Generalized Gibbs ensemble and work statistics of a quenched Luttinger liquid

Abstract

We analyze the probability distribution function (PDF) of work done on a Luttinger liquid for an arbitrary finite duration interaction quench and show that it can be described in terms of a generalized Gibbs ensemble. We construct the corresponding density matrix with explicit intermode correlations, and determine the duration and interaction dependence of the probability of an adiabatic transition and the PDF of nonadiabatic processes. In the thermodynamic limit, the PDF of work exhibits a non-Gaussian maximum around the excess heat, carrying almost all the spectral weight. In contrast, in the small system limit most spectral weight is carried by a delta peak at the energy of the adiabatic process, and an oscillating PDF with dips at energies commensurate to the quench duration and with an exponential envelope develops. Relevance to cold atom experiments is also discussed.

Full counting statistics of time-of-flight images

“…The non-thermal nature of the state is reflected by the fact that in contrast to a thermal state characterized by a single inverse temperature, β p strongly depend on the momentum p [68]. Similar pre-thermalization phenomena are encountered in some quench experiments on closed, cold atomic systems, where the long-time expectation value of local observables can be well described by a thermal ensemble, despite the non-equilibrium state of the system [69,70].…”

“…After long enough holding times, τ h , the distribution of the particle numberN p looks thermal for each momentum p. Based on this thermal, exponential distribution ofN p , one can define an effective inverse temperature β p [68],…”

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

“…The work distribution is thus simply a continuous version of the distribution of squared overlaps, with the zero of the energy shifted by E i . The work statistics in non-equilibrium processes in various quantum many-body systems is a topic of much recent interest [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Another quantity of wide current interest, closely related to the overlap distribution and work distribution, is the return probability,…”

Time evolution during and after finite-time quantum quenches in Luttinger liquids

“…Further aspects that were investigated include the excitation energy, the work statistics, finite-temperature initial states, the Loschmidt echo and the diagonal ensemble reached at late times. [46][47][48][49][50][51][52] In this article we aim at obtaining a complete understanding of finite-time quenches in Luttinger liquids. To this end we consider the time evolution during and after the quench and derive exact, analytical results to go beyond the perturbative regime.…”