Work analysis of one-dimensional driven quantum systems

Abstract

In recent years there has been a tremendous advance in the techniques to trap and control systems of a few bosonic and fermionic atoms [1,2]. In these systems the trap properties are usually tunable, thus allowing one to study how the quantum system adapts to the new trap properties. In particular one can consider a simple scenario in which a particle is trapped in a harmonic oscillator potential which trapping frequency is varied in time with a given time dependence. This system represents a simple example where the concepts of work [3] need to be adapted to quantum

We introduce the probability distribution of work performed on a one- dimensional quantum system and study the cases of a single particle in a harmonic or finite well potential and of a Bose-Einstein condensate in a finite well potential. The irreversible work is generalised for the case of Bose-Einstein condensates, described in the mean-field theory by the Gross-Pitaevskii equation. The properties of the ground state are analysed for each case, finding two di?erent static regimes for the finite well potential (with a third one for a BEC with attractive interactions) and one for the harmonic well. Finally, the irreversible work is studied for a linear ramping protocol where the potential is widened, and a relation between the static regimes and the dynamics of the system is identified. The evolution of the system is obtained by numerically solving either the time-dependent Gross-Pitaevskii or Schrödinger equation through the Crank-Nicolson method.