The 2008 proposal of Cramer, Flesch, McCulloch, Schollwöck and Eisert (Phys. Rev. Lett. 101, 063001) established the period-2 optical-superlattice quench — bosonic atoms prepared in the density-wave state ∣1,0,1,0,… ⟩|1,0,1,0,\dots\rangle of the one-dimensional Bose-Hubbard model with the tunnel coupling suddenly switched on — as an experimentally realizable probe of local relaxation in an isolated quantum many-body system, and left open what the locally relaxed state is at general finite interaction strength UU (Gibbs, generalized Gibbs, or otherwise), how its ensemble parameters are fixed by the initial state, whether and how the quasi-momentum distribution S(q,t)S(q,t) relaxes for sizable UU, and how to access the late-time window despite linearly growing entanglement. Later work substantially resolved the late-time-access question by exactly the proposed route: the experiment of Trotzky et al. (Nature Physics 8, 325 (2012)) realized the setting and found all measured local observables relaxing to values compatible with a global maximum-entropy (Gibbs) state, and subsequent numerics confirmed ensemble agreement in related Bose-Hubbard quenches. The surviving open core is a complete, quantitative, and ideally rigorous characterization of local relaxation at finite UU: determination, by proof or controlled analytic theory, of whether the relaxed state is the Gibbs ensemble — or a generalized Gibbs ensemble built from additional conserved quantities — with the ensemble parameters mapped from the initial state, an explanation of the interaction-enhanced damping at intermediate UU, and the long-time fate of S(q,t)S(q,t) for the density-wave quench.

Background

A closed quantum many-body system evolving unitarily never reaches thermal equilibrium in the strict sense — a pure state stays pure — yet observables restricted to a small subsystem can become indistinguishable from those of a thermal (maximum-entropy) state for very long times. Whether and when this local relaxation occurs, and which statistical ensemble describes the relaxed state, is the central question of quantum thermalization. The canonical dichotomy is between the Gibbs ensemble, fixed only by the conserved total energy and particle number, and the generalized Gibbs ensemble (GGE) — the maximum-entropy state built from the complete set of conserved quantities that exist in integrable models — a distinction introduced for hard-core lattice bosons by Rigol, Dunjko, Yurovsky and Olshanii (Phys. Rev. Lett. 98, 050405 (2007)). For free models, exact local relaxation can be proven: Cramer, Dawson, Eisert and Osborne (Phys. Rev. Lett. 100, 030602 (2008)) showed, for Bose-Hubbard systems quenched into the non-interacting regime, that the reduced states of blocks of sites converge in trace norm to a maximum-entropy state, without any time average.

The specific setting is the quench proposed by Cramer, Flesch, McCulloch, Schollwöck and Eisert (Phys. Rev. Lett. 101, 063001 (2008)) and analyzed in depth by Flesch, Cramer, McCulloch, Schollwöck and Eisert (Phys. Rev. A 78, 033608 (2008)): bosonic atoms in a period-2 optical superlattice, a standing-wave light potential whose unit cell contains two lattice sites, which allows an alternating energy bias and hence the preparation of a product state with exactly one atom on every odd site and none on the even sites. When the superlattice is suddenly switched off, the state ∣ψ(0)⟩=∣1,0,1,0,… ⟩|\psi(0)\rangle=|1,0,1,0,\dots\rangle evolves under the one-dimensional Bose-Hubbard Hamiltonian

H^=−J∑i=1L(b^i+1†b^i+b^i†b^i+1)+U2∑i=1Ln^i(n^i−1)−μ∑i=1Ln^i,\hat H=-J\sum_{i=1}^{L}\left(\hat b^\dagger_{i+1}\hat b_i+\hat b^\dagger_i\hat b_{i+1}\right)+\frac{U}{2}\sum_{i=1}^{L}\hat n_i(\hat n_i-1)-\mu\sum_{i=1}^{L}\hat n_i,

where b^i\hat b_i annihilates a boson on site ii, n^i=b^i†b^i\hat n_i=\hat b_i^\dagger\hat b_i, JJ is the hopping matrix element, UU the on-site repulsion and μ\mu the chemical potential. In the two exactly solvable limits the local density relaxes by dephasing as ⟨n^i(t)⟩=12−(−1)i2J0(4t)\langle\hat n_i(t)\rangle=\tfrac{1}{2}-\tfrac{(-1)^i}{2}J_0(4t), a Bessel function with an asymptotic power-law decay t−1/2t^{-1/2}: for U=0U=0 this follows from the free-boson solution, and for U→∞U\to\infty the bosons become hard-core (at most one per site) and map via the Jordan-Wigner transformation (the standard mapping of hard-core bosons to free spinless fermions) to free fermions with the same decay. At finite UU, time-dependent DMRG (t-DMRG) — a matrix-product-state algorithm that follows coherent time evolution of one-dimensional quantum systems — shows local relaxation over the accessible window t≲6/Jt\lesssim 6/J, a strongly enhanced damping at intermediate U/J≈3U/J\approx 3, and a nearest-neighbor correlator whose relaxed value follows the J/UJ/U dependence that thermal perturbation theory predicts for the Gibbs state ρ^β∝e−βH^\hat\rho_\beta\propto e^{-\beta\hat H} of the post-quench Hamiltonian. The experiment of Trotzky et al. (Nature Physics 8, 325 (2012)) realized this proposal with 87^{87}Rb atoms and found densities, currents and coherences relaxing, for U/J≈2.4U/J\approx 2.4–1010, to steady values compatible with a global maximum-entropy state specified only by energy and particle number.

Four limitations were left open by the two 2008 papers and frame the problem. First, the nature of the locally relaxed state at general finite UU beyond the numerically accessible times — Gibbs, generalized Gibbs, or something else — including the role of any further conserved quantities; the finite-UU Bose-Hubbard chain is believed non-integrable (its level statistics are those of quantum-chaotic systems away from the integrable limits, as shown for the extended Bose-Hubbard model by Kollath, Roux, Biroli and Läuchli, J. Stat. Mech. (2010) P08011), so the standard expectation is Gibbs relaxation via the eigenstate thermalization hypothesis (ETH) — the conjecture that individual high-energy eigenstates of non-integrable Hamiltonians already carry thermal values of local observables — but ETH itself remains unproven, as surveyed by Gogolin and Eisert (Rep. Prog. Phys. 79, 056001 (2016)). Second, the long-time fate of the quasi-momentum distribution (structure factor) S(q,t)=L−1∑i,jeiq(i−j)⟨b^i†(t)b^j(t)⟩S(q,t)=L^{-1}\sum_{i,j}e^{iq(i-j)}\langle\hat b_i^\dagger(t)\hat b_j(t)\rangle: at U=0U=0 the momentum-mode occupations are constants of motion, so S(q,t)S(q,t) does not relax at all, while at sizable UU relaxation requires mode coupling whose strength, timescale and endpoint were left undetermined. Third, the entanglement bottleneck: the entanglement entropy grows linearly in time after the quench, so exact matrix-product-state simulations become exponentially expensive, limiting early studies to t∼6/Jt\sim 6/J. Fourth, the quantitative map from the initial state to the ensemble parameters — the effective inverse temperature β\beta (and chemical potential) — was not established for the interacting case, β\beta entering only as an external parameter of the perturbative comparison.

Problem Statement

Consider the one-dimensional Bose-Hubbard Hamiltonian

H^=−J∑i=1L(b^i+1†b^i+b^i†b^i+1)+U2∑i=1Ln^i(n^i−1)−μ∑i=1Ln^i,\hat H=-J\sum_{i=1}^{L}\left(\hat b^\dagger_{i+1}\hat b_i+\hat b^\dagger_i\hat b_{i+1}\right)+\frac{U}{2}\sum_{i=1}^{L}\hat n_i(\hat n_i-1)-\mu\sum_{i=1}^{L}\hat n_i,

where b^i\hat b_i annihilates a boson on site ii, n^i=b^i†b^i\hat n_i=\hat b_i^\dagger\hat b_i counts the particles on site ii, J>0J>0 is the hopping matrix element, U>0U>0 the finite on-site repulsion, and μ\mu the chemical potential, on lattices of even length LL (open or periodic boundary conditions, to be declared by the solution, since the statements sought are thermodynamic-limit statements), initialized in the product (density-wave) state ∣ψ(0)⟩=∣1,0,1,0,…,1,0⟩|\psi(0)\rangle=|1,0,1,0,\dots,1,0\rangle — the state prepared by a period-2 optical superlattice before the tunnel coupling is suddenly switched on — and evolved unitarily, in the standard thermodynamic ordering (relaxation windows [trel,trec][t_{\mathrm{rel}},t_{\mathrm{rec}}] with L→∞L\to\infty taken before t→∞t\to\infty, up to finite-size recurrences). For general finite interaction strength U/JU/J — not only the exactly solvable free limits U=0U=0 and U→∞U\to\infty — characterize local relaxation completely, as one coherent objective with three jointly supporting deliverables:

(1) Nature of the relaxed state and the ensemble map. Determine whether the reduced states of blocks of consecutive sites converge, in trace norm (the sum of the singular values of the difference of the two reduced density operators) and up to recurrences, to the reduction of a maximum-entropy state, and identify that ensemble: is it the Gibbs ensemble — the maximum-entropy state fixed only by the initial state's conserved energy and particle number — or a generalized Gibbs ensemble built from additional conserved quantities? Determine also the quantitative map from the initial state to the ensemble parameters — the effective inverse temperature β\beta (and chemical potential) as functions of U/JU/J — such that the relaxed values of quasi-local observables (operators supported on finitely many consecutive sites) are predictable ab initio from the initial condition.

(2) Relaxation dynamics and mechanism. Provide a controlled analytic or numerical account of how relaxation proceeds at finite UU: the time dependence of quasi-local observables such as ⟨n^i(t)⟩\langle\hat n_i(t)\rangle and ⟨b^i+1†(t)b^i(t)⟩\langle\hat b^\dagger_{i+1}(t)\hat b_i(t)\rangle, their power-law exponents or exponential rates and their dependence on U/JU/J, and in particular the mechanism of the interaction-enhanced damping at intermediate U/J≈3U/J\approx 3–55, where relaxation is observed to be much faster than the t−1/2t^{-1/2} decay of both free limits.

(3) Long-time fate of the momentum distribution. Settle whether, on what timescale, and by what mechanism the quasi-momentum distribution S(q,t)=L−1∑i,jeiq(i−j)⟨b^i†(t)b^j(t)⟩S(q,t)=L^{-1}\sum_{i,j}e^{iq(i-j)}\langle\hat b_i^\dagger(t)\hat b_j(t)\rangle relaxes at finite UU — where the momentum-mode occupations are no longer conserved — and determine its final distribution, including whether it reaches the Gibbs prediction or retains memory of the initial density wave.

The three deliverables are jointly posed and, together, target a quantitative (and, where possible, rigorous) characterization of local relaxation for this canonical quench, at the level of a controlled analytic or numerical determination with quantified uncertainties. Proving (or refuting) that the relaxed state is the Gibbs ensemble for the non-integrable Bose-Hubbard chain would be a concrete instance of the eigenstate-thermalization program rather than the general ETH problem itself.

References

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