Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians

TL;DR

This paper explores the complex geometry of coherent states in non-Hermitian quantum systems, comparing two approaches and revealing a unifying complex symplectic structure that links complex and real phase space dynamics.

Contribution

It introduces a comprehensive analysis of complexified coherent states, resolving apparent contradictions between different dynamical approaches and identifying a natural complex structure connecting complex and real phase spaces.

Findings

Complexified coherent states are equivalent on specific complex Lagrangian manifolds.

A natural complex structure maps complex phase space to real phase space.

The complex symplectic geometry underpins the dynamics of non-Hermitian quantum systems.

Abstract

The complex geometry underlying the Schr\"odinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.