Volume 126, Number 2, April 2019
|Number of page(s)||7|
|Published online||07 June 2019|
Nonequilibrium dynamics of noninteracting fermions in a trap
1 Univ. Bordeaux, CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798 F-33400 Talence, France
2 Laboratoire de Physique de l'Ecole Normale Supérieure, PSL University, CNRS, Sorbonne Universités 24 rue Lhomond, F-75231 Paris, France
3 LPTMS, CNRS, Univ. Paris- Sud, Université Paris-Saclay - F-91405 Orsay, France
Received: 24 March 2019
Accepted: 9 May 2019
We consider the real-time dynamics of N noninteracting fermions in d = 1. They evolve in a trapping potential V(x), starting from the equilibrium state in a potential V0(x). We study the time evolution of the Wigner function W(x, p, t) in the phase space (x, p), and the associated kernel which encodes all correlation functions. At t = 0 the Wigner function for large N is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale eN being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several V0(x). In the large-N limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time-dependent rescaling. For nonharmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a nontrivial time- and space-dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large-time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the “classical” regime . Using the diagonal ensemble we compute the Wigner function in the quantum case (large N, fixed ℏ ) and show that it agrees with the GGE. We also obtain the higher order (nonlocal) correlations in the diagonal ensemble.
PACS: 05.30.Fk – Fermion systems and electron gas / 05.30.-d – Quantum statistical mechanics / 71.10.-w – Theories and models of many-electron systems
© EPLA, 2019
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