One-particle density matrix and momentum distribution of the out-of-equilibrium 1D Tonks-Girardeau gas: Analytical results at large $N$

Abstract

In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms $N$, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large $N$, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $⟨{\hat{\Psi }}^{†}(x)\hat{\Psi }(x^{\prime })⟩$ of the out-of-equilibrium TG gas in the $N\to \infty$ limit, valid on distances $|x-x^{\prime }|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well,which mimics a `quantum Newton cradle' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.

13pages, 6 figures. v2: minor changes; v3: fixed layout issues in appendices

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• Inhomogeneous quenches and GHD in the $\nu =1$ QSSEP model