Thermalization in closed quantum systems: semiclassical approach
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov Thermalization in closed quantum systems: Semiclassical approach
J. G. Cosme and O. Fialko Institute for Advanced Study and Centre for Theoretical Chemistry and Physics, Massey University, Auckland, New Zealand Institute of Natural and Mathematical Sciences and Centre for TheoreticalChemistry and Physics, Massey University, Auckland, New Zealand (Dated: September 18, 2018)Thermalization in closed quantum systems can be understood either by means of the eigenstatethermalization hypothesis or the concept of canonical typicality. Both concepts are based on quan-tum mechanical formalism such as spectral properties of the eigenstates or entanglement betweensubsystems respectively. Here we study instead the onset of thermalization of Bose particles in atwo-band double well potential using the truncated Wigner approximation. This allows us to usethe familiar classical formalism to understand quantum thermalization in this system. In particular,we demonstrate that sampling of an initial quantum state mimics a statistical mechanical ensemble,while subsequent chaotic classical evolution turns the initial quantum state into the thermal state.
I. INTRODUCTION
Thermalization is regarded as one of the most funda-mental facts in physics. Its foundations and basic pos-tulates are still the subject of debates. Recent advancesin addressing this issue have brought us two quantumconcepts that shed light on the onset of thermal equi-librium inside closed quantum systems. One of them isthe so-called eigenstate thermalization hypothesis (ETH,[1, 2]). ETH conjectures that under certain initial condi-tions the expectation values of observables of a quantumsystem behave as if they were thermal during later timesof its evolution. In another approach, so-called canonicaltypicality (CT, [3]), the process of thermalization is ex-plained via entanglement between subsystems [4]. How-ever, the quantum ideas can be hard to imagine. Semi-classical ideas, on the other hand, may provide intuitivephysical insights into quantum mechanics [5]. Here westudy the onset of thermalization in a quantum systemusing the semiclassical truncated Wigner approximation.This allows us to use the familiar classical formalism toexplain how quantum fluctuations in an initial state turninto thermal fluctuations at later times during chaoticclassical evolution of Wigner trajectories.We introduce notations to be used throughout the restof the paper by reviewing briefly recent progress on quan-tum thermalization. ETH was tested against anotherhypothesis numerically in [6] (see also [7]). To under-stand ETH, consider the initial state | φ i = P k α k | k i ,where | k i are the eigenstates of a Hamiltonian ˆ H witheigenvalues E k . The eigenstates are thermal, which isreflected in the spectral properties as follows. The stateevolves as | φ ( t ) i = e − i ˆ Ht/ ~ | φ i . As a prerequisite forthermalization, the expectation value of an observable h ˆ Oi = P k,l α ∗ l α k e i ( E l − E k ) t/ ~ O lk with O lk = h l | ˆ O| k i ,after sufficient time, must relax to the long-time aver-age h ˆ Oi = P k | α k | O kk . ETH then states that thisoccurs if O kk is a smooth function of E k , while theoff-diagonal elements O kl are negligible. Moreover, theenergy E = P k | α k | E k must have small uncertainty∆ E = P k | α k | ( E k − E ) [2, 6] to ensure that the re- laxed state is thermal and depends only on the energy.CT, on the other hand, replaces the need for any en-semble averaging, the main postulate of statistical me-chanics considered to be artificial. Here, thermaliza-tion is reached on the level of a subsystem. The re-duced density matrix of the subsystem is canonical forthe majority of the possible pure states of the entiremany-body system under global constraint such as en-ergy. This was established under great generality by in-voking Levy’s lemma [4]. Then, the properties of thesubsystem can be calculated using the reduced densitymatrix constructed from the pure density matrix of theentire system ˆ ρ p = | φ ( t ) ih φ ( t ) | giving the same predic-tion as if the entire system was in the microcanonicalstate ˆ ρ m = N − E ,δ P k | k ih k | . The sum is over N E ,δ eigenstates | k i , with energies lying within some window[ E − δ, E + δ ] such that δ ≪ E .In classical physics, thermalization is explained bymeans of classical chaos. Being chaotic, the system wan-ders all over the constant energy surface in the phasespace, becoming ergodic. Averaged properties of the sys-tem over a long time can then be estimated by averagingover the accessible phase space [8]. To relate that to ther-malization in closed quantum systems, it was originallyargued that ETH can be explained if the quantum systemis chaotic in the classical limit [2]. However, many quan-tum systems do not have classical counterparts; neverthe-less, 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.we write the density matrix corresponding to | φ ( t ) i asˆ ρ p = X k | α k | | k ih k | + X k = l α k α ∗ l e − i ( E k − E l ) t/ ~ | k ih l | . (1)Here the coefficients α k are assumed to be random andindependent, implying that the second term quickly av-erages to zero at long times. In the remaining first term,the factor | α k | is assumed to be a smooth function withnarrow width ∆ E ≪ E , yielding ˆ ρ p ≈ ˆ ρ m . II. THE MODEL
We study thermalization in a quantum system whichis amenable to the semiclassical analysis. Bosons aretrapped in a double-well potential shown in Fig. 1. Theyoccupy four energy levels (we will refer to them alsoas modes) and are described by the following two-bandBose-Hubbard Hamiltonian [13]:ˆ H = − X r = r ′ ,l J l ˆ b l † r ˆ b lr ′ + X r,l U l ˆ n lr (ˆ n lr −
1) + X r,l E lr ˆ n lr + U X r,l = l ′ (2ˆ n lr ˆ n l ′ r + ˆ b l † r ˆ b l † r ˆ b l ′ r ˆ b l ′ r ) , (2)where ˆ b l † r and ˆ b lr are the bosonic creation and annihi-lation operators respectively of an atom in well r andenergy level l . The parameters in the Hamiltonian canbe easily evaluated for a specific double-well potential[13]. The ground and first excited-state energies are E lr = R dxφ l ∗ r ( x ) ˆ H sp φ lr ( x ). The tunneling term betweenwells is J l = − R dxφ l ∗ L ( x ) ˆ H sp φ lR ( x ). The interactionterm between atoms in the same well and on the sameenergy level is U l = g R dx | φ lr | and on different en-ergy levels is U = g R dx | φ r ( x ) | | φ r ( x ) | . This termcontributes to atoms changing energy levels. We con-sider a harmonic potential with oscillator frequency ω ,which is split by a focused laser beam located at thecenter of the trap and described by a Gaussian poten-tial V exp( − x / σ ). The barrier height is chosen to be V = 5 ~ ω with width σ = 0 . l ho , where the harmonic os-cillator length is l ho = p ~ /mω . The localized functions φ lr are obtained by numerically solving the eigenstatesof the single particle Hamiltonian ˆ H sp . The coupling g can be varied by Feshbach resonance in an experiment.Parameters in units of the harmonic confinement ~ ω are J = 0 . J = 0 . E r = 1 . E r = 3 . U = 4 /N , U = 3 U /
4, and U = U /
2. For N = 40 particles, thesystem is complex enough to show thermalization and FIG. 1. Schematic of the double-well potential with two en-ergy bands. The diagram shows the tunneling of particles andhow the energy levels change due to the interactions betweenthem. The interband coupling U makes the system complexenough to show quantum thermalization of the particles. can be studied by exact diagonalization. A small num-ber of modes and large number of particles allow us tostudy the system also in the semiclassical limit. III. TRUNCATED WIGNER APPROXIMATION
For large N , we can use the semiclassical truncatedWigner approximation (TWA). The leading correctionsdue to finiteness of N is of the order 1 /N [14]. For N = 40 we can safely ignore such terms and the resultscan be compared to exact diagonalization (see below).The use of TWA is justified since the initial state sam-pling naturally mimics the ensemble averaging in statis-tical mechanics. As a matter of fact, it was conjecturedthat it is probable that each Wigner trajectory approxi-mately corresponds to a single realization of experiments[14]. Within TWA the operators are treated as complexnumbers, b lr and b l ∗ r , satisfying the following set of non-linear equations: i ~ ∂b lr ∂t = ∂H W ∂b l ∗ r , (3)where H W = h β | ˆ H | β i is the Weyl-ordered Hamiltonianoperator. It is calculated by replacing ˆ b lr → (cid:18) b lr + ∂∂b l ∗ r (cid:19) and ˆ b l † r → (cid:18) b l ∗ r − ∂∂b lr (cid:19) in Eq. (2) [14]. We solveEq. (3) by sampling appropriately an initial quantumstate. If the initial state is a Fock state, b lr are sam-pled for large N with fixed amplitude p n lr and uni-formly random phase θ , i.e., b lr = p n lr exp( iθ ) [15]. Wealso study dynamics from the coherent state sampled as b lr = p n lr +1 / η + iη ), where η j are real normal Gaus-sians. These have the correlations η j = 0 and η j η k = δ jk ,where the overline denotes an average over many samples[15]. The occupation numbers can be calculated by aver-aging | b lr ( t ) | over many realizations from the initial sam-pling. To ensure convergence the number of trajectoriesand number of particles are taken to be large, N = 10 .The dynamics of ultracold atoms in a single-band dou-ble well potential has been studied extensively [16]. Thissystem can be mapped onto the classical pendulum withsmall Josephson oscillations and the self-trapping regimeof atoms being identified with small-amplitude oscilla-tions and full rotation of the pendulum around its pivotpoint, respectively. Similarly, a two-band double-well po-tential filled with cold atoms can be described as twonontrivially coupled nonrigid pendulums in the semiclas-sical limit [17]. As a result, there are regimes, where thesystem is chaotic [18]. With our choice of parameters, theWigner trajectories are indeed chaotic as shown below.Thermal equilibration through chaotic semiclassicaldynamics has been studied before, e.g., in the Dickemodel [19] and in trapped atomic gases with spin-orbitcoupling [20]. In this paper we extend that by showing indetail how an initial quantum state turns into the thermal tω h n i / N FockCoherentDiagonal
FIG. 2. TWA dynamics of the occupation numbers ineach mode from Fock and coherent initial states with thesame energy E /N ≈ . ~ ω and narrow energy variances∆ E /E = 0 .
09 and 0 .
02 respectively. They equilibrate atvalues which are in excellent agreement with the quantum di-agonal ensembles prediction for N = 40. A single TWA tra-jectory is shown in semitransparent gray; it exhibits chaoticbehavior. Only averaging over many such trajectories leadsto the correct relaxed values of the occupation numbers. state through chaotic evolution. The proper sampling ofan initial quantum state mimics a statistical mechanicalensemble, while ergodicity of Wigner trajectories drivesthe initial pure distribution to the thermal one. IV. EXACT DIAGONALIZATION
The semiclassical analysis will be compared with thepredictions of the full quantum dynamics for consistency.We use the Fock basis | n i = | n L i ⊗ | n R i ⊗ | n L i ⊗ | n R i ,where | n lr i is a number state on level l and site r .The eigenstates are | k i = P n C kn | n i , where C kn are ex-tracted from exact diagonalization of the Hamiltonian(2) for N = 40 particles. Assume that the initial state | φ i = | n i is a Fock state with fixed energy E , im-plying that the coefficients α k ≡ C k ∗ n are obtained fromthe exact diagonalization (ED). The details of the initialstate are irrelevant for subsequent evolution if its energyvariance is small, as we discussed above. The system forany such initial state with the same energy E relaxes tothe diagonal ensemble ˆ ρ m = P k | C kn | | k ih k | .Another commonly used state albeit quite differ-ent from the Fock state, is a coherent state, whichis a superposition of all possible Fock states, | β i = e −| b | / P ∞ n =0 ( b n / √ n !) | n i . Inserting the resolution ofidentity expressed via the Fock basis, we get h β | ˆ H | β i = P n,m h β | n ih m | β i E nm , where E nm = h n | ˆ H | m i . We splitit into the sums over diagonal and off-diagonal terms, h β | ˆ H | β i = P n e −| b | | b | n n ! E n + P n = m e −| b | b ∗ n b m √ n ! m ! E nm . Since we require the energy variance in the Fock basis −1 −0.5 0 0.5 1−1−0.500.51
Re( b L / √ N ) I m ( b L / √ N ) t ω =4 Fock −1 −0.5 0 0.5 1−1−0.500.51 Re( b L / √ N ) I m ( b L / √ N ) t ω =4 Coherent −1 −0.5 0 0.5 1−1−0.500.51 Re( b L / √ N ) I m ( b L / √ N ) t ω =100 −1 −0.5 0 0.5 1−1−0.500.51 Re( b L / √ N ) I m ( b L / √ N ) t ω =100 FIG. 3. Ergodicity in TWA. As an example we show Wignerdistributions of one of the lower levels. Initially the systemis prepared in the Fock state (green cycle shown on the left)or in the coherent state (green bump shown on the right).Trajectories sampled from the initial states fill the availablephase space as time evolves (blue dots). The distributions be-come essentially the same at some time and after that they donot change, which suggests thermalization is reached withinTWA. In this sense, quantum fluctuations of the initial stateturn into thermal fluctuations in the course of evolution. to be small, which translates to qP m = n E nm ≪ E n ,the off-diagonal elements in the Hamiltonian matrix E nm are negligibly small as compared to the diagonal elements E n . Therefore we neglect the terms with n = m . For aquantum system with large number of particles in eachmode, | b | ≫
1, the Poissonian factor e −| b | | b | n n ! becomessharply peaked at around n = | b | . This allows us toapproximate h β | ˆ H | β i ≈ h n | ˆ H | n i with n = | b | . The en-ergy variance of the coherent state can be shown to getprogressively small when we increase the number of par-ticles. Therefore, it is supposed to relax to the diagonalensemble ˆ ρ m if it has the same energy E . We examinethis within the semiclassical approach. V. RESULTS
To compare ED and TWA, we scale all quantities with N. We study the dynamics from initial Fock and coher-ent states with the same energy E /N ≈ . ~ ω . Therelaxation dynamics from TWA calculations is shown inFig. 2. The final relaxed values of the occupation num-bers are in excellent agreement with the quantum di-agonal ensembles prediction. For the two different ini-tial conditions the upper and lower modes thermalizeat n L ≈ n R ≈ . N, n L ≈ n R ≈ . N . The indepen-dence on initial conditions is the hallmark of thermaliza-tion. The Wigner trajectories exhibit chaotic behaviorby quickly filling the available phase space, as shown inFig. 3. This is quite similar to classical thermalization,where such behavior of classical trajectories is the sourceof ergodicity in an ensemble of identical systems [8]. Thequantum-mechanical fuzziness is the key to understandergodicity in the semiclassical language: averaging overthe initial sampling replaces ensemble averaging.Within the realm of ED the diagonal ensemble ˆ ρ m contains all necessary information about the relaxedstate. Our aim is to calculate the population distri-bution in a mode P n and compare it with the anal-ogous distribution derived from the TWA approxima-tion. This quantity simply gives the occupancy of thatmode, h ˆ n i = P n ′ n ′ P n ′ , and the reduced density ma-trix of that mode, P n ′ P n ′ | n ′ ih n ′ | . To calculate thisdensity distribution we express ˆ ρ m via the Fock basisˆ ρ m ≈ P k,n | C kn | | C kn | | n ih n | , where again we rely on thechaoticity of the eigenstates, leaving only the diagonalcontribution, yielding P n = P k | C kn | | C kn | . We founda very good agreement with the analogous distributionobtained from the TWA calculations, as it is shown inFig. 4. The later is obtained by noting that n lr → | b lr | .The resulting population distributions can be inferredfrom the ergodicity of the Wigner trajectories originat-ing from the initial sampling which is narrow in en-ergy. The Wigner distribution function of the entiresystem can thus be represented as some sharply peakedfunction ¯ δ ( H W − E ), the microcanonical analog of thequantum case. We approximate it with the Gaussian¯ δ ( x ) = σ √ π exp( − x / σ ) with σ ∼ ∆ E . The Wignerdistribution of a mode b lr is then given by integratingout over all modes but one, W m ( b lr , b l ∗ r ) = R ¯ δ ( H W − E ) Q l ′ = l, r ′ = r db l ′ r ′ ∗ db l ′ r ′ , which is numerically evaluatedusing the Monte Carlo integration. We found good agree-ment with the exact Wigner distributions as shown inFig. 4. This is conceptually analogous to CT, althoughthe underlying basis of our approach is quite distinct.Having observed equilibration in each of the fourmodes, we now examine the states of each mode in moredetail. Our aim is to demonstrate that the equilibriumstates are thermal. In studying thermalization in closedquantum systems, the Hamiltonian of a system is usuallysplit into several parts, ˆ H = ˆ H S + ˆ H B + ˆ H int , represent-ing the subsystem, the bath, and interactions betweenthem leading to their mutual thermalization. Equation(2) represents such a situation: Each of the four modescan be regarded as a subsystem (= ˆ H S ) coupled to therest of the system (= ˆ H B ), via the tunneling and couplingterms (= ˆ H int ). The modes may exchange energy andparticles via tunneling; therefore thermal states of eachmode are expected to be described by grand-canonicalensembles ˆ ρ GC ∼ e − β ( ˆ H S − µ ˆ n S ) , where for a given modeˆ H S = U l ˆ n lr (ˆ n lr −
1) + E lr ˆ n lr and ˆ n S = ˆ n lr . To ensurethat the resulting distributions are thermal, we fittedthem with the grand-canonical distribution ˆ ρ GC . Wehave extracted β − ≈ ~ ω , µ ≈ ~ ω and β − ≈ ~ ω , n L /N P DiagonalTWAGC n L /N P DiagonalTWAGC | b L | / √ N W W m | b L | / √ N W W m FIG. 4. (Upper panels) Comparison of the distributions P n derived from the diagonal ensemble with the correspond-ing distributions derived from the grand-canonical ensembles.Both are compared with the corresponding data extractedfrom TWA. (Lower panels) The good agreement between theexact Wigner distribution and the distribution W m . µ ≈ . ~ ω for the lower and upper modes, respectively.In passing, we note that if we chose, e.g., the inter-bandcoupling U to be much smaller than the current one,the situation is different: the equilibration is lost alongwith the chaotic behavior of the Wigner trajectories. VI. DISCUSSIONS AND CONCLUSIONS
We have demonstrated that each of the four modesthermalizes with the rest of the system. We did thisas follows. On the one hand, each mode is weakly cou-pled to each other via tunneling and interaction termsin the Hamiltonian so that they can exchange particlesand energy. As a result, the corresponding distributionsare expected (and were shown) to be well described bythe grand-canonical ensembles with appropriate temper-atures and chemical potentials. On the other hand, theinitial state independence of the final states provides fur-ther evidence of thermalization of each mode. Moreover,the final states of each mode can be inferred from themicrocanonical ensemble of the whole system.We have also shown that the semiclassical truncatedWigner approach and full quantum description agree inreproducing the states of each mode after they have beenthermalized. While the quantum description of thermal-ization has been elucidated extensively in the literature[1–4, 7], in this work we analyzed the semiclassical ap-proach in order to seek further insights into the physicsof quantum thermalization. The truncated Wigner ap-proach has revealed deep connection between thermal-ization in closed quantum systems and classical thermal-ization. Although the main postulate of classical sta-tistical 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 ergodic-ity of Wigner trajectories leads to thermal relaxation.Being conceptually different, our study adds to the un-derstanding of quantum thermalization and to the recentadvances in pushing the limits of quantum thermalizationand its understanding in the macroscopic limit [10, 19– 21].
ACKNOWLEDGMENTS
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