PT symmetry and the evolution speed in open quantum systems 1

Dorje C Brody

doi:10.1088/1742-6596/2038/1/012005

Abstract

The dynamics of an open quantum system with balanced gain and loss is not described by a PT-symmetric Hamiltonian but rather by Lindblad operators. Nevertheless the phenomenon of PT-symmetry breaking and the impact of exceptional points can be observed in the Lindbladean dynamics. Here we briefly review the development of PT symmetry in quantum mechanics, and the characterisation of PT-symmetry breaking in open quantum systems in terms of the behaviour of the speed of evolution of the state.

Highlights

These Hamiltonians, while not Hermitian, are invariant under the action of parityand-time (PT) reversal: p → p, x → −x, and i → −i
While Hermiticity may be replaced by the condition of PT symmetry to enforce the reality of observables, a Hilbert space endowed with a PT-conjugation inner product is problematic, for, the parity operator is trace free and squares to the identity so that half of its eigenvalues are negative
In other words, such Hilbert spaces are of the Pontryagin-type that come equipped with indefinite metrics [2]

Summary

PT symmetry and quantum physics

We begin with a brief account of the development of PT symmetry in quantum mechanics. For many model Hamiltonians studied in the literature, either gor g−1 (or both) are unbounded, and in these cases it is not simple to determine whether the two descriptions are equivalent: It may be the case that there are genuinely new physics of closed quantum systems modelled on infinite-dimensional Hilbert spaces, possibly with a partially broken PT symmetry, this remains to be further investigated. The system is open, it can behave in a manner similar to a closed one because the overall energy is conserved Such an open system can be described by a PT-symmetric Hamiltonian, where the degrees freedom associated with the operator gcan be controlled in a laboratory by adjusting, for instance, the gain and loss strengths. The purpose here is to explore this question by focusing on the behaviour of the speed of the evolution of the state that has been examined in [20]

Open quantum system dynamics

Embedding mixed states in Euclidean space