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arXiv:1305.5496 · Physical Review B

Sudden expansion of Mott insulators in one dimension

Lev Vidmar, S. Langer, Ian P. McCulloch, Ulrich Schneider, Ulrich Schollwöck, Fabian Heidrich‐Meisner

Published Dec 16, 2013• 70 citations
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Abstract

We investigate the expansion of bosons and fermions in a homogeneous lattice after a sudden removal of the trapping potential using exact numerical methods. As a main result, we show that in one dimension, both bosonic and fermionic Mott insulators expand with the same velocity, irrespective of the interaction strength, provided the expansion starts from the ground state of the trapped gas. Furthermore, their density profiles become identical during the expansion; the asymptotic density dynamics is identical to that of initially localized, noninteracting particles, and the asymptotic velocity distribution is flat. The expansion velocity for initial correlated Mott insulating states is therefore independent of the interaction strength and particle statistics. Interestingly, this nonequilibrium dynamics is sensitive to the interaction driven quantum phase transition in the Bose-Hubbard model; while being constant in the Mott phase, the expansion velocity decreases in the superfluid phase and vanishes for large systems in the noninteracting limit. These results are compared to the setup of a recent experiment [Ronzheimer et al., Phys. Rev. Lett. 110, 205301 (2013)], where the trap opening was combined with an interaction quench from infinitely strong interactions to finite values. In the latter case, the interaction quench breaks the universal dynamics in the asymptotic regime and the expansion depends on the interaction strength. We carry out an analogous analysis for a two-component Fermi gas, with similar observations. In addition, we study the effect of breaking the integrability of hard-core bosons in different ways; while the fast ballistic expansion from the ground state of Mott insulators in one dimension remains unchanged for finite interactions, we observe strong deviations from this behavior on a two-leg ladder even in the hard-core case. This change in dynamics bares similarities with the dynamics in the dimensional crossover from one to two dimensions observed in the aformentioned experimental study.

DOI
10.1103/physrevb.88.235117

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We investigate the expansion of bosons and fermions in a homogeneous lattice after a sudden removal of the trapping potential using exact numerical methods. As a main result, we show that in one dimension, both bosonic and fermionic Mott insulators expand with the same velocity, irrespective of the interaction strength, provided the expansion starts from the ground state of the trapped gas. Furthermore, their density profiles become identical during the expansion; the asymptotic density dynamics is identical to that of initially localized, noninteracting particles, and the asymptotic velocity distribution is flat. The expansion velocity for initial correlated Mott insulating states is therefore independent of the interaction strength and particle statistics. Interestingly, this nonequilibrium dynamics is sensitive to the interaction driven quantum phase transition in the Bose-Hubbard model; while being constant in the Mott phase, the expansion velocity decreases in the superfluid phase and vanishes for large systems in the noninteracting limit. These results are compared to the setup of a recent experiment [Ronzheimer et al., Phys. Rev. Lett. 110, 205301 (2013)], where the trap opening was combined with an interaction quench from infinitely strong interactions to finite values. In the latter case, the interaction quench breaks the universal dynamics in the asymptotic regime and the expansion depends on the interaction strength. We carry out an analogous analysis for a two-component Fermi gas, with similar observations. In addition, we study the effect of breaking the integrability of hard-core bosons in different ways; while the fast ballistic expansion from the ground state of Mott insulators in one dimension remains unchanged for finite interactions, we observe strong deviations from this behavior on a two-leg ladder even in the hard-core case. This change in dynamics bares similarities with the dynamics in the dimensional crossover from one to two dimensions observed in the aformentioned experimental study.
We numerically investigate the expansion of clouds of hard-core bosons in the two-dimensional square lattice using a matrix-product-state--based method. This nonequilibrium setup is induced by quenching the trapping potential to zero and our work is specifically motivated by a recent experiment with interacting bosons in an optical lattice [Ronzheimer et al., Phys. Rev. Lett. 110, 205301 (2013)]. As the anisotropy of the amplitudes Jx{J}_{x}Jx​ and Jy{J}_{y}Jy​ for hopping in different spatial directions is varied from the one- to the two-dimensional case, we observe a crossover from a fast ballistic expansion in the one-dimensional limit ${J}_{x}\ensuremath{\gg}{J}_{y}$ to much slower dynamics in the isotropic two-dimensional limit Jx=Jy{J}_{x}={J}_{y}Jx​=Jy​. We further study the dynamics on multileg ladders and long cylinders. For these geometries we compare the expansion of a cloud to the melting of a domain wall, which helps us to identify several different regimes of the expansion as a function of time. By studying the dependence of expansion velocities on both the anisotropy Jy/Jx{J}_{y}/{J}_{x}Jy​/Jx​ and the number of legs, we observe that the expansion on two-leg ladders, while similar to the two-dimensional case, is slower than on wider ladders. We provide a qualitative explanation for this observation based on an analysis of the rung spectrum.
We develop an exact approach to study the quench dynamics of hard-core bosons initially in thermal equilibrium in one-dimensional lattices. This approach is used to study the sudden expansion of thermal states after confining potentials are switched off. We find that a dynamical fermionization of the momentum distribution occurs at all temperatures. This phenomenon is studied for low initial site occupations, for which the expansion of the cloud is self-similar. In this regime, the occupation of the natural orbitals allows one to distinguish hard-core bosons from noninteracting fermions. We also study the free expansion of initial Mott insulating domains at finite temperature and show that the emergence of off-diagonal one-body correlations is suppressed gradually with increasing temperature. Surprisingly, the melting of the Mott domain is accompanied by an effective cooling of the system. We explain this phenomenon analytically using an equilibrium description based on an emergent local Hamiltonian.
We experimentally and numerically investigate the expansion of initially localized ultracold bosons in homogeneous one- and two-dimensional optical lattices. We find that both dimensionality and interaction strength crucially influence these nonequilibrium dynamics. While the atoms expand ballistically in all integrable limits, deviations from these limits dramatically suppress the expansion and lead to the appearance of almost bimodal cloud shapes, indicating diffusive dynamics in the center surrounded by ballistic wings. For strongly interacting bosons, we observe a dimensional crossover of the dynamics from ballistic in the one-dimensional hard-core case to diffusive in two dimensions, as well as a similar crossover when higher occupancies are introduced into the system.
After release from the trap the momentum distribution of an impenetrable gas asymptotically approaches that of a spinless noninteracting Fermi gas in the initial trap. This phenomenon is called dynamical fermionization and, very recently, has been experimentally confirmed in the case of the Lieb-Liniger model in the Tonks-Girardeau regime. We prove analytically and confirm numerically that following the removal of axial confinement the strongly interacting Bose-Fermi mixture exhibits dynamical fermionization and the asymptotical momentum distribution of each component has the same shape as its density profile at t=0t=0t=0. Under a sudden change of the trap frequency to a new nonzero value the dynamics of both fermionic and bosonic momentum distributions presents characteristics which are similar to the case of single component bosons experiencing a similar quench. Our results are derived using a product representation for the correlation functions which, in addition to analytical considerations, can be implemented numerically very easily with complexity which scales polynomially in the number of particles.
Abstract We study the expansion of a one-dimensional boson gas by initialising it in a small region of a chain, and then suddenly allowing it to expand into the remainder of the chain. We consider three initial ground-state configurations: the Mott insulator, the conventional superfluid, whose momentum density is sharply peaked at zero momentum, and the cat-like state with momentum peaks at ±π/2, produced by kinetic driving, the latter being a particular case of a flat-band system. In turn, we consider three types of expansion: spectroscopic (with interactions tuned to zero), dynamic (with standard short-range repulsive interactions), and under kinetic driving. The numerical calculations are exact. We compute the momentum and real-space one-particle densities, as well as the two-particle momentum correlations. We find that the spectroscopic time-of-flight experiment reflects the initial momentum distribution except for the larger number of momentum states and at high momenta. For the dynamic expansion starting from an insulator, we recover the non-equilibrium quasi-condensation into momenta ±π/2, provide a physical explanation in terms of interacting bosons that is confirmed by the numerical simulation, and note the existence of nontrivial correlations in the momentum distribution. Under kinetic driving the expansion is comparatively slow, but we conjecture that at high densities it will be much faster. We compare various measures of the two-particle momentum correlations, noting that some of them tend to conceal the possible cat-like structure of a many-body state.
We experimentally and numerically investigate the sudden expansion of fermions in a homogeneous one-dimensional optical lattice. For initial states with an appreciable amount of doublons, we observe a dynamical phase separation between rapidly expanding singlons and slow doublons remaining in the trap center, realizing the key aspect of fermionic quantum distillation in the strongly interacting limit. For initial states without doublons, we find a reduced interaction dependence of the asymptotic expansion speed compared to bosons, which is explained by the interaction energy produced in the quench.