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In one-dimensional (1D) quantum gases, the momentum distribution (MD) of theatoms is a standard experimental observable, routinely measured in variousexperimental setups. The MD is sensitive to correlations, and it is notoriouslyhard to compute theoretically for large numbers of atoms $N$, which oftenprevents direct comparison with experimental data. Here we report significantprogress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptoticlimit of large $N$, at zero temperature and driven out of equilibrium by aquench of the confining potential. We find an exact analytical formula for theone-particle density matrix $\langle \hat{\Psi}^\dagger(x) \hat{\Psi}(x')\rangle$ of the out-of-equilibrium TG gas in the $N \rightarrow \infty$ limit,valid on distances $|x-x'| $ much larger than the interparticle distance. Bycomparing with time-dependent Bose-Fermi mapping numerics, we demonstrate thatour analytical formula can be used to compute the out-of-equilibrium MD withgreat accuracy for a wide range of momenta (except in the tails of thedistribution at very large momenta). For a quench from a double-well potentialto a single harmonic well,which mimics a `quantum Newton cradle' setup, ourmethod predicts the periodic formation of peculiar, multiply peaked, momentumdistributions.

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