Quantum-Classical Correspondence Principle for Work Distributions in a Chaotic System

0.28 MB·

by  Long Zhu

Quantum-Classical Correspondence Principle for Work Distributions in a Chaotic
System
Long Zhu (朱隆),1 Zongping Gong (龚宗平),1 Biao Wu (吴飙),2,3,4,5 and H. T. Quan(全海涛)1,3,
∗
1School of Physics, Peking University, Beijing 100871, China
2International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
3Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
6 4Wilczek Quantum Center, College of Science, Zhejiang University of Technology, Hangzhou 310014, China
1 5Synergetic Innovation Center for Quantum Effects and Applications (SICQEA),
0
Hunan Normal University, Changsha 410081, China
2
(Dated: May 19, 2016)

y
a We numerically study the work distributions in a chaotic system and examine the relationship
M between quantum work and classical work. Our numerical results suggest that there exists a cor-
respondence principle between quantum and classical work distributions in a chaotic system. This
correspondencewasprovedforone-dimensional(1D)integrablesystemsinarecentwork[Jarzynski,
8
1 Quan,andRahav,Phys. Rev. X5,031038(2015)]. Ourinvestigationfurtherjustifiesthedefinition
of quantumwork via two point energy measurements.

]
h PACSnumbers: 05.70.Ln,05.30.-d,05.45.Pq
c
e
m
I. INTRODUCTION respondence principle for work distributions (if there is
- one) can be a good evidence to justify this definition.
t
a Inthepasttwodecades,substantialdevelopmentshave
t
s been made in the field of nonequilibrium statistical me- Recently,therelationshipbetweenquantumandclassi-
.
t chanicsinsmallsystems[1,2]. Asetofexactrelationsof calworkdistributionsinonedimensional(1D)integrable
a
fluctuationsregardingwork[3–8],heat[8,9],andentropy systems has been carefully studied [17]. By employing
m
production [10] have been discovered. They are now col- the semiclassical method [19, 20], it is rigorously proved
- lectively known as Fluctuation Theorems (FT) [1–3, 8]. that such a correspondence principle exists for work dis-
d
n These theorems are valid in processes that are arbitrar- tributions in 1D integrable systems when the quantum
o ily far from equilibrium, and have significantly advanced workisdefinedviatwopointenergymeasurements. Nev-
c our understanding about the physics of nonequilibrium ertheless, for a generic system, especially a chaotic one,
[ processesinsmallsystems. TheseFTnotonlyimplythe the correspondence principle for work distributions has

2 second law of thermodynamics but also predict quanti- not been explored so far. In this article, we try to ad-
v tatively the probabilities of the events, which “violate” dress this issue following the efforts of Ref. [17]. If the
2 the second law in small systems. Despite these develop- correspondence principle for work distributions also ex-
6 ments, therearestillsome aspectsofthese FT that have ists in chaotic systems, the justification of the definition
2
not been fully understood. One of these aspects is the ofquantumworkviatwopointenergymeasurementscan
5
definition of quantum work. For an isolated quantum be extended to chaotic systems.
0
system, there are various definitions of quantum work
.
1 [11]. However,itisfoundthatwithinalargeclassofdef-
0 Amongvariouschaoticsystems,oneofthemostexten-
initions, only one [12, 13] of them satisfies the FT. That
6 sively studied systems is the billiard systems [21, 22]. In
is, the work defined through two point energy measure-
1 this article, we numerically study the work distribution
v: ments: one at the beginning and the other at the end in a driven billiard system—a ripple billiard [23] with
of a driving process [14, 15]. This definition of quan-
i moving boundaries. We numerically compute the time
X tum work, though leading to FT, seems ad hoc because
evolutionofthe chaotic billiardsystemin both quantum
the collapseofthe wavefunction [16], which playsa cen-
r and classical regimes, and then study the relationship
a tral role in determining the work [13], brings profound
between the distributions of quantum work and classical
interpretational difficulty to the definition of quantum
work. Our numerical results suggest that the correspon-
work[17]. Itisthusimportanttojustify thedefinitionof
dence principle applies in this context.
quantum work, i.e., to find other independent evidences
(besides the validity of FT) to support the definition of
quantumworkviatwopointenergymeasurements. Since The paper is organizedas follows. In Sec. II we intro-
the correspondence principle [18] is a bridge connecting duce the quantum and the classical transition probabili-
quantumandclassicalmechanics,webelievethatthecor- ties, which are used in the calculationof the work distri-
butions. In Sec. III, we introduce the billiard model. In
Sec. IV, we present our numerical results and our anal-
ysis. In Sec. V, we make some concluding remarks. The
∗Electronicaddress: [email protected] numerical method is presented in the Appendix.
2
II. CLASSICAL AND QUANTUM TRANSITION where N is the number of representative points which
in
PROBABILITIES fall into the energy window (EB,EB ) at t = τ, and
n n+1
N is the total number of the representative points.
total
We consider a quantum system, which is driven in a So, PC(nB|mA) is the probability of a classical particle
nonequilibrium process from time t = 0 to t = τ. This whose energy is initially EmA and finally falls into the
is usually characterized by the work parameter of the energy window (EnB,EnB+1) at t=τ.
systembthatchangesfromAtoB. Theworkparameter In order to study the initial thermal distribution
b can be the position of a piston or the spring constant PC(m) and PQ(m), we need to clarify the density of
A A
of a harmonic oscillator or else. For this nonequilibrium states. The density of states for the classical system is
process, its work distribution function can be expressed given by
as [17, 24]:
ddqddp
ρ¯(E)= δ(E−H(p,q)), (5)
PQ(W)= PQ(m)PQ(nB|mA)δ(W−EB+EA). (1) (2π~)d
A n m Z
m,n where q and p are the coordinate and the momentum of
X
the system, d is the spatialdimension, H(p,q)=H(0)is
It is clear that the work distribution function is deter-
the initial Hamiltonian. Accordingly, the initial thermal
mined by two factors. The first one is the initial ther-
distribution for this classical system reads
mPstaamtleed−iwsβthEreimAnb,ubtwio=hnerAfeu,nEcZmAtAiQoins≡tPhAQe(emmn)ee−r=gβyEemAo−fβitEshmAteh/ZmeAQtph,arZetiiAQgtieon=n- PAC(m)=ZEEmAmA+1ρ¯(E)Z1ACe−βEdE, (6)
function and β is the inverse temperature of the initial
where
P
thermal state. The second one is the transition proba-
ddqddp
bility PQ(nB|mA) between the initial and the final en- ZC = e βH(p,q) (7)
ergy eigenstates during the driving process. Similarly, A (2π~)d −
Z
the workdistribution function ofthe correspondingclas- is the classical partition function.
sical system is [17] For the quantum system, according to Gutzwiller [21]
thesemiclassicaldensityofstatesisequaltothe summa-
PC(W)≈ PAC(m)PC(nB|mA)δ(W−EnB+EmA), (2) tion of the classical density of states ρ¯(E) and an oscil-
Xm,n lating correction term ρ˜(E) [18, 21, 26–28]
where PC(m) (PC(nB|mA)) is the classical counterpart ρ(E)=ρ¯(E)+ρ˜(E). (8)
A
of PQ(m) (PQ(nB|mA)).
A The oscillation part ρ˜(E) has an origin in the classical
The transition probability of the quantum system is
period orbits, and is absent for classicalsystems. To the
defined as
first order approximation, or on an energy scale larger
2 than the period of ρ˜(E), we can ignore the oscillation
PQ(nB|mA)≡ nB Uˆ(t) mA , (3)
part ρ˜(E) and keep only the average density of states
where mA and nB , re(cid:12)(cid:12)(cid:12)s(cid:10)pect(cid:12)(cid:12)ively,(cid:12)(cid:12)repr(cid:11)e(cid:12)(cid:12)(cid:12)sent the mth qρ¯(uEan).tuTmhesryesftoerme, itsheapinpirtoiaxlimthaetremlyaelqduisatlritboutitiosnclfaosrsitchael
eigenstateattheinitialtimet=0andthentheigenstate
counterpart
atthefi(cid:12)(cid:12)nal(cid:11)timet=(cid:12)(cid:12) τ.(cid:11)Accordingly,PQ(nB|mA)denotes
the quantum transition probability from the mth eigen- EmA+1 1
PQ(m)= ρ(E) e βEdE
sUˆta(tt)e iwshtehnebu=nitAar,ytooptehreatnotrhseaitgiesfnysitnagtei~w∂hUˆen(t)b/∂=t B=. A ZEmA ZAQ − (9)
Hˆ(t)Uˆ(t),whereHˆ(t)isthetime-dependentHamiltonian ≈ EmA+1ρ¯(E) 1 e βEdE =PC(m).
of the system. ZEmA ZAC − A
While the definition of the quantum transition prob-
As a result, the comparison between the quantum (1)
ability is straightforward, the definition of its classical
and the classical(2) work distributions is reduced to the
counterpartisabitsubtle. Theclassicaltransitionprob-
comparison between the quantum PQ(nB|mA) (3) and
abilityPC(nB|mA)isdefinedasfollows[17,25]. Initially
the classical PC(nB|mA) (4) transition probabilities.
the microscopic states are evenly sampled from the en-
Both the quantum transition probability (3) and its
ergy shell E = EA in the classical phase space (see Fig.
m classical counterpart (4) are computable. However, un-
1). Each microscopic state is represented by a phase-
like the integrable systems [17], for chaotic systems, we
space point. The initial states then undergo Newtonian
have to resort to the numerical method as the analyti-
dynamics governed by H(t), when the work parameter
calsemiclassical(WKB)wavefunction[29]oftheenergy
b(t) is varied according to a given protocol. The corre-
eigenstateinafullychaoticsystemisusuallyunavailable
sponding classical transition probability is defined as
[18, 22, 27, 30–32]. We will introduce the model in the
N next section. Our numerical results will be presented in
PC(nB|mA)≡ in , (4)
N the Sec. IV.
total
3
III. THE RIPPLE BILLIARD MODEL
a
Aprototypemodelwidelystudiedinquantumchaosis
a static billiard system whose boundaries are fixed and
the Hamiltonian is time-independent. For our study, we L
choose a ripple billiard [23] whose sinusoidal boundaries v v
move in opposite directions. The advantage of choos-
ing the ripple billiard instead of other more extensively
studied systems in literature, such as the stadium bil-
liard[21,33],isthateachentryofitsHamiltonianmatrix
can be expressed in terms of elementary functions [23]. b
Thus the eigenenergies and eigenstates of the system at
any moment of time can be obtained through exact nu-
merical diagonalization, which is usually not doable in (a)
chaotic systems. As a result, we can accurately simulate P
y
the quantum dynamical evolution in the chaotic system,
which is usually a big challenge [34]. In addition, the
degree of chaoticity of the model can be controlled by
tuning the geometric parameters a, b and L, which en-
ablesustostudytheinfluence ofthedegreeofchaoticity
easily.
Theripplebilliard[23]withbothboundariesmovingat
P
thesamespeedandintheoppositedirectionisillustrated x
in Fig. 1. In ourmodel, the workparameteris the (half)
lengthbofthebilliard. Wemoveboththecurvedbound-
aries because the symmetry canhelp us simplify the cal-
culation. Thepositionandthemomentumoftheparticle
inside the billiard are denoted by (x,y) and (p ,p ), re-
x y
spectively. This model system can be characterized by
the followingparameters: b ,a,L, v, andτ. b is the ini- (b)
0 0
tiallength,aandLaretheparameterscharacterizingthe
boundaryshapeoftheripplebilliard,whilevisthespeed FIG. 1: Ripple billiard with moving boundaries. The coor-
of the moving boundary and τ is the total driving time. dinates in the position space and the momentum space are
Inanappropriatelychosencoordinate,the boundariesof denoted by (x,y) and (px,py), respectively. Parameter b is
the ripple billiard can be depicted by varied in time b(t) = b0 +vt, where b0 = A. The parame-
ter a represents the ripple amplitude. The red dots in the
f(y,t)=±[b(t)−acos(2πy/L)], (10) Fig. 1(a) show that the initial states are evenly sampled in
the coordinate space. Fig. 1(b) shows that the initial states
where a represents the ripple amplitude and b(t)=b + are evenly sampled in the momentum space from an energy
0
vt denotes the length at time t. When a is decreased, shellEmA =(p2x+p2y)/2M,whereM isthemassofthebilliard
the system becomes “less chaotic”. When a = 0, the ball.
system becomes integrable. Since we are interested in
the dynamical evolution of a chaotic system, we fix a at
a finite number and vary b in time. For a large a the the position components of the representative points
system is always in the deep chaotic regime. (x,y) are evenly sampled inside the potential well (see
This kindofdrivenquantumsystemsareofinterestin Fig. 1(a)). In the momentum space, the momentum
the field ofmesoscopic physics and have been studied by components (p ,p ) are evenly sampled from the energy
x y
Cohen et al. [35]. For 1D system, work distributions in shell EA = (p2 +p2)/2M (see Fig. 1(b)). Such a choice
m x y
a quantum and a classicalbilliard have been obtained in of sampling assures a uniform (isotropic) distribution in
Refs.[36, 37]. For2Dsystems,somebriefresults regard- thecoordinate(momentum)spaceafterthelocalaverage
ing time-dependent integrable quantum billiards (rect- overavicinity whichissmallcomparedtothe sizeofthe
angular and elliptical billiards) have been reported by potential well but large compared to the quantum wave
Shmelcher et al. [38], but no results about chaotic bil- length [30, 32, 39]. For simplicity, we set the mass of
liards were reported there. the billiard ball to be M = 0.5. In the next section, we
It should be emphasized that the counterpart of the numerically simulate the classical and the quantum evo-
energy eigenstate mA in the classical system is a mi- lution of the driven ripple billiard system and compare
crocanonicalensemble,namelythecollectionofrepresen- these two transition probabilities to check if there ex-
(cid:12) (cid:11)
tative points evenl(cid:12)y sampled from a 3D “energy shell” ists the correspondence principle between the transition
E = EA in a 4D phase space. In the coordinate space, probabilities in this chaotic system.
m
4
IV. NUMERICAL RESULTS quantumandtheclassicaltransitionprobabilities,wewill
study the effects of the degree of chaoticity (character-
ized by a), the speed (v) of the moving boundary, and
Wedevelopamethodtoaccuratelycalculatethequan-
the value of the Planck constant (~) on the convergence
tum transition probabilities in a ripple billiard system
of the two transition probabilities. To this end, we in-
with moving boundaries. Technical details can be found
troduce a measure to quantify the distance between the
in the Appendix. We only present the numerical results
classical and the quantum cumulative transition proba-
in the main text.
bilities. The measure we choose is the root-mean-square
An example of the comparison between the quan-
error(RMSE)(seetextbooksonmathematicalstatistics,
tum and the classicaltransitionprobabilities is shownin
for example Ref. [40]). At time t, the RMSE R(t) be-
Fig.2. Theparametersareb =0.5,a=0.2,L=1.0,v=
0
2.0,τ =0.4, and the Planck’s constant ~=1.0. The ini- tween the two cumulative transition probabilities is de-
fined as
tial state is the 100th eigenstate (m = 100). In Fig. 2
we notice that (i) both the quantum and the classical
transition probabilities are less regular than those in the (SC(t)−SQ(t))2
1D integrable case [17] and (ii) the quantum transition R(t)≡v n;SnC6=SnQ n n , (11)
probabilities are sparse and discrete, while the classical uuP N(t)
t
transition probabilities are quasi-continuous and spread
n
inawiderangeofthe energyspectrum. Inordertocom- where SC(t) = PC(kB(t)|mA) is the classical cu-
n
pare these two transition probabilities in a better way, k=1
we plot the cumulative transition probabilities in Fig. 3 mulative transitioPn probability at time t and SQ(t) =
n
for different sets of parameters. From Fig. 3 we can see n
PQ(kB(t)|mA) is its quantum counterpart. We use
that the quantum and the classical transition probabil-
k=1
ities do not collapse onto the same curve, but are very kPB(t) instead of kB to emphasize that the work parame-
close to each other. Especially the quantum cumulative ter B is time-dependent. The sum of the squared differ-
transitionprobabilitycurveoscillatesaroundthesmooth ence between SC(t) and SQ(t) is taken over all n where
n n
classical cumulative transition probability curve. This thesetwoquantitiesarenotequal. N(t)isthetotalnum-
phenomenonisreminiscentoftheresultsin1Dintegrable ber ofthe quantumnumbers atwhichthese twocumula-
systems [17], where it has been explained as a conse- tivetransitionprobabilitiesaredifferent. Roughlyspeak-
quence of the interference of different classical trajecto- ing, R(t) is the average of the local deviations between
ries. Although we cannot clearly see the correspondence these two cumulative transition probabilities at time t.
between the quantum and the classicaltransition proba- R(t) = 0 means that the two distributions are identical.
bilitiesinFig.2,weobviouslyobservetheconvergencein ThelargerR(t)is,themoredistinctthetwoprobabilities
Fig. 3, which implies a corresponding principle between SC(t) and SQ(t) are. We note that the correspondence
n n
the transition probabilities. principle implies the convergence between the classical
and the quantum transition probabilities in some aver-
agesense[17]. TheRMSEquantifiestheaveragedistance
0.5 betweentwoprobabilitydistributions. Hence,webelieve
y 0.1
bilit that the RMSE can be a good measure to quantify the
a 0.4 applicability of the correspondence principle.
b
Pro 0.05 InFigs.4and5weshowthenumericalresultsofRMSE
0.3 for different values of a and v. For the convenience of
comparison,the horizontalaxes inFigs. 4 and5 arecho-
0
50 100 150 sen to be the moving distance instead of the moment of
0.2
time t. In other words,we compare the RMSE when the
movingboundariesreachthesamelocation. InFig.4,the
Classical Transition Probability
0.1 Quantum Transition Probability fixed parameters are b0 = 0.5,L = 1.0,v = 2.0,τ = 0.4,
and a is chosen to be a = 0.1,0.2,0.3. In Fig. 5, a is
00 200 n 400 fixed at 0.2 while v is set to be v = 1,8,40, and accord-
ingly the total moving times are τ =0.80,0.10,0.02. All
the other parameters in Fig. 5 are the same as those in
FIG.2: ComparisonbetweenquantumPQ(nB|mA)andclas- Fig. 4. In all these cases, R is initially equal to zero,
sical PC(nB|mA) transition probabilities. Here the horizon- andincreasesvery rapidly (the topis notvisible in some
talaxislabelsquantumnumbern. Theparametersaresetas figures), and then begins to decrease. We observe that
b0 = 0.5,a = 0.2,L = 1.0,v = 2.0,τ = 0.4,~ = 1.0. The ini- R decreases with oscillations in all the cases, and finally
tial state is set to be the 100th eigenstate, namely m=100. saturate at a finite value. The initial jump of RMSE
The partial enlarged details are shown in the insetting. from 0 to a large number is probably due to the local
nature of the classicaldynamics andthe nonlocalnature
Having demonstrated the correspondencebetween the of the quantum dynamics. In a short time, the classical
5
y 1 1
bilit
ba 0.8 0.8
o
Pr
e 0.6 0.6
v
ati
mul 0.4 0.4
u
C Classical Cumulative Transition Probability
0.2 Quantum Cumulative Transition Probability 0.2 Classical Cumulative Transition Probability
Quantum Cumulative Transition Probability
00 100 200 300 400 00 100 200 300 400
(a) (b)
1 1
0.8 0.8
0.6 0.6
0.4 0.4
Classical Cumulative Transition Probability Classical Cumulative Transition Probability
0.2 Quantum Cumulative Transition Probability 0.2 Quantum Cumulative Transition Probability
00 100 200 300 400 00 100 200 300 400
(c) (d) n
FIG.3: ComparisonbetweenquantumSQ andclassicalSC cumulativetransitionprobabilitieswithvarioussetsofparameters
n n
a,vand~. Herethehorizontalaxislabelsquantumnumbernandtheverticalaxislabelsthecumulativetransitionprobability
SnQ(SnC). The parameters in the four figures are respectively set as: (a) b0 = 0.5,a = 0.2,L = 1.0,v = 2.0,τ = 0.4,~ = 1.0;
(b) b0 = 0.5,a = 0.2,L = 1.0,v = 8.0,τ = 0.1,~ = 1.0; (c) b0 = 0.5,a = 0.3,L = 1.0,v = 2.0,τ = 0.4,~ = 1.0; (d)
b0 =0.5,a=0.2,L=1.0,v=2.0,τ =0.4,~=1.5571.
transitionprobabilitiescannotfullyreflecttheglobalfea-
0.5
ture of the system. However,even on a short time scale,
the quantum transitionprobabilities can fully reflect the a=0.1
E
global feature of the system. Hence, at the initial stage S0.4 a=0.2
M
of the driving process, these two transition probabilities R a=0.3
differ substantially. Fig. 4 shows that the larger the pa- 0.3
rameterais,whichmeansthatthesystembecomesmore
chaotic,themorerapidlyRfalls,andthesmallervalueR
0.2
saturatesat. Thisresultindicatesthatthesetwocumula-
tiveprobabilitiesbecomecloserwhenthesystembecomes
0.1
morechaotic,andcouldbeexplainedasfollows: Fora2D
system, the more chaotic it is, the better the quantum-
classical correspondence principle applies [41]. Fig. 5 00 0.2 0.4 0.6 0.8
Moving Distance
shows that as the boundary-moving speed increases, the
twocumulativetransitionprobabilitiesbecomemoredis-
tinct. A similar result was obtained in the 1D piston FIG. 4: Comparison of RMSE as a function of boundary-
system[36]. Thisresultcanalsobeexplainedbythefact moving distance (one side) at different degrees of chaoticity
that the quantum transition probabilities can always re- a=0.1,0.2,0.3. Theotherparametersaresetasb0=0.5,L=
1.0,v = 2,τ = 0.4,~ = 1.0 and the initial state is the 100th
flect the global property of the system, while the classi-
eigenstate (m=100).
cal transition probabilities cannot unless the boundaries
move slowly and the classical particles collide frequently
with the boundaries.
pose, we should keep the energy,instead of the quantum
WehavealsostudiedtheeffectofthePlanck’sconstant number, of the initial state as a constant. Except that,
~ on the convergence of the quantum and the classical all the other parameters b , a, v, L and τ are fixed. As
0
transitionprobabilities. As is known, quantum andclas- we cannot guarantee the accuracy of numerical results
sical predictions must agree when the Planck constant when we decrease the value of ~ (see the Appendix), we
approaches zero (~ −→ 0) [18]. Therefore, it is interest- increase the value of ~ and present the results in Fig. 6.
ingto seehowR changesinthis chaoticsystemwhenwe We clearly observe the increase of the saturated value
adjust the value of the Planck constant ~. For this pur- of the RMSE when ~ increases. This result implies that
6
denceprincipleoftransitionprobabilitiesinachaoticsys-
0.5
tem. The more chaotic the systemis, or the more slowly
v=1 theboundariesmove,thebettertheconvergencebetween
E
S0.4 v=8 thequantumandtheclassicaltransitionprobabilitiesbe-
M
R v=40 comes. Also, we indirectly show that the smaller the
0.3 value of the Planck constant ~ is, the better the conver-
gence between these two cumulative transition probabil-
itiesis. Lastbutnotleast,wewouldliketomentionthat
0.2
the correspondence principle between these two transi-
tion probabilities may break down in the long time limit
0.1
(after the so-called Ehrenfest time) in a chaotic system
[31, 42–44]. But in the case of nonequilibrium driving,
00 0.2 0.4 0.6 0.8 especially a transient driving process as is usually the
Moving Distance
case in the study of FT, the correspondence principle is
still valid.
FIG. 5: Comparison of RMSE as a function of boundary-
moving distance (one side) at different boundary-moving
speeds v = 1,8,40. The other parameters are set as b0 =
0.5,a=0.2,L=1.0,~=1.0 andtheinitial stateisthe100th V. CONCLUSION
eigenstate (m = 100). τ is set to be τ = 0.80,0.10,0.02 to
ensure the total moving distance equals 0.80 in all three ex- Inthisarticlewenumericallystudythecorrespondence
periments. principle for work distributions in a prototype model of
quantum chaos—a driven ripple billiard system. The
quantum (or classical) work distribution function is de-
the difference between these two cumulative transition
termined by two factors: (1) the initial thermal equilib-
probabilities becomes more prominent with the increase rium distribution PQ(m) (or PC(m)) and (2) the tran-
of ~. Although we cannot give accurate numerical re- A A
sition probabilities PQ(nB|mA) (or PC(nB|mA)). Since
sults for a smaller value of ~ for computational reasons,
the initialdistribution functions for the classicalandthe
our results in Fig. 6 serve as an indirect evidence that,
quantum cases are approximately equal, the correspon-
in this chaotic system, the distance between these two
dence principle between work distributions is simplified
transition probabilities will diminish when the value of
tothe correspondenceprinciplebetweentransitionprob-
~ decreases. This is in accordance with the well-known
abilities. Unlike the 1D integrable systems [17], we can-
correspondence principle that quantum mechanics is re-
not employ analytical approaches due to the lack of the
duced to classical mechanics in the limit of ~−→0.
semiclassical(WKB)wavefunctioninafullychaoticsys-
tem [18, 22, 27, 30–32]. Instead, we numerically calcu-
0.7 late both the classical and the quantum transition prob-
abilities. Compared with the 1D integrable systems, the
E0.6
S ¯h=1.0 transitionprobabilitiesinthechaoticsystemarelessreg-
M ¯h=1.5571
R0.5 ¯h=2.9259 ular. In particular, the quantum transition probabilities
are sparse and discrete, while the classical ones are dif-
0.4 fusive and quasi-continuous. While these features make
the correspondence principle in the chaotic system less
0.3
“evident”, we still observe the convergence from the cu-
0.2 mulative transition probabilities, thus demonstrate that
the correspondence principle of the transition probabili-
0.1
ties applies in the ripple billiard system. Our numerical
results indicate that the convergence, which is quanti-
00 0.2 0.4 0.6 0.8
Moving Distance fied by the statistical quantity RMSE, becomes better
when the system is more chaotic or the driving speed
gets slower. We also provide indirect evidences that the
FIG. 6: Comparison of RMSE as a function of boundary-
movingdistance(oneside)fordifferentvaluesof~. Theother convergence becomes better when ~ decreases.
parameters are set as b0 = 0.5,a = 0.2,L = 1.0,v = 2,τ = We would like to emphasize that, similar to integrable
0.4. The initial energy of the system is set to be the 100th systems [17], this correspondence principle is a dynamic
eigen energy when ~ = 1.0. The quantum number of the one (the quantum and classical transition probabilities
initial states are equal to m = 100, m = 40 and m = 10 converge when ~ → 0) instead of the usual static one
respectively.
(probability distributions in position space converge for
large quantum number) [45]. Hence, in the context of
Inthis section,we haveshownthatunder variouscon- semiclassical physics, our work complements extensive
ditions there does exist a quantum-classical correspon- previous studies on the static correspondence principle.
7
Inthecontextofnonequilibriumquantumthermodynam- with the time-independent boundary
ics, our work complements the recent progress made in
Ref. [17] and further justifies the definition of quantum −f(y)≤x≤f(y), (A.5)
work via two point energy measurements.
where
In the future it would be interesting to explore the
dynamic correspondence principle in a quantum many-
f(y)=b−acos(2πy). (A.6)
bodysystem,whereindistinguishability[46]andthespin
statistics effect will make the quantum-classical corre-
To solve equation (A.4), they suggest to straighten the
spondence principle even more elusive. These problems
boundaries by introducing a pair of curvilinear coordi-
are left for our future works.
nates (u,v)
Acknowledgments HTQ gratefully acknowledges
support from the National Science Foundation of China x
u= ,v =y. (A.7)
under grants 11375012,11534002,and The Recruitment 2f(y)
Programof Global Youth Experts of China. BW is sup-
portedbytheNationalBasicResearchProgramofChina Intermsofcoordinates(u,v),theripplebilliardistrans-
(Grants No. 2013CB921903 and No. 2012CB921300) formed into a square billiard. Let us introduce a set of
and the National Natural Science Foundation of China orthogonaland complete wave functions
(Grants No. 11274024, No. 11334001, and No.
11429402). 2 x 1
φ (x,y)= sin mπ + sin(nπy).
m,n
sf(y) 2f(y) 2
(cid:20) (cid:18) (cid:19)(cid:21)
(A.8)
APPENDIX: NUMERICAL METHOD
In the basis of φ (x,y), Eq. (A.4) can be transformed
m,n
into a matrix equation
We consider the quantum and the classical dynamics
of the ripple billiard system with both boundaries mov-
∞
ing at the same speed v and in the opposite directions. Hm′n′mnBml n =ElBml ′n′, (A.9)
Forthe classicalcase,wenumericallysimulatethe evolu- m,n=1
X
tionofaclassicalparticleundergoingNewtoniandynam-
ics. Then we repeat the simulation while changing the and the hamiltonian matrix elements are
initial location and the initial direction of the velocity m2π2~2
onfumthbeerpaorftpicaler.ticWlese(mNakeinaEhqis.to(4g)r)amwhbicyhcfoaullntinintgo tthhee Hm′n′mn = 4 δm′,m(In2′−n−In2′+n)
energywindow(EB,EBin )atthe momentoftime t=τ. +n2π2~2δm′,mδn′,n+nπ2~2aδm′,mJn2′,n
n n+1
This histogram gives the classical transition probability 3
PC(nB|mA). Inournumericalexperiment,werepeatthe +aπ2~2δm′,mJn3′,n− 2a2π2δm′,mJn5′,n
(A.10)
simulationfor 1milliontimes (Ntotal inEq.(4)) foreach +2mnaπ3~2(Km1′+m+Km1′ m)Jn2′,n
set of parameters. We recall that the initial locations in −
thecoordinatespaceandthedirectionsinthemomentum +2maπ3~2(Km1′+m+Km1′ m)Jn3′,n
−
space are evenly sampled. −6ma2π3~2(Km1′+m+Km1′ m)Jn5′,n
For the quantum case, the evolution is described by −
the solutionofthe followingtime-dependentSchr¨odinger +2m2a2π4~2(Km2′ m−Km2′+m)Jn5′,n,
−
equation:
where,
~2
− (∂2+∂2)ψ(x,y,t)=i~∂ ψ(x,y,t), (A.1)
2m x y t I1 = 0, n is odd (A.11)
n 1 (b √b2 a2)n/2, n is even
which is subjected to the boundary condition ψ|∂D = 0, ( √b2 a2 − a−
−
where
D ={(x,y):−f(y,t)≤x≤f(y,t),0≤y ≤L}, (A.2)
0, n is odd
I2 = (A.12)
and n 2b+n√b2 a2I1, n is even
( 2(b2 a2−) n
−
f(y,t)=b(t)−acos(2πy/L). (A.3)
InthefollowingwesetL=1andM =0.5forsimplicity. 0, n=0
One of us B. Wu and collaborators [23] have given the Kn1 = −(−1)n+1, n6=0 (A.13)
solution to the Schr¨odinger equation of the static ripple (cid:26) 2nπ
billiard system
∂2 ∂2 1/12, n=0
−~2(∂x2 + ∂y2)ψ(x,y)=Enψ(x,y), (A.4) Kn2 =(cid:26) (−n12)πn2+1, n6=0 (A.14)
8
This is an ordinary differential equation and we solve it
numerically by using the Crank-Nicolsonmethod, where
Jn2′,n =In1′+n 2+In1′ n 2−In1′+n+2−In1′ n+2, (A.15) Eq. (A.22) is approximated by
− − − −
Jn3′,n =In1′ n+2+In1′ n 2−In1′+n+2−In1′+n 2, (A.16)
− − − −
cm′n′(tk+1)−cm′n′(tk) 1 1
1 tk+1−tk = 2 {i~[H(tk)]m′n′mn
Jn5′,n =In2′−n−In2′+n− 2(In2′−n+4+In2′−n−4 (A.17) 1Xm,n 1
−In2′+n+4−In2′+n 4). +Bm′n′mn(tk)}cmn(tk)+ 2 {i~[H(tk+1)]m′n′mn
− m,n
X
Fortheripplebilliardwithmovingboundaries,wechoose +Bm′n′mn(tk+1)}cmn(tk+1),
φm,n(x,y,t) as a set of orthonormalbasis, (A.23)
2 mπ x
φ (x,y,t)= sin +1 sin(nπy).
m,n
sf(y,t) 2 f(y,t)
(cid:20) (cid:18) (cid:19)(cid:21)
According to Eq. (A.23), we can solve the coefficients of
(A.18)
In comparison to Eq. (A.8), here we only replace f(y) the (k+1)th timestep cm′n′(tk+1) from those of the kth
with f(y,t). We expand the wave function of time t timestepcm′n′(tk). The initialcoefficientscmn(0)canbe
easily calculated by decomposing the initial state in the
ψ(x,y,t) in the basis of φ (x,y,t)
m,n
basis of φ(x,y,0).
ψ(x,y,t)= c (t)φ (x,y,t). (A.19)
mn mn
The biggest challenge in our numerical calculation is
m,n
X the computational resources. Notice that in Eq. (A.23),
Substituting ψ(x,y,t) (A.19) into Eq. (A.1), and taking we need to multiply matrices and inverse the matrices
the inner product with φm′n′(x,y,t), we obtain in each timestep. If the size of the matrix is too large,
the computational resources required will be unaccept-
[Hm′n′mn(t)+i~Bm′n′mn(t)]cmn(t)=i~c˙m′n′(t), ably huge. Hence, properly choosing the cutoff of the
matrix is the key point in the numerical calculation. In
m,n
X
(A.20) our calculation, the initial state is the 100th eigenstate
where Hm′n′mn(t) is the Hm′n′mn in Eq. (A.10) with b (m = 100), the total timestep is ∼ 106, and the cutoff
replaced by b(t), and Bm′n′mn(t) is dimension of the matrix is 2000 when ~ = 1.0. When
we change the parameter ~, the quantum number of the
1 initial state needs to be changed accordinglyto keep the
Bm′n′mn(t)= 2δm′m+mπ(Km1′+m+Km1′−m) initialenergyasaconstant. When~decreases,thequan-
(cid:20) (cid:21)
tumnumberoftheinitialstatechangesintoalargernum-
·(In1′ n−In1′+n)b˙(t). ber,sothesizeofthematrixmustbechosenlarger,which
− (A.21)
meanstherequiredcomputationalresourceswillincrease
exponentially and this method will soon run out of com-
Here K1 and I1 are defined in Eqs. (A.11) and
m′ m n′ n putational resources. This is the reason why we cannot
(A.13). W±e rewrite E±q. (A.20) as guaranteethe accuracywhen we decreasethe value of ~.
This method may also fail when the moving speed is too
1
c˙m′n′(t)= i~Hm′n′mn(t)+Bm′n′mn(t) cmn(t). fast or the quantum number of the initial state is too
m,n(cid:20) (cid:21) large,for that the cutoff dimension of the matrix 2000is
X
(A.22) too small in these situations.
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