Quantum many-body real-time dynamics of the trap-quench expansion of the Fermi–Hubbard gas in two and three dimensions, beyond the semiclassical Boltzmann picture

When a confined gas of ultracold fermions in a homogeneous optical lattice is released by a sudden trap quench, its expansion directly probes nonequilibrium transport in the Hubbard model. The 2012 experiment of Schneider et al. established the phenomenology — ballistic expansion without interactions, a strongly slowed, almost bimodal expansion with interactions, and an exact symmetry between attractive and repulsive interactions — but could compare it only with a semiclassical Boltzmann equation, while exact quantum real-time methods (DMRG) existed in one dimension alone. This task asks for a quantitative quantum many-body theory of this expansion in two and three dimensions: how entanglement spreading and (repulsively) bound pairs modify the kinetic picture, where the relaxation-time and weak-coupling scattering description fails at intermediate-to-strong coupling and long times, what the long-time equilibration and entropy/mass-transport mechanisms are, and whether the observed entropy separation yields a practical cooling protocol (quantum distillation). Partial progress exists — real-space nonequilibrium dynamical mean-field theory, nonequilibrium Green functions in $d=1,2,3$, one-dimensional and bosonic experiments, and bad-metal transport measurements — but no controlled quantum treatment of the fermionic expanding cloud in $d = 2$ and $d = 3$ exists.

Background

The Fermi–Hubbard model is the paradigmatic lattice model of strongly correlated electrons: spin-1/2 fermions hop between nearest-neighbor sites with amplitude $J$ and interact on the same site with strength $U$. Ultracold two-component atomic gases trapped in periodic light fields (optical lattices) realize this model with independently tunable $J$, $U$, density and dimensionality, without the impurities and phonons of a solid, and are therefore widely used as quantum simulators of strongly correlated matter.

A particularly direct nonequilibrium probe is the sudden-expansion (trap-quench) experiment introduced by Schneider et al. (2012): a degenerate cloud of fermionic potassium atoms in a homogeneous, box-shaped optical lattice is released by switching off the confining potential, and the subsequent expansion of the density distribution is monitored. The experiment found ballistic expansion for non-interacting atoms, and, even at weak interactions, an almost bimodal expansion — a dense, slow core plus fast ballistic tails — with a dramatically reduced expansion velocity. Unexpectedly, the density dynamics was identical for attractive and repulsive interactions of the same magnitude, a consequence of an exact dynamical symmetry of the Hubbard model that holds when the initial state and the observed quantity are invariant under time reversal. Assuming fast local thermalization, the interacting core at repulsive coupling is described by a negative absolute temperature (a population-inverted equilibrium state, well defined because the kinetic energy of the lowest Hubbard band is bounded above).

The authors compared their data with a semiclassical kinetic description: the Boltzmann transport equation for the quasimomentum distribution in the relaxation-time approximation, with transport scattering rates estimated from weak-coupling ($U^2$) theory. The agreement was qualitative to semi-quantitative only. They furthermore observed a dynamical separation of entropy: the ballistic tails carry high entropy per particle while a high-density, low-entropy core remains. They suggested exploiting this for local cooling via quantum distillation — a dynamical mechanism proposed by Heidrich-Meisner et al. (2009) in which, after release, rapidly moving unpaired atoms (singlons) leave the cloud while nearly immobile paired atoms (doublons, two fermions on one site) stay behind, locally producing a lower-entropy state.

Exact quantum many-body real-time calculations were available only in one dimension, where the density-matrix renormalization group (DMRG) — a variational method whose accuracy relies on the moderate entanglement of one-dimensional states — reproduces the expansion (Kajala, Massel and Törmä, 2011). The paper's discussion states the resulting gap verbatim: 'so far no methods are available to calculate the dynamics quantum-mechanically in higher dimensions', and notes that the full quantum dynamics also includes the formation of entanglement between distant atoms and the existence of bound or repulsively bound states (pairs that are stable even for repulsive interactions, because the bounded band energy prevents their decay into free atoms). This task is the closure of that gap together with the quantitative questions attached to it in the same paper: the validity range of the semiclassical kinetic description, the long-time equilibration and entropy/mass transport of the expanding cloud, and the use of its entropy separation for cooling. The paper additionally proposes disorder in the two-dimensional dynamics, other dimensionalities, and Bose–Fermi mixtures as natural extensions of the same program; these are not required here.

Problem Statement

Obtain a quantum-mechanical many-body account of the sudden (trap-quench) expansion of a two-component fermionic gas in the lowest band of a homogeneous optical lattice, described by the Fermi–Hubbard model with hopping $J$ and onsite interaction $U$, in spatial dimensions $d = 2$ and $d = 3$ — the regime for which the source experiment stated that 'so far no methods are available to calculate the dynamics quantum-mechanically in higher dimensions'. The problem comprises four jointly connected deliverables:

1. A controlled real-time quantum many-body method, valid from weak to strong coupling, for the expanding inhomogeneous fermionic cloud in $d = 2$ and $d = 3$, quantifying how genuine quantum correlations — entanglement spreading between distant atoms, and the formation of bound or repulsively bound states — modify the semiclassical kinetic picture of the expansion.
2. A quantitative determination of the range of validity of the relaxation-time Boltzmann description and of the weak-coupling ($U^2$) scattering-time estimates used to interpret the expansion experiment, in particular at intermediate-to-strong coupling and for long times.
3. The long-time fate of the expanding cloud with respect to global equilibration, including the mechanisms and timescales of its entropy and mass transport (which set lower bounds for adiabatic loading and cooling strategies in lattice fermion systems).
4. An assessment, at the level of a demonstrated or quantitatively predicted protocol, of whether the observed separation into high-entropy ballistic tails and a low-entropy high-density core can be used for practical local cooling via quantum distillation in $d = 2$ and $d = 3$.

All four deliverables are jointly required; together they constitute one overarching objective: replacing the semiclassical kinetic description of this expansion by a quantitative quantum many-body one and extracting its physical consequences for equilibration and cooling. The source paper additionally proposes disorder in the two-dimensional dynamics, further dimensionalities, and Bose–Fermi mixtures as natural extensions of the same program; these are not required here.

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