Classical limit of non-Hermitian quantum dynamics—a generalized canonical structure

Abstract

We investigate the classical limit of non-Hermitian quantum dynamics arising from a coherent state approximation, and show that the resulting classical phase space dynamics can be described by generalised "canonical" equations of motion, for both conservative and dissipative motion. The dynamical equations combine a symplectic flow associated with the Hermitian part of the Hamiltonian with a metric gradient flow associated with the anti-Hermitian part of the Hamiltonian. We derive this structure of the classical limit of quantum systems in the case of a Euclidean phase space geometry. As an example we show that the classical dynamics of a damped and driven oscillator can be linked to a non-Hermitian quantum system, and investigate the quantum classical correspondence. Furthermore, we present an example of an angular momentum system whose classical phase space is spherical and show that the generalised canonical structure persists for this nontrivial phase space geometry.

Quantum-classical correspondence for a non-Hermitian Bose-Hubbard dimer

“…However, there is a rapidly growing interest in the use of non-Hermitian Hamiltonians arising from different areas. The first is the field of open quantum systems where complex energies with negative imaginary parts are used to describe an overall probability decrease that models decay, transport or scattering phenomena (see, e.g., [17][18][19][20][21][22] and references therein). Although in most cases these non-Hermitian Hamiltonians are introduced heuristically, they can be derived in a mathematically satisfactory way starting from a system coupled to a continuum of states (see, e.g., [19,23] and references cited therein).…”

“…Non-Hermitian quantum dynamics differ drastically from their unitary counterparts, and their generic features are far from being fully understood. In particular, the investigation of the quantum classical correspondence for non-Hermitian systems is only at its beginning [22,[36][37][38][39][40].…”

“…In section III the non-Hermitian Bose-Hubbard dimer is introduced as a many-boson generalization of the non-Hermitian two-level system. In section IV we review the generalized mean-field approximation introduced in [51] and show that it can be expressed in a canonical form of dissipative classical mechanics suggested in [22]. We analyze the resulting mean-field dynamics in detail in section V and compare it to the full many-particle system in section VI.…”

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

“…This is not surprising from the point of view of complex geometry. It might seem natural to assume the metric to be given by the standard euclidean metric G = I [4]. As has been shown in [5,6], however, starting from the quantum dynamics in the semiclassical limit, G is itself time dependent.…”

“…A first step towards classical non-Hermitian dynamics has been made by the authors in [4,5]. The interesting complex structure of the resulting metriplectic flow in phase space was revealed in [6].…”