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arXiv:1810.09198 · Journal of Statistical Mechanics Theory and Experiment

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

P. L. Krapivsky, J. M. Luck, Kirone Mallick

Published Feb 1, 2019• 2 citations
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Abstract

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.

DOI
10.1088/1742-5468/aaf8a2

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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.
Abstract We consider N non-interacting fermions performing continuous-time quantum walks on a one-dimensional lattice. The system is launched from a most compact configuration where the fermions occupy neighboring sites. We calculate exactly the quantum return probability (sometimes referred to as the Loschmidt echo) of observing the very same compact state at a later time t . Remarkably, this probability depends on the parity of the fermion number—it decays as a power of time for even N , while for odd N it exhibits periodic oscillations modulated by a decaying power law. The exponent also slightly depends on the parity of N , and is roughly twice smaller than what it would be in the continuum limit. We also consider the same problem, and obtain similar results, in the presence of an impenetrable wall at the origin constraining the particles to remain on the positive half-line. We derive closed-form expressions for the amplitudes of the power-law decay of the return probability in all cases. The key point in the derivation is the use of Mehta integrals, which are limiting cases of the Selberg integral.
We consider a system of non-interacting fermions prepared in the many-body ground state of an isotropic trapping potential in any dimension, and investigate the ballistic expansion of the fermionic cloud after the potential is suddenly released. Using semi-classical techniques, we derive the full late-time profile of the expanding cloud in the regime when the fermion number is large. We thus obtain explicit expressions for power-law potentials with arbitrary exponent aaa and in all dimensions ddd. The momentum distribution and the spatial profile of the cloud exhibit a universal edge exponent d/ad/ad/a, thus generalizing the Wigner semi-circle law and the Thomas-Fermi distribution.
We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.
In the first part of the article, we study one-dimensional noninteracting fermions in the continuum and in the presence of the repulsive inverse power law potential, with an emphasis on the Wigner function in the semiclassical limit. In this limit, the Wigner function exhibits an edge called the Fermi surf that depends only on the classical one-particle Hamiltonian. Around the Fermi surf, under a well-defined semiclassical limit, the Wigner function can be expressed in terms of Airy functions which yield a smooth matching between the two regions delimited by the Fermi surf. In the second part of the article, the system is prepared in the ground state of the inverse power law potential where only the left half line is filled with fermions. Then the potential is switched off, resulting in the emergence of a propagating quantum front. We show that the power law decay of the pre-quench potential that separates the left and right half systems leads to the emergence of the Airy kernel (well known in Random Matrix Theory) at the quantum front in the long-time limit. This also comes with anomalous diffusive spreading around the front.
Abstract We study transport in the free fermionic one-dimensional systems subjected to arbitrary local potentials. The bias needed for the transport is modeled by the initial highly non-equilibrium distribution where only half of the system is populated. Additionally to that, the local potential is also suddenly changed when the transport starts. For such a quench protocol we compute the full counting statistics (FCS) of the number of particles in the initially empty part. In the thermodynamic limit, the FCS can be expressed via the Fredholm determinant with the kernel depending on the scattering data and Jost solutions of the pre-quench and the post-quench potentials. We discuss the large-time asymptotic behavior of the obtained determinant and observe that if two or more bound states are present in the spectrum of the post-quench potential the information about the initial state manifests itself in the persistent oscillations of the FCS. On the contrary, when there are no bound states the asymptotic behavior of the FCS is determined solely by the scattering data of the post-quench potential, which for the current (the first moment) is given by the Landauer–Büttiker formalism. The information about the initial state can be observed only in the transient dynamics.
Free expansion following the removal of axial confinement represents a fundamental nonequilibrium scenario in the study of many-body ultracold gases. Using the stationary phase approximation, we analytically demonstrate that for all one-dimensional spinor gases with repulsive contact interactions, whether bosonic or fermionic, the asymptotic density and momentum distribution can be directly determined from the quasimomentum distribution (Bethe rapidities) of the trapped gas. We efficiently obtain the quasimomentum distribution numerically by solving the integral equations that characterize the ground state of the integrable system within the local density approximation. Additionally, we derive analytical solutions for both weakly and strongly interacting regimes. Unlike in bosonic gases, where rapidity distributions and density profiles vary significantly across interaction regimes, fermionic gases maintain similar profiles in both weakly and strongly interacting limits. Notably, the gas expands self-similarly only when released from a harmonic trap. For other power-law trapping potentials, the asymptotic density profile is strongly influenced by the initial confinement geometry. Our results extend readily to Bose-Fermi mixtures and finite temperatures.