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

TL;DR

This paper analyzes the long-time decay of the return probability for a system of non-interacting fermions released from a 1D trap, revealing universal power-law behavior and explicit amplitude formulas.

Contribution

It provides a universal power-law decay law for the return probability of fermions after a trap release, with explicit amplitude calculations for specific potentials.

Findings

Power-law decay of return probability with universal exponent

Explicit amplitude formulas in terms of wavefunction moments

Scaling laws for large fermion numbers involving Fermi energy

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.