PHYSICAL REVIEW A 90, 053602 (2014)

Thermalization in closed quantum systems: Semiclassical approach

J. G. Cosme1 and O. Fialko2

1Institute for Advanced Study and Centre for Theoretical Chemistry and Physics, Massey University, Auckland, New Zealand2Institute of Natural and Mathematical Sciences and Centre for Theoretical Chemistry and Physics, Massey University,

Auckland, New Zealand(Received 27 May 2014; revised manuscript received 27 August 2014; published 3 November 2014)

Thermalization in closed quantum systems can be understood either by means of the eigenstate thermalizationhypothesis or the concept of canonical typicality. Both concepts are based on quantum-mechanical formalism,such as spectral properties of the eigenstates or entanglement between subsystems, respectively. Here westudy instead the onset of thermalization of Bose particles in a two-band double-well potential using thetruncated Wigner approximation. This allows us to use the familiar classical formalism to understand quantumthermalization in this system. In particular, we demonstrate that sampling of an initial quantum state mimics astatistical mechanical ensemble, while subsequent chaotic classical evolution turns the initial quantum state intothe thermal state.

DOI: 10.1103/PhysRevA.90.053602 PACS number(s): 67.85.−d, 05.30.−d, 03.65.Sq, 05.45.Mt

I. INTRODUCTION

Thermalization is regarded as one of the most fundamentalfacts in physics. Its foundations and basic postulates are stillthe subject of debates. Recent advances in addressing thisissue have brought us two quantum concepts that shed lighton the onset of thermal equilibrium inside closed quantumsystems. One of them is the so-called eigenstate thermalizationhypothesis (ETH, [1,2]). ETH conjectures that under certaininitial conditions the expectation values of observables ofa quantum system behave as if they were thermal duringlater times of its evolution. In another approach, so-calledcanonical typicality (CT, [3]), the process of thermalization isexplained via entanglement between subsystems [4]. However,the quantum ideas can be hard to imagine. Semiclassical ideas,on the other hand, may provide intuitive physical insights intoquantum mechanics [5]. Here we study the onset of thermal-ization in a quantum system using the semiclassical truncatedWigner approximation. This allows us to use the familiarclassical formalism to explain how quantum fluctuations in aninitial state turn into thermal fluctuations at later times duringchaotic classical evolution of Wigner trajectories.

We introduce notations to be used throughout the rest ofthe paper by reviewing briefly recent progress on quantumthermalization. ETH was tested against another hypothesisnumerically in [6] (see also [7]). To understand ETH, considerthe initial state |φ0〉 = ∑

k αk|k〉, where |k〉 are the eigenstatesof a Hamiltonian H with eigenvalues Ek . The eigenstates arethermal, which is reflected in the spectral properties as follows.The state evolves as |φ(t)〉 = e−iH t/�|φ0〉. As a prerequisite forthermalization, the expectation value of an observable 〈O〉 =∑

k,l α∗l αke

i(El−Ek )t/�Olk with Olk = 〈l|O|k〉, after sufficient

time, must relax to the long-time average 〈O〉 = ∑k |αk|2Okk .

ETH then states that this occurs if Okk is a smooth functionof Ek , while the off-diagonal elements Okl are negligible.Moreover, the energy E0 = ∑

k |αk|2Ek must have smalluncertainty �E0 = ∑

k |αk|2(Ek − E0)2 [2,6] to ensure thatthe relaxed state is thermal and depends only on the energy.

CT, on the other hand, replaces the need for any ensembleaveraging, the main postulate of statistical mechanics con-sidered to be artificial. Here, thermalization is reached on

the level of a subsystem. The reduced density matrix of thesubsystem is canonical for the majority of the possible purestates of the entire many-body system under global constraintsuch as energy. This was established under great generalityby invoking Levy’s lemma [4]. Then, the properties of thesubsystem can be calculated using the reduced density matrixconstructed from the pure density matrix of the entire systemρp = |φ(t)〉〈φ(t)|, giving the same prediction as if the entiresystem was in the microcanonical state ρm = N−1

E0,δ

∑k |k〉〈k|.

The sum is over NE0,δ eigenstates |k〉, with energies lyingwithin some window [E0 − δ,E0 + δ] such that δ � E0.

In classical physics, thermalization is explained by means ofclassical chaos. Being chaotic, the system wanders all over theconstant energy surface in the phase space, becoming ergodic.Averaged properties of the system over a long time can thenbe estimated by averaging over the accessible phase space [8].To relate that to thermalization in closed quantum systems,it was originally argued that ETH can be explained if thequantum system is chaotic in the classical limit [2]. However,many quantum systems do not have classical counterparts;nevertheless, they thermalize [6,9,10]. Thermalization in suchsystems is believed to be related to the onset of chaoticeigenstates in the system [11,12]. To demonstrate this, wewrite the density matrix corresponding to |φ(t)〉 as

ρp =∑

k

|αk|2|k〉〈k| +∑k �=l

αkα∗l e

−i(Ek−El )t/�|k〉〈l|. (1)

Here the coefficients αk are assumed to be random andindependent, implying that the second term quickly averagesto zero at long times. In the remaining first term, the factor|αk|2 is assumed to be a smooth function with narrow width�E0 � E0, yielding ρp ≈ ρm.

II. THE MODEL

We study thermalization in a quantum system which isamenable to the semiclassical analysis. Bosons are trappedin a double-well potential as shown in Fig. 1. They occupyfour energy levels (we will refer to them also as modes)and are described by the following two-band Bose-Hubbard

1050-2947/2014/90(5)/053602(5) 053602-1 ©2014 American Physical Society

J. G. COSME AND O. FIALKO PHYSICAL REVIEW A 90, 053602 (2014)

FIG. 1. (Color online) Schematic of the double-well potentialwith two energy bands. The diagram shows the tunneling of particlesand how the energy levels change due to the interactions betweenthem. The interband coupling U 01 makes the system complex enoughto show quantum thermalization of the particles.

Hamiltonian [13]:

H = −∑r �=r ′,l

J l bl†r bl

r ′ +∑r,l

U lnlr

(nl

r − 1) +

∑r,l

Elr n

lr

+U 01∑r,l �=l′

(2nl

r nl′r + bl†

r bl†r bl′

r bl′r

), (2)

where bl†r and bl

r are the bosonic creation and annihilationoperators, respectively, of an atom in well r and energy level l.The parameters in the Hamiltonian can be easily evaluatedfor a specific double-well potential [13]. The ground andfirst excited-state energies are El

r = ∫dxφl∗

r (x)Hspφlr (x). The

tunneling term between wells is J l = − ∫dxφl∗

L (x)HspφlR(x).

The interaction term between atoms in the same well and on thesame energy level is Ul = g

∫dx|φl

r |4 and on different energylevels is U 01 = g

∫dx|φ0

r (x)|2|φ1r (x)|2. This term contributes

to atoms changing energy levels. We consider a harmonicpotential with oscillator frequency ω0, which is split by afocused laser beam located at the center of the trap anddescribed by a Gaussian potential V0exp(−x2/2σ 2). Thebarrier height is chosen to be V0 = 5�ω0 with width σ =0.1lho, where the harmonic oscillator length is lho = √

�/mω0.The localized functions φl

r are obtained by numerically solvingthe eigenstates of the single-particle Hamiltonian Hsp. Thecoupling g can be varied by Feshbach resonance in anexperiment. Parameters in units of the harmonic confinement�ω0 are J 0 = 0.26, J 1 = 0.34, E0

r = 1.25, E1r = 3.17, U 0 =

4/N , U 1 = 3U 0/4, and U 01 = U 0/2. For N = 40 particles,the system is complex enough to show thermalization and canbe studied by exact diagonalization. A small number of modesand large number of particles allow us to study the system alsoin the semiclassical limit.

III. TRUNCATED WIGNER APPROXIMATION

For large N , we can use the semiclassical truncatedWigner approximation (TWA). The leading corrections dueto finiteness of N is of the order 1/N2 [14]. For N = 40 wecan safely ignore such terms and the results can be comparedto exact diagonalization (see below). The use of TWA isjustified, since the initial state sampling naturally mimics the

ensemble averaging in statistical mechanics. As a matter offact, it was conjectured that it is probable that each Wignertrajectory approximately corresponds to a single realizationof experiments [14]. Within TWA the operators are treated ascomplex numbers, bl

r and bl∗r , satisfying the following set of

nonlinear equations:

i�∂bl

r

∂t= ∂HW

∂bl∗r

, (3)

where HW = 〈β|H |β〉 is the Weyl-ordered Hamiltonian op-erator. It is calculated by replacing bl

r → (blr + 1

2∂

∂bl∗r

) and

bl†r → (bl∗

r − 12

∂∂bl

r) in Eq. (2) [14]. We solve Eq. (3) by

sampling appropriately an initial quantum state. If the initialstate is a Fock state, bl

r are sampled for large N with fixedamplitude

√nl

r and uniformly random phase θ , i.e., blr =√

nlr exp(iθ ) [15]. We also study dynamics from the coherent

state sampled as blr = √

nlr + 1/2(η1 + iη2), where ηj are

real normal Gaussians. These have the correlations ηj = 0and ηjηk = δjk , where the overline denotes an average overmany samples [15]. The occupation numbers can be calculatedby averaging |bl

r (t)|2 over many realizations from the initialsampling. To ensure convergence the number of trajectoriesand number of particles are taken to be large, N = 104.

The dynamics of ultracold atoms in a single-band double-well potential has been studied extensively [16]. This systemcan be mapped onto the classical pendulum with smallJosephson oscillations and the self-trapping regime of atomsbeing identified with small-amplitude oscillations and fullrotation of the pendulum around its pivot point, respectively.Similarly, a two-band double-well potential filled with coldatoms can be described as two nontrivially coupled nonrigidpendulums in the semiclassical limit [17]. As a result, thereare regimes where the system is chaotic [18]. With our choiceof parameters, the Wigner trajectories are indeed chaotic asshown below.

Thermal equilibration through chaotic semiclassical dy-namics has been studied before, e.g., in the Dicke model [19]and in trapped atomic gases with spin-orbit coupling [20]. Inthis paper we extend that by showing in detail how an initialquantum state turns into the thermal state through chaoticevolution. The proper sampling of an initial quantum statemimics a statistical mechanical ensemble, while ergodicity ofWigner trajectories drives the initial pure distribution to thethermal one.

IV. EXACT DIAGONALIZATION

The semiclassical analysis will be compared with thepredictions of the full quantum dynamics for consistency. Weuse the Fock basis |n〉 = |n0

L〉 ⊗ |n0R〉 ⊗ |n1

L〉 ⊗ |n1R〉, where

|nlr〉 is a number state on level l and site r . The eigenstates

are |k〉 = ∑n Ck

n|n〉, where Ckn are extracted from exact

diagonalization of the Hamiltonian (2) for N = 40 particles.Assume that the initial state |φ0〉 = |n0〉 is a Fock state withfixed energy E0, implying that the coefficients αk ≡ Ck∗

n0are

obtained from the exact diagonalization (ED). The details ofthe initial state are irrelevant for subsequent evolution if itsenergy variance is small, as we discussed above. The system

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THERMALIZATION IN CLOSED QUANTUM SYSTEMS: . . . PHYSICAL REVIEW A 90, 053602 (2014)

for any such initial state with the same energy E0 relaxes tothe diagonal ensemble ρm = ∑

k |Ckn0

|2|k〉〈k|.Another commonly used state, albeit quite different from

the Fock state, is a coherent state, which is a superposition of allpossible Fock states, |β〉 = e−|b|2/2 ∑∞

n=0(bn/√

n!)|n〉. Insert-ing the resolution of identity expressed via the Fock basis, weget 〈β|H |β〉 = ∑

n,m〈β|n〉〈m|β〉Enm, where Enm = 〈n|H |m〉.We split it into the sums over diagonal and off-diagonalterms, 〈β|H |β〉 = ∑

n e−|b|2 |b|2n

n! En + ∑n�=m e−|b|2 b∗nbm√

n!m!Enm.

Since we require the energy variance in the Fock basis

to be small, which translates to√∑

m�=n E2nm � En, the

off-diagonal elements in the Hamiltonian matrix Enm arenegligibly small as compared to the diagonal elements En.Therefore we neglect the terms with n �= m. For a quantumsystem with large number of particles in each mode, |b|2 � 1,the Poissonian factor e−|b|2 |b|2n

n! becomes sharply peaked ataround n = |b|2. This allows us to approximate 〈β|H |β〉 ≈〈n|H |n〉 with n = |b|2. The energy variance of the coherentstate can be shown to get progressively small when we increasethe number of particles. Therefore it is supposed to relax tothe diagonal ensemble ρm if it has the same energy E0. Weexamine this within the semiclassical approach.

V. RESULTS

To compare ED and TWA, we scale all quantities withN. We study the dynamics from initial Fock and coherentstates with the same energy E0/N ≈ 2.25�ω0. The relaxationdynamics from TWA calculations is shown in Fig. 2. Thefinal relaxed values of the occupation numbers are in excellentagreement with the quantum diagonal ensemble prediction.For the two different initial conditions, the upper and lowermodes thermalize at n0

L ≈ n0R ≈ 0.4N,n1

L ≈ n1R ≈ 0.1N . The

independence on initial conditions is the hallmark of thermal-ization. The Wigner trajectories exhibit chaotic behavior by

0 20 40 60 80 1000

0.2

0.4

0.6

tω0

n/N

FockCoherentDiagonal

FIG. 2. (Color online) TWA dynamics of the occupation num-bers in each mode from Fock and coherent initial states with the sameenergy E0/N ≈ 2.25�ω0 and narrow energy variances �E0/E0 =0.09 and 0.02, respectively. They equilibrate at values which are inexcellent agreement with the quantum diagonal ensembles predictionfor N = 40. A single TWA trajectory is shown in semitransparentgray; it exhibits chaotic behavior. Only averaging over many suchtrajectories leads to the correct relaxed values of the occupationnumbers.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Re(b0L/

√N)

Im(b

0 L/√

N)

tω0=4 Fock

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Re(b0L/

√N)

Im(b

0 L/√

N)

tω0=4 Coherent

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Re(b0L/

√N)

Im(b

0 L/√

N)

tω0=100

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Re(b0L/

√N)

Im(b

0 L/√

N)

tω0=100

FIG. 3. (Color online) Ergodicity in TWA. As an example weshow Wigner distributions of one of the lower levels. Initially thesystem is prepared in the Fock state (green cycle shown on the left)or in the coherent state (green bump shown on the right). Trajectoriessampled from the initial states fill the available phase space as timeevolves (blue dots). The distributions become essentially the sameat some time and after that they do not change, which suggeststhermalization is reached within TWA. In this sense, quantumfluctuations of the initial state turn into thermal fluctuations in thecourse of evolution.

quickly filling the available phase space, as shown in Fig. 3.This is quite similar to classical thermalization, where suchbehavior of classical trajectories is the source of ergodicityin an ensemble of identical systems [8]. The quantum-mechanical fuzziness is the key to understand ergodicity inthe semiclassical language: averaging over the initial samplingreplaces ensemble averaging.

Within the realm of ED the diagonal ensemble ρm containsall necessary information about the relaxed state. Our aimis to calculate the population distribution in a mode Pn

and compare it with the analogous distribution derived fromthe TWA approximation. This quantity simply gives theoccupancy of that mode, 〈n〉 = ∑

n′ n′Pn′ , and the reduceddensity matrix of that mode,

∑n′ Pn′ |n′〉〈n′|. To calculate

this density distribution we express ρm via the Fock basis,ρm ≈ ∑

k,n |Ckn0

|2|Ckn|2|n〉〈n|, where again we rely on the

chaoticity of the eigenstates, leaving only the diagonal contri-bution, yielding Pn = ∑

k |Ckn0

|2|Ckn|2. We found a very good

agreement with the analogous distribution obtained from theTWA calculations, as it is shown in Fig. 4. The later is obtainedby noting that nl

r → |blr |2.

The resulting population distributions can be inferredfrom the ergodicity of the Wigner trajectories originatingfrom the initial sampling, which is narrow in energy. TheWigner distribution function of the entire system can thus berepresented as some sharply peaked function δ(HW − E0), themicrocanonical analog of the quantum case. We approximate itwith the Gaussian δ(x) = 1

σ√

2πexp(−x2/2σ 2) with σ ∼ �E0.

The Wigner distribution of a mode blr is then given by

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J. G. COSME AND O. FIALKO PHYSICAL REVIEW A 90, 053602 (2014)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

n0L/N

P

DiagonalTWAGC

0 0.2 0.4 0.6 0.8 10

2

4

6

8

n1L/N

P

DiagonalTWAGC

0 0.2 0.4 0.6 0.8 10

1

2

3

4

|b0L|/

√N

W

Wm

0 0.2 0.4 0.6 0.8 10

1

2

3

|b1L|/

√N

WWm

FIG. 4. (Color online) (Upper panels) Comparison of the distri-butions Pn derived from the diagonal ensemble with the correspond-ing distributions derived from the grand-canonical ensembles. Bothare compared with the corresponding data extracted from TWA.(Lower panels) The good agreement between the exact Wignerdistribution and the distribution Wm.

integrating out over all modes but one, Wm(blr ,b

l∗r ) =∫

δ(HW − E0)∏

l′ �=l,r ′ �=r dbl′r ′

∗dbl′

r ′ , which is numerically eval-uated using the Monte Carlo integration. We found goodagreement with the exact Wigner distributions as shown inFig. 4. This is conceptually analogous to CT, although theunderlying basis of our approach is quite distinct.

Having observed equilibration in each of the four modes,we now examine the states of each mode in more de-tail. Our aim is to demonstrate that the equilibrium statesare thermal. In studying thermalization in closed quantumsystems, the Hamiltonian of a system is usually split intoseveral parts, H = HS + HB + Hint, representing the sub-system, the bath, and interactions between them, leadingto their mutual thermalization. Equation (2) represents sucha situation: Each of the four modes can be regarded as asubsystem (=HS) coupled to the rest of the system (=HB)via the tunneling and coupling terms (=Hint). The modes mayexchange energy and particles via tunneling; therefore thermalstates of each mode are expected to be described by grand-canonical ensembles ρGC ∼ e−β(HS−μnS ), where for a givenmode HS = Ulnl

r (nlr − 1) + El

r nlr and nS = nl

r . To ensure thatthe resulting distributions are thermal, we fitted them with the

grand-canonical distribution ρGC . We have extracted β−1 ≈6�ω0, μ ≈ 3�ω0 and β−1 ≈ 12�ω0, μ ≈ 0.25�ω0 for thelower and upper modes, respectively. In passing, we notethat if we chose, e.g., the interband coupling U 01 to be muchsmaller than the current one, the situation is different: theequilibration is lost along with the chaotic behavior of theWigner trajectories.

VI. DISCUSSION AND CONCLUSIONS

We have demonstrated that each of the four modes thermal-izes with the rest of the system. We did this as follows. Onthe one hand, each mode is weakly coupled to each othervia tunneling and interaction terms in the Hamiltonian sothat they can exchange particles and energy. As a result, thecorresponding distributions are expected (and were shown)to be well described by the grand-canonical ensembles withappropriate temperatures and chemical potentials. On theother hand, the initial state independence of the final statesprovides further evidence of thermalization of each mode.Moreover, the final states of each mode can be inferred fromthe microcanonical ensemble of the whole system.

We have also shown that the semiclassical truncated Wignerapproach and full quantum description agree in reproducingthe states of each mode after they have been thermalized.While the quantum description of thermalization has beenelucidated extensively in the literature [1–4,7], in this workwe analyzed the semiclassical approach in order to seekfurther insights into the physics of quantum thermalization.The truncated Wigner approach has revealed deep connectionbetween thermalization in closed quantum systems and clas-sical thermalization. Although the main postulate of classicalstatistical mechanics is considered to be artificial and wasreplaced in quantum formalism [4], we have shown thatquantum-mechanical fuzziness of initial states within thesemiclassical formalism naturally supports the concept ofstatistical mechanical ensembles. Moreover, the ergodicityof Wigner trajectories leads to thermal relaxation. Beingconceptually different, our study adds to the understanding ofquantum thermalization and to the recent advances in pushingthe limits of quantum thermalization and its understanding inthe macroscopic limit [10,19–21].

ACKNOWLEDGMENTS

We thank Anatoli Polkovnikov and Joachim Brand foruseful and insightful comments. O.F. was supported by theMarsden Fund (Project No. MAU1205), administrated by theRoyal Society of New Zealand.

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