � nT ): (3)

It is more convenient to study this system by rescalingvariables, in terms of which the new Hamiltonian be-

comes [15]

H(p; q; t) =1

2p2 + � cos q

Xn

Æ(t � n): (4)

where q and p are dimensionless position and momentum,satisfying [q; p] = i�k, for a dimensionless �k = 4�hk2LT=m,and � is the scaled kick strength. Following the exper-imental situation, we will use a typical value of � = 10and open boundary conditions on q. The value of �k is ameasure of the system action relative to �h. When �k issmall, the system action is large compared to �h and thesystem can be considered to be e�ectively macroscopic.Conversely, when �k is large, the system is microscopicand will behave quantum mechanically under weak envi-ronmental interaction. The ability to change the systemaction relative to �h (i.e., changing �k) allows a systematicexperimental study of the quantum-classical transition.The parameter ranges we have studied numerically be-low have been chosen to be more or less typical of thoseutilized in present experiments. Finally, we note that thescaling required to bring the equation to this dimension-less form hides the fact that increasing the dimensionlesse�ective Planck's constant �k at �xed � involves increas-ing the period between the kicks and decreasing theirstrength | thus, all else being equal, this system is ex-pected to behave more clasically under observation whenthe kicks are harder and spaced closer in time.The e�ect of random momentum kicks due to spon-

taneous emission can be modeled approximately by aweak coupling to a thermal bath [25]. In current exper-iments the atom interacts with a standing wave of laserlight leading to a (sinusoidal) spatial modulation of thebath coupling. This modulation in turn produces a cor-responding spatial variation in the di�usion coeÆcient ofthe Master equation describing the evolution of the re-duced density matrix for the position of the atom. Inthe language of continuous measurements, spontaneousemission can be regarded as a measurement of a func-tion of the position x, namely cos(x). While one cancertainly study this class of measurement processes, inthis paper we will study the case of continuous measure-ments of position. In general, as far as the study of thequantum-classical transition is concerned, the exact na-ture of the measurement process is not expected to beimportant provided that it yields suÆcient informationto enable the observer to localize the system in phasespace.A continuous measurement of position is described by

the (nonlinear) stochastic Schr�odinger equation [26]

j ~ (t+ dt)i =

�1� 1

�k

�iH + �kkq2

�dt

+4kqR(t)dt] j ~ (t)i; (5)

where the continuous measurement record obtained bythe observer is

R(t)dt = hq(t)idt + dW=p8k; (6)

4

dW being Wiener noise. The noise represents the in-herent randomness in the outcomes of measurements.Aside from the unitary evolution, this equation describeschanges in the system wave function as a result of mea-surements made by the observer. The parameter k char-acterizes the rate at which information is extracted fromthe system [27].Averaging over all possible results of measurements

leads to a Master equation describing the evolution ofthe reduced density matrix for the system. This Masterequation has the form

_� = � i

�k[H; �]� k[q; [q; �]] (7)

where the di�usion coeÆcient is given by Denv = k�k2,with Denv the di�usion constant describing the rate atwhich the momentum of a free particle would di�usedue to a thermal environment. From the relationshipbetween k and Denv given above, we see that �k deter-mines the relationship between the information providedabout q, and the resulting momentum disturbance. Theimportant point to note for the following is that for a�xed Denv , the measurement strength is reduced as �k in-creases (conversely, for �xed measurement strength Denv

increases with �k). The stochastic Schr�odinger equation(5) is said to represent an unraveling of the Master equa-tion (7) with averages over the Schr�odinger trajectoriesreproducing the expectation values computed using thereduced density matrix �.All dynamical systems that one can build in the lab-

oratory are necessarily observed in order to investigatetheir motion. For suÆciently small �k, and a reasonablevalue of k, the measurement maintains localization ofthe wavefunction, while generating an insigni�cantDenv .For suÆciently localized wavefunctions, expectation val-ues of products of operators are very close to the prod-ucts of the expectation values of the individual operators.Ehrenfest's theorem then implies that the classical equa-tions of motion are satis�ed. On the other hand, forsuÆciently large �k (and the same insigni�cant Denv ), kis suÆciently small that the quantum dynamics is essen-tially preserved. It is the detailed nature of this transitionthat we now wish to investigate.

IV. QUANTUM-CLASSICAL TRANSITION

AND THE LATE-TIME DIFFUSION

COEFFICIENT

Our strategy in examining the quantum to classicaltransition is to �x the level of noise (i.e., the di�usion co-eÆcient Denv ) resulting from the measurement at somesuÆciently small value, so that classical behavior is ob-tained for small �k and then to study systematically howquantum behavior emerges as �k is increased. The keydiagnostic is the behavior in time of the expectationvalue of momentum squared, hp2(t)i. Previous studies of

the QDKR have shown that in the presence of decoher-ence/measurement, dynamical localization is lost in thesense that there exists a non-zero late-time momentumdi�usion coeÆcient, but that this di�usion coeÆcient isnot necessarily the same as the intrinsic classical di�u-sion coeÆcient Dcl [10, 11]. The existence of this late-time di�usion coeÆcient provides a particularly conve-nient means of characterizing and studying the quantum-classical transition: The late-time di�usion coeÆcient isan unambiguous, theoretically well-de�ned quantity and,moreover, is also measurable in present experiments.In the classical regime (when �k is suÆciently small),Dp

attains the classical value (Dp ' Dcl ) withDenv � Dcl .This should not be confused with the noisy classical limitwhich arises under strong driving by the noise (largeDenv ) in which case Dp ' Dcl + Denv [28]. At suf-�ciently large values of �k, on the other hand, one expectsquantum e�ects to be dominant and therefore one shouldvery nearly obtain dynamical localization (Dp ' 0).Thus, one of the key questions is the behavior of Dp

as a function of �k in the transition regime in betweenDp ' Dcl and Dp ' 0, and the variation of Dp as afunction of decoherence or measurement strength as setby the value of Denv .Analytical investigation of the transition regime is

made diÆcult by the nonlinearity of the dynamical equa-tions and the lack of a small parameter in which to carryout a perturbative analysis [29]. The nonlinearity re-sults from the fact that the measurement record R(t)drives the evolution of the wave function in equation (5)and, at the same time, is itself dependent on the expec-tation value of the position. We solved the nonlinearSchr�odinger equations (5) numerically.Numerical investigation of the dynamics of the envi-

ronmentally coupled QDKR requires the solution of astochastic, nonlinear partial di�erential equation. In or-der to obtain the desired ensemble averages, one needs toaverage over many noise realizations for each set of pa-rameters considered. We implemented a split-operatorspectral algorithm on a parallel supercomputer to solveEq. (5), and then averaged over the resulting trajecto-ries to obtain the solution to Eq. (7). (Direct solutionof Eq. (7) to the desired accuracy is still a major chal-lenge for supercomputers.) Grid sizes for computing thewave function ranged from 1 � 64K depending on thevalue of Denv and �k. One thousand realizations were av-eraged over for each data point. Our numerical resultsfor the transition regime contain a variety of interestingphenomena which we discuss below.

V. THE STRUCTURE OF THE

QUANTUM-CLASSICAL TRANSITION

The generic behavior of quantum trajectories is thefollowing. If we construct a trajectory starting with aminimumuncertainty wavepacket, the nonlinearity of thesystem dynamics tends to spread it out. However, the

5

05

1015

2025

300

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

t

⟨ p2⟩

Classical

Quantum

Quantum

with D

env =0.1

FIG

.1:

Thespread

inmom

entum,measu

redbyhp

2i,as

afunction

oftim

efor

thenoiseless

classicalsystem

,thenoiseless

quantum

system

with

�k=

3,andthesam

equantum

system

with

Denv

=0:1.

Toobtain

thelatter

theMaster

equation

has

been

solvedbyaveragin

g1000

trajectories.

localizin

gin uence

ofmeasu

rementlim

itsthespread

(inboth

qandp),

andastead

ystate

iseventually

attained.

Thestro

nger

themeasurem

ent,thesooner

thespread

ischecked

.Thisbehavior

ofindividualtra

jectories

appears

tobethekey

toundersta

ndingDpeven

though,in

this

case,weare

interested

notin

single

trajectories,

butin

thebehavior

ofanensem

bleofthese

trajecto

ries.In

Fig.

1weplothp

2i(t)forthecla

ssicalsystem

,and

forthequantum

system

with

andwith

outmeasurem

ent

for�k=

3.(U

singtheresu

ltsof

[4],cla

ssicalbehavior

isnot

expected

unless

Denv�

3and�k�

2 pD

env.)

Atearly

times,

thequantum

valueof

thedi�usion

rateismuch

high

erthanthecla

ssicalvalu

e,alth

ough,con

sis-ten

twith

dynamica

llocalization

,thisdecrea

seswith

time

(eventually

fallin

gto

zero).

Ontheoth

erhand,when

the

system

isunder

observation

,theinitia

levolu

tionof

the

system

ishardlya�ected

;itisonlythat

thedi�usion

ratereach

esacon

stantvalueat

aboutt=10(fo

rthisvalu

eof

Denv)atwhich

poin

tapurely

di�usive

evolu

tiontakes

over.

Thus,themeasurem

entappears

toinduce

a`pre-

mature'

(time-d

ependent)stea

dy-sta

te,justas

itinduces

a`prem

ature'

steady-sta

tewidth

[4].Thusthedi�usion

rategets

frozenin

toits

earlytim

evalu

ewhich

,in

this

case,

issubsta

ntia

llylarger

than

thecla

ssicalresu

lt.In

Fig.2weplotDpasafunctio

nof�kforD

env=0:1

andD

env=10�4,with

�=10.Forsm

all�k,Dpisessen

-tia

llygiven

bythecla

ssicalvalu

e(D

cl=

31:2),

andfor

large�k,Dpisclose

tothequantum

value(D

p=0)

when

DenvissuÆcien

tlysm

all.Thuswesee

theexpected

tran-

sitionfro

mcla

ssicalto

quantum

behavior.

How

ever,the

transitio

nreg

ionissurprisin

glycomplex

:thevalueofDp

inthisreg

ionvaries

widely

asafunction

of�kand

rises

tomorethan

twice

Dclatits

peak

!Another

remarkab

lefact

isthatfea

tures

andplacem

entofthetran

sitionre-

gion(asafunctio

nof�k)isrela

tiveinsen

sitiveto

thevalu

e

12

34

56

78

910

0 10 20 30 40 50 60 70 80 90

100

110

k

d⟨ p2⟩ / dt

12

34

56

78

910

0 10 20 30 40 50 60 70 80 90

100

110

k

d⟨ p2⟩ / dt

FIG

.2:

Thelate-tim

e(t=

30-50)di�usion

coeÆ

cientas

afunc-

tionof

�kfor

twovalu

esof

themeasu

rementinduced

di�usion

coeÆ

cient,D

env.Thecircles

arethecalcu

latedpoin

ts;the

solidcurves

aremean

tsim

ply

toaid

theeye.

(a)D

env=0:1;

(b)D

env=10�4.

Thedash

edlinerep

resents

theShepelyan

-skycurvediscu

ssedin

thetex

t.

ofD

envas

itchanges

overthree

orders

inmagn

itude.

Inwhat

follows,for

conven

ience

wewill

referto

theplots

inFig.

2sim

ply

astran

sitioncurves.

Som

eunderstan

dingof

thiscom

plex

structu

recan

be

obtain

edbycom

parison

with

theearly

timedi�usion

ratefor

theunmeasu

redquantum

system

asderived

analy

ti-cally

byShepely

ansky,

Dinit=�2

2(1

+2J2 (�

e� )

+2J22(�

e�)+���);

(8)

where

�e�=2�sin

(�k=2)=�k

.Thisapprox

imate

expression

forD

initisalso

plotted

inFig.

2.Asthisform

ulaisonly

validfor

��

�k,acon

dition

not

met

overmost

ofthis

range,

weuse

itonly

asaqualitative

indicator

(itisa

particu

larlypoor

indicator

oftheactu

albehavior

near

thequantum

resonance

at�k=

2�).

Neverth

eless,the

trendin

thedata

isobviou

s:wesee

that

forsuÆcien

tly

6

large Denv , the transition curve follows the early timequantum di�usion rate fairly closely over the region ofthe �rst peak: measurement is e�ective at `freezing in'the early time value, and it is from this that the complexstructure originates.

The structures in the transition region can thereforebe qualitatively understood in terms of this expression.Comparison of Eq. 8 with an expression [16] for Dcl (�)shows that this initial di�usion rate in units of the squareof the kick strength, �2, are identical except for a renor-malization of � to �e�. The classical system, however,possesses regimes of increased di�usion due to the pres-ence of accelerator modes at certain values of �. In thequantum system, we can scan through these values of �e�by tuning �k, provided � is large enough. This is the ori-gin of the �rst peak in the transition region for our valueof � = 10: at �k � 3, �e� � 6:907, which is the positionof the �rst such accelerator mode.

The scaled classical di�usion constant, Dcl =�2, also in-

creases as �! 0. This leads to an interesting behaviourin the quantum expression because by choosing �k = 2�,we can tune �e� = 0 even when � remains non-zero.Correspondingly the quantum system with this value of�k shows an enhanced early di�usion which has no sim-ple classical counterpart. In fact, in a purely quantummechanical approach, this very narrow quantum reso-nance [14] arises due to quantum mechanical interferencee�ects with no classical analog. It is remarkable thatthese inherently quantum mechanical e�ects survive and,in fact, are stabilized by continuous measurement and theassociated decoherence in the Master equation (7). Thiscounter-intuitive behavior results in part from the factthat the QDKR strongly resists decoherence to classical-ity as explained in Ref. [12].

For smaller values of Denv , the transition curve dropsbelow the Shepelyansky predictions for the di�usion rate,which is consistent with the notion that the weaker mea-surement takes longer to stabilize the falling quantumdi�usion rate. As is evident from Fig. (1) locking in ata later time on the noiseless quantum curve will clearlyproduce a smaller value of the di�usion coeÆcient.

One consequence of this complex behavior is that, inthe transition region, the cross-over from the classical toquantum regimes can also lead to e�ects that are quitecounter-intuitive. An example of such behavior is ex-posed by plotting the late-time Dp as a function of Denv

for di�erent values of �k as we have done in Fig. 3. Oneinteresting open question that can be addressed this wayis whether the quantum evolution, as a function ofDenv ,�rst goes over to the classical limit (with small noise) orreaches the classical value only in the fully noise domi-nated limit. At suÆciently small �k and a �nite Denv , itfollows from the results of Ref. [4] that the classical limitwill exist and thus the quantum evolution will go over tothe small-noise classical limit. However, as Fig. 3 shows,in the intermediate regime this is not the case. Indeed,for a range of values of �k, an inversion of what is usuallyexpected occurs: The system di�uses slower (intuitively,

0 0.5 1 1.5 20

10

20

30

40

50

60

70

d ⟨ p

2 ⟩ / d

t

Denv

Classical

= 2

= 3

= 5

k

k

k

FIG. 3: The late time momentum di�usion coeÆcient Dp

as a function of the di�usion coeÆcient Denv in the Masterequation for the QDKR. Results at di�erent values of �k showthe nontrivial nature of the approach to the classical noise-dominated result (solid line).

a `more quantum' behavior) at smaller values of �k (e.g.,�k = 3 vs. �k = 5 in Fig. 3).In summary, we would like to emphasize certain im-

portant points regarding the transition curve. The �rstis that examining only the transition curve, one mightconclude that the classical limit has been achieved whenthe classical value ofDp has been reached (e.g. at �k � 0:2forDenv = 10�4), especially since it remains at this valuefor smaller �k. However, examining the position probabil-ity distribution of the particle for a typical trajectorywith �k = 0:2, and Denv = 10�4, we �nd that far frombeing well localized, the particle position is spread signif-icantly over four periods of the potential, and the distri-bution contains a great deal of complex structure. As aresult, the individual trajectories, which are in principlemeasurable, are still far from classical [30]. Evolving forsmall �k reveals that true classical motion emerges at thetrajectory level for Denv = 10�4 only when �k <� 10�3,as expected from the conditions in Ref. [4]. The secondpoint is that since the transition curves rise above theclassical value in the intermediate regime, they necessar-ily cross this value again during their descent into thequantum region. Hence it is important not to sample thecurve only in a limited region, in which case one couldmistakenly conclude that the transition from quantum toclassical behavior of the di�usion constant had alreadytaken place.Experimental veri�cation of our predictions should be

within reach of the present state of the art. Either spon-taneous emission or continuous driving with noise shouldbe, in principle, suÆcient to observe the anticipated dif-fusive behavior in hp2(t)i (see, e.g., Ref. [15]). Measuringhp2(t)i accurately can, however, still be complicated byproblems with spurious tails in the momentum distribu-tion, nevertheless, these problems can likely be overcome

7

especially since the predicted e�ects do not require theexperiments to be run for long times (see Ref. [31] fora recent measurement of the behavior near the quantumresonances including decoherence e�ects).

VI. CONCLUSION

To conclude, we reemphasize a few key points: We haveshown that it is possible to characterize the quantum-classical transition in the QDKR by �xing the environ-mentally induced di�usion (Denv ) at some suÆcientlysmall value, and examining the late-time di�usion coeÆ-cient as the size of the system is increased (�k decreased).In doing so, we have shown that the late-time behaviorin the transition region is strikingly complex and di�er-ent from both the classical and quantum behavior, andthat this dynamics follows, instead, the early time quan-tum di�usion rates. Remarkably, the temporary natureof the early-time quantum di�usion rates is in fact sta-

bilized by the continuous measurement allowing for theirpossible measurement. These predictions for a distinct,experimentally accessible, `transition dynamics' providean interesting area for investigation in experiments cur-rently being performed on the quantum delta-kicked ro-tor.

VII. ACKNOWLEDGEMENTS

We thank Doron Cohen, Andrew Daley, Andrew Do-herty, Scott Parkins, Daniel Steck, and Bala Sundaramfor helpful and stimulating discussions. K.J. would liketo thank Scott Parkins for hospitality at the University ofAuckland during the development of this work. Numer-ical calculations were performed at the Advanced Com-puting Laboratory, Los Alamos National Laboratory, andat the National Energy Research Scienti�c ComputingCenter, Lawrence Berkeley National Laboratory.

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8

[23] Actually, if the nonlinearity is large on the quantum

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[24] We are assuming that both 4[mF 2=(@xF )2] jF=pj and

E jp=F j evaluated at a typical point of the trajectory arecomparable to the action of the system, and de�ne thisto be �hs.

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[30] It is unknown whether or not the non-classicality ofthe trajectories leaves a signature detectable in ensem-ble measurements.

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