Microscopic theory of the slow, diffusion-dominated expansion of interacting lattice bosons in two dimensions and across the 1D-to-2D crossover

When a cloud of strongly interacting bosons initially localized in a homogeneous optical lattice is released by quenching the trap to zero, its expansion is dramatically suppressed away from the integrable limits of the Bose-Hubbard model, producing bimodal density profiles with a slow high-density core and ballistic wings. Ronzheimer et al. (Phys. Rev. Lett. 110, 205301 (2013)) conjectured, without proof, that the common microscopic reason for the slow expansions observed in one and two dimensions and with multiply occupied initial sites is the emergence of diffusive dynamics in the central high-density region. They also found a qualitative difference between one- and two-dimensional dynamics at strong interactions that contradicts time-dependent Gutzwiller mean-field predictions, and their one-dimensional t-DMRG benchmarks were limited to small particle numbers. The open problem is to establish quantitatively and microscopically that diffusion in the core underlies the observed slow expansion: derive the transport coefficients and core expansion velocities as functions of interaction strength, initial occupancy (including doublons), and dimensionality, using controlled beyond-mean-field methods for interacting two-dimensional lattice dynamics, and determine the role of doublons, quantum distillation, and the dimensional crossover. Later work has provided size-limited tensor-network simulations, approximate beyond-Gutzwiller approaches that reproduce the qualitative ballistic-halo-plus-diffusive-core structure, and a direct observation of quantum distillation in a one-dimensional doublon sea, but no controlled quantitative microscopic theory of the two-dimensional strongly interacting expansion or derivation of the diffusive-core mechanism exists, so the problem remains open.

Background

Ultracold atoms in optical lattices (periodic light-shift potentials created by interfering laser beams, which trap atoms at their minima) realize the Bose-Hubbard model, in which bosons hop between neighboring lattice sites with amplitude $J$ and repel each other on the same site with interaction energy $U$. In the sudden-expansion protocol, an initially localized cloud (for example, an array of one-dimensional Mott insulators, i.e., regions with exactly one boson per lattice site, prepared by dephasing a deep lattice) is released by suddenly removing the confining potential, and the subsequent density profiles are recorded. In the integrable limits of this model — non-interacting bosons ($U=0$) in any dimension, and infinitely strong (hard-core) interactions in one dimension, where bosons map to free fermions (the Tonks-Girardeau limit) — the expansion is ballistic: the cloud radius grows linearly in time. Ronzheimer et al. (Phys. Rev. Lett. 110, 205301 (2013)) showed experimentally and numerically (with one-dimensional time-dependent density-matrix renormalization group, t-DMRG, a method that represents one-dimensional quantum states as matrix product states and is limited to small particle numbers and short times) that away from these limits the expansion slows dramatically: finite interactions in one dimension, the crossover to two dimensions, or an admixture of multiply occupied sites (doubly occupied sites are called doublons) all suppress the core expansion velocity, and in two dimensions the cloud develops a bimodal shape with a nearly stationary round core on top of a square ballistic background. In the experiment, the core expansion velocity $v_c$ is extracted from the half width at half maximum (HWHM) of the density profile; the t-DMRG version of this diagnostic suffers large finite-size effects, so an alternative radius-based velocity $v_r$ with an extrapolation to infinite particle number was introduced. The authors conjectured that the common microscopic origin of all these slow expansions is the emergence of diffusive dynamics (density spreading proportional to $\sqrt{t}$) in the central high-density region, analogous to the diffusive expansion previously established for strongly interacting fermions in two-dimensional lattices, but they did not derive this connection. The contrast with theory is sharp: the time-dependent Gutzwiller ansatz, a mean-field decoupling of the lattice into independent sites, predicts for the expansion of Mott insulators a qualitatively similar behavior independent of dimension, including dynamical condensation even in two dimensions (Jreissaty, Carrasquilla, Wolf and Rigol, Phys. Rev. A 84, 043610 (2011); Hen and Rigol, Phys. Rev. Lett. 105, 180401 (2010)), whereas the experiment shows a qualitative one- versus two-dimensional difference at strong interactions. Multiply occupied sites introduce further physics: at large $U/J$, doublons are long-lived because their effective hopping is of order $J^2/U$, and expansion in their presence can exhibit quantum distillation — a demixing of doublons and single atoms in which singly occupied sites are expelled from the doublon-rich core, originally identified by Heidrich-Meisner, Manmana, Rigol, Muramatsu, Feiguin and Dagotto (Phys. Rev. A 80, 041603 (2009)). What is missing is a quantitative microscopic theory connecting these observations: transport coefficients derived from the microscopic dynamics, controlled numerical methods for interacting two-dimensional lattice dynamics on experimentally relevant scales, and a systematic account of the doublon and higher-occupancy dynamics and its interplay with the dimensional crossover.

Problem Statement

Establish, quantitatively and microscopically, the mechanism and dynamics of the slow expansion of interacting bosons released from a homogeneous optical lattice, as observed by Ronzheimer et al. (Phys. Rev. Lett. 110, 205301 (2013)). Specifically: (i) prove or derive, rather than conjecture, whether the emergence of diffusive dynamics in the central high-density region is the common microscopic reason for the slow expansions observed in one dimension at finite interactions, in the presence of multiply occupied initial sites, and in two dimensions at strong interactions, jointly across all three regimes, and derive the effective diffusion constants and the core expansion velocity $v_c$ of the high-density core as functions of the interaction strength $U/J$, the initial higher-occupancy fraction, and the lattice dimensionality, and compare them with the measured values and bimodal density profiles; (ii) develop a controlled theoretical or numerical method for the real-time dynamics of the interacting two-dimensional Bose-Hubbard model far from equilibrium, beyond the time-dependent Gutzwiller mean-field ansatz whose dimension-independent predictions (including dynamical condensation in two dimensions) contradict the observed qualitative difference between one- and two-dimensional dynamics at strong interactions, and determine whether dynamical condensation at finite quasimomenta (macroscopic occupation of single-particle states at nonzero crystal momentum, rather than at the band minimum, as predicted for one dimension) survives in two dimensions; and (iii) determine quantitatively the dynamics and lifetimes of the doublons and higher occupancies formed during the expansion, including quantum distillation processes and the stability of the doublon core, and their role in the crossover from one- to two-dimensional expansion. These three parts are jointly the elements of a single predictive, quantitative theory of interacting-boson lattice expansion connecting microscopic dynamics to the measured cloud profiles, expansion velocities and their dependence on dimension and initial occupancy.

References

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2. M. Jreissaty, J. Carrasquilla, F. A. Wolf, and M. Rigol, Expansion of Bose-Hubbard Mott insulators in optical lattices, Phys. Rev. A 84, 043610 (2011), DOI: 10.1103/PhysRevA.84.043610, https://doi.org/10.1103/PhysRevA.84.043610 (time-dependent Gutzwiller prediction challenged by the experiment)
3. I. Hen and M. Rigol, Strongly Interacting Atom Lasers in Three-Dimensional Optical Lattices, Phys. Rev. Lett. 105, 180401 (2010), DOI: 10.1103/PhysRevLett.105.180401, https://doi.org/10.1103/PhysRevLett.105.180401 (time-dependent Gutzwiller prediction of dimension-independent dynamics and dynamical condensation)
4. F. Heidrich-Meisner, S. R. Manmana, M. Rigol, A. Muramatsu, A. E. Feiguin, and E. Dagotto, Quantum distillation: Dynamical generation of low-entropy states of strongly correlated fermions in an optical lattice, Phys. Rev. A 80, 041603 (2009), DOI: 10.1103/PhysRevA.80.041603, https://doi.org/10.1103/PhysRevA.80.041603 (origin of the quantum distillation concept)
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9. J. Schwingel, M. Turaev, J. Kroha, and S. Ray, Expansion dynamics of strongly correlated lattice bosons: A self-consistent density-matrix approach, Phys. Rev. A 112, 043322 (2025), DOI: 10.1103/bzvn-sp2h, https://doi.org/10.1103/bzvn-sp2h, arXiv:2507.09397, https://arxiv.org/abs/2507.09397 (self-consistent density-matrix method beyond Gutzwiller for the strongly interacting 2D expansion)