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arXiv:1204.3371 · Physical Review A

Quantum dynamics and entanglement in one-dimensional Fermi gases released from a trap

Ettore Vicari

Published Jun 26, 2012• 40 citations
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Abstract

We investigate the entanglement properties of the nonequilibrium dynamics of one-dimensional noninteracting Fermi gases released from a trap. The gas of $N$ particles is initially in the ground state within a hard-wall or a harmonic trap, and then it expands after dropping the trap. We compute the time dependence of the von Neumann and R\'enyi entanglement entropies and the particle fluctuations of spatial intervals around the original trap in the limit of a large number $N$ of particles. The results for these observables apply to one-dimensional gases of impenetrable bosons as well. We identify different dynamical regimes at small and large times, depending also on the initial condition, i.e., whether it is that of a hard-wall or a harmonic trap. In particular, we show analytically that the expansion from hard-wall traps is characterized by the asymptotic small-time behavior $S\ensuremath{\approx}(1/3)\mathrm{ln}(1/t)$ of the von Neumann entanglement entropy and the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ (where $V$ is the particle variance), which are analogous to the equilibrium behaviors whose leading logarithms are essentially determined by the corresponding conformal field theory with central charge $c=1$. The time dependence of the entanglement entropy of extended regions during the expansion from harmonic traps has a remarkable property in that it can be expressed as a global time-dependent rescaling of the space dependence of the initial equilibrium entanglement entropy.

DOI
10.1103/physreva.85.062324

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We investigate the entanglement properties of the nonequilibrium dynamics of one-dimensional noninteracting Fermi gases released from a trap. The gas of NNN particles is initially in the ground state within a hard-wall or a harmonic trap, and then it expands after dropping the trap. We compute the time dependence of the von Neumann and R'enyi entanglement entropies and the particle fluctuations of spatial intervals around the original trap in the limit of a large number NNN of particles. The results for these observables apply to one-dimensional gases of impenetrable bosons as well. We identify different dynamical regimes at small and large times, depending also on the initial condition, i.e., whether it is that of a hard-wall or a harmonic trap. In particular, we show analytically that the expansion from hard-wall traps is characterized by the asymptotic small-time behavior S\ensuremath≈(1/3)ln(1/t)S\ensuremath{\approx}(1/3)\mathrm{ln}(1/t)S\ensuremath≈(1/3)ln(1/t) of the von Neumann entanglement entropy and the relation S\ensuremath≈\ensuremathπ2V/3S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3S\ensuremath≈\ensuremathπ2V/3 (where VVV is the particle variance), which are analogous to the equilibrium behaviors whose leading logarithms are essentially determined by the corresponding conformal field theory with central charge c=1c=1c=1. The time dependence of the entanglement entropy of extended regions during the expansion from harmonic traps has a remarkable property in that it can be expressed as a global time-dependent rescaling of the space dependence of the initial equilibrium entanglement entropy.
We investigate the entanglement properties of the equilibrium and nonequilibrium quantum dynamics of two-dimensional (2D) and three-dimensional (3D) Fermi gases, by computing entanglement entropies of extended space regions, which generally show multiplicative logarithmic corrections to the leading power-law behaviors, corresponding to the logarithmic corrections to the area law. We consider 2D and 3D Fermi gases of NNN particles constrained within a limited space region, for example, by a hard-wall trap, at equilibrium at T=0T=0T=0, i.e., in their ground state, and compute the first few terms of the asymptotic large-NNN behaviors of entanglement entropies and particle fluctuations of subsystems with some convenient geometries, which allow us to significantly extend their computation. Then we consider their nonequilibrium dynamics after instantaneously dropping the hard-wall trap, which allows the gas to expand freely. We compute the time dependence of the von Neumann entanglement entropy of space regions around the original trap. We show that at small time it is characterized by the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ with the particle variance, and multiplicative logarithmic corrections to the leading power law, i.e., $S\ensuremath{\sim}{t}^{1\ensuremath{-}d}\mathrm{ln}(1/t)$.
We investigate quantum correlations in the ground state of noninteracting Fermi gases of NNN particles trapped by an external space-dependent harmonic potential, in any dimension. For this purpose we compute one-particle correlations, particle fluctuations, and bipartite entanglement entropies of extended space regions, and study their large-NNN scaling behaviors. The half-space von Neumann entanglement entropy is computed for any dimension, obtaining ${S}_{\mathrm{HS}}\ensuremath{\approx}{c}_{l}{N}^{(d\ensuremath{-}1)/d}\mathrm{ln}N$ analogously to homogenous systems, with cl=1/6,1/(62),1/(66){c}_{l}=1/6,\phantom{\rule{0.222222em}{0ex}}1/(6\sqrt{2}),\phantom{\rule{0.222222em}{0ex}}1/(6\sqrt{6})cl​=1/6,1/(62​),1/(66​) in one, two, and three dimensions, respectively. We show that the asymptotic large-NNN relation ${S}_{A}\ensuremath{\approx}{\ensuremath{\pi}}^{2}{V}_{A}/3$, between the von Neumann entanglement entropy SA{S}_{A}SA​ and particle variance VA{V}_{A}VA​ of an extended space region AAA, holds for any subsystem AAA and in any dimension, analogously to homogeneous noninteracting Fermi gases.
Abstract We consider the non-equilibrium dynamics of the entanglement entropy of a one-dimensional quantum gas of hard-core particles, initially confined in a box potential at zero temperature. At t = 0 the right edge of the box is suddenly released and the system is let free to expand. During this expansion, the initially correlated region propagates with a non-homogeneous profile, leading to the growth of entanglement entropy. This setting is investigated in the hydrodynamic regime, with tools stemming from semi-classical Wigner function approach and with recent developments of quantum fluctuating hydrodynamics. Within this framework, the entanglement entropy can be associated to a correlation function of chiral twist-fields of the conformal field theory that lives along the Fermi contour and it can be exactly determined. Our predictions for the entanglement evolution are found in agreement with and generalize previous results in literature based on numerical calculations and heuristic arguments.
We derive exact relations between the Rényi entanglement entropies and the particle-number fluctuations of (connected and disjoint) spatial regions in systems of N non-interacting fermions in arbitrary dimension. We prove that the asymptotic large- N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large- N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Rényi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems.
We address the effects of dissipative defects giving rise to a localized particle loss, in one-dimensional noninteracting lattice fermionic gases confined within a region of size $\ensuremath{\ell}$. We consider homogeneous systems within hard walls and inhomogeneous systems where the particles are trapped by space-dependent external potentials, such as harmonic traps. We model the dissipative particle-decay mechanism by Lindblad master equations governing the time evolution of the density matrix. The resulting quantum dynamics is analyzed in protocols starting from the ground state of the Hamiltonian for N0{N}_{0}N0​ particles, then evolving under the effect of one dissipative particle-loss defect, for example, at the center of the system. We study the interplay between time, size $\ensuremath{\ell}$, and the number N0{N}_{0}N0​ of initial particles, considering two different situations: (i) fixed number N0{N}_{0}N0​ of initial particles; and (ii) fixed ratio ${N}_{0}/\ensuremath{\ell}$, corresponding to the thermodynamic limit of the initial equilibrium state. We show that the quantum evolutions of the particle number and density develop various intermediate and asymptotic dynamic regimes, and nontrivial large-time states when the dissipative mechanism acts at the center of the system.
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.