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

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.

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