Large deviations, central limit, and dynamical phase transitions in the atom maser

Abstract

The theory of quantum jump trajectories provides a new framework for understanding dynamical phase transitions in open systems. A candidate for such transitions is the atom maser, which for certain parameters exhibits strong intermittency in the atom detection counts and has a bistable stationary state. Although previous numerical results suggested that the “free energy” may not be a smooth function, we show that the atom detection counts satisfy a large deviations principle and, therefore, we deal with a phase crossover rather than a genuine phase transition. We argue, however, that the latter occurs in the limit of an infinite pumping rate. As a corollary, we obtain the central limit theorem for the counting process. The proof relies on the analysis of a certain deformed generator whose spectral bound is the limiting cumulant generating function. The latter is shown to be smooth so that a large deviations principle holds by the Gärtner–Ellis theorem. One of the main ingredients is the Krein–Rutman theory, which extends the Perron–Frobenius theorem to a general class of positive compact semigroups.

Averaging of random affine transformations of functions domain

“…Mathematical models with compositions of random operator-valued functions arise in problems of classical and quantum mechanics for systems in random nonstationary fields [6]- [13]. The averaged dynamics of such systems is of both theoretical and practical interest from the point of view of analyzing the mean values of observables.…”

Invariant Projections for Covariant Quantum Markov Semigroups

“…As for the classical Ornstein-Uhlenbeck process, assuming that the matrix Z is stable, namely all eigenvalues have strictly negative real part, every initial state converges to the invariant one [18]. It is therefore natural to study the speed of convergence towards that invariant state in view of applications to problems such as return to equilibrium and limit theorems and dynamical phase transitions (see [23]).…”