Return probability of N fermions released from a 1D confining potential

Abstract

We consider N non-interacting fermions prepared in the ground state of a 1D confining potential and submitted to an instantaneous quench consisting in releasing the trapping potential. We show that the quantum return probability of finding the fermions in their initial state at a later time falls off as a power law in the long-time regime, with a universal exponent depending only on N and on whether the free fermions expand over the full line or over a half-line. In both geometries, the amplitudes of this power-law decay are expressed in terms of finite determinants of moments of the one-body bound-state wavefunctions in the potential. These amplitudes are worked out explicitly for the harmonic and square-well potentials. At large fermion numbers they obey scaling laws involving the Fermi energy of the initial state. The use of the Selberg–Mehta integrals stemming from random matrix theory has been instrumental in the derivation of these results.