When a charge-density wave of ultracold bosons in decoupled one-dimensional optical-lattice chains (a Bose-Hubbard model with finite on-site interaction U/JU/J) is released and evolves in isolation, its quasi-local densities and currents relax to steady values far faster than in the analytically solvable limits of non-interacting (U/J=0U/J=0) and hard-core (U/J→∞U/J\to\infty) bosons, where damping follows a t−1/2t^{-1/2} law: measured short-time power-law exponents reach values around 0.860.86 instead of 0.50.5, and the relaxed values of the measured observables are compatible with a maximum-entropy (grand-canonical) state fixed only by the total energy and particle number. The originating experiment (Trotzky et al., Nature Physics 8, 325 (2012)) explicitly identifies the exact origin of this enhanced relaxation as one of the major open problems it poses, together with the questions of whether the locally relaxed state is fully thermal and of how single-chain-resolved measurements could settle it. The problem is to give a microscopic, quantitative account of the relaxation law of quasi-local observables after this interacting one-dimensional Bose-Hubbard quench — including the relative roles of interaction-driven scattering, the harmonic trapping potential, and next-nearest-neighbor hopping — for an identification of the ensemble that describes the relaxed state for all few- and multi-site observables, and for an experimental test of both at the level of individual chains.

Background

Ultracold atoms in optical lattices realize the Bose-Hubbard model, in which bosons hop between neighboring lattice sites with amplitude JJ and repel each other on the same site with interaction energy UU. A bichromatic optical superlattice (two superposed standing-wave lattices of commensurate period) can prepare a high-fidelity charge-density-wave initial state ∣1,0,1,0,… ⟩|1,0,1,0,\dots\rangle in which every other site of each one-dimensional chain is occupied by one atom; suddenly lowering the barriers on every second site is a quantum quench that switches on the tunnel coupling JJ and lets the closed system evolve unitarily. In the experiment of Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert and Bloch (Nature Physics 8, 325 (2012)), many parallel one-dimensional chains with different atom numbers were trapped in a weak harmonic potential KK, and the time evolution was tracked through quasi-local observables: the even- and odd-site populations, the quasi-local density current, and the nearest-neighbor coherence visible in the time-of-flight interference pattern (the standard ultracold-atom measurement in which the trap is switched off, the atoms expand ballistically, and the momentum distribution is imaged). Time-dependent density-matrix renormalization group (t-DMRG) simulations — a variational tensor-network method able to follow one-dimensional many-body dynamics for short to intermediate times, limited at long times by the growth of entanglement — agreed quantitatively with the data for the accessible times. In the two exactly solvable limits U/J=0U/J=0 and U/J→∞U/J\to\infty (hard-core bosons, which cannot share a site and can be mapped to free fermions), local densities oscillate as zeroth-order Bessel functions damped by a power law t−1/2t^{-1/2}, a mechanism of local relaxation by ballistically propagating excitations. For finite U/JU/J the experiment found markedly faster damping, with fitted power-law exponents up to α=0.86(7)\alpha=0.86(7) at U/J=5.16U/J=5.16, and numerical simulations of homogeneous chains had already seen the same trend. After relaxation, the steady-state values of the measured quasi-local observables were compatible with a maximum-entropy state in the sense of Jaynes: a Gibbs (grand-canonical) ensemble fixed only by the conserved total energy and total particle number, with the nearest-neighbor visibility (the contrast of the time-of-flight interference peaks) peaking near U/J=4U/J=4 as perturbation theory for a thermal state suggests. Whether this compatibility constitutes full thermalization — equality of all few- and multi-site observables with their equilibrium-ensemble values, as predicted for generic non-integrable systems by the eigenstate thermalization hypothesis (ETH), the proposal that individual high-energy eigenstates of a chaotic many-body Hamiltonian already encode thermal values of local observables — was left open, as was the physical identification of any additional conserved quantities that a maximum-entropy description should include. A key experimental limitation is that all signals were averages over an inhomogeneous ensemble of many chains with different atom numbers, which masks global observables and any chain-to-chain differences; the authors proposed preparing or selecting a single chain as the way to answer what global state is actually reached, including possible prethermalization (long-lived near-thermal transient states that precede true thermalization) and experimental tests of ETH.

Problem Statement

Provide a microscopic, quantitative account of the local relaxation following the interacting one-dimensional Bose-Hubbard density-wave quench realized by Trotzky et al. (Nature Physics 8, 325 (2012)), resolving the questions the experiment itself posed as open. Concretely, for the chain with Hamiltonian H^=−J∑j(b^j+1†b^j+h.c.)−J′∑j(b^j+2†b^j+h.c.)+U2∑jn^j(n^j−1)+K∑jj2n^j\hat H=-J\sum_j(\hat b^\dagger_{j+1}\hat b_j+\mathrm{h.c.})-J'\sum_j(\hat b^\dagger_{j+2}\hat b_j+\mathrm{h.c.})+\frac{U}{2}\sum_j \hat n_j(\hat n_j-1)+K\sum_j j^2\hat n_j, where the weak next-nearest-neighbor hopping J′J' (with J′/JJ'/J small but fixed by the bichromatic superlattice geometry) is retained because it is present in the experiment, quenched from the two-periodic state ∣1,0,1,0,… ⟩|1,0,1,0,\dots\rangle: (1) derive the decay law of the quasi-local densities, currents, and nearest-neighbor coherences — its functional form and parameter dependence on U/JU/J, trap strength KK, and longer-range hopping — and give the mechanism by which finite interactions accelerate local relaxation relative to the t−1/2t^{-1/2} damping of the U/J=0U/J=0 and U/J→∞U/J\to\infty limits, quantitatively separating interaction-induced relaxation from trap- and ensemble-induced dephasing, so that the observed exponents (e.g. α≈0.86\alpha\approx0.86 versus 0.50.5) are explained rather than only reproduced numerically; (2) identify the ensemble describing the fully relaxed state — whether a grand-canonical Gibbs state in the total energy and particle number, or a maximum-entropy state involving additional conserved quantities — and demonstrate that it reproduces all few- and multi-site observables, not only the measured nearest-neighbor correlator and quasi-local densities and currents; and (3) propose a single-chain (or few-chain) measurement protocol for these predictions, beyond the ensemble-averaged signal, whose predicted single-chain signals would discriminate the global state actually reached, including the presence or absence of prethermalization and of eigenstate-thermalization behavior. These three parts share one objective — a complete microscopic characterization of how, how fast, and to what equilibrium this paradigmatic isolated interacting quantum many-body system relaxes.

References

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