Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians

Abstract

The complex geometry underlying the Schrödinger 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.

18 pages, small improvements made, similar to published version

Cited by in corpus (9)

• Non-Gaussianity as a signature of a quantum theory of gravity
• Time correlation functions for non-Hermitian quantum systems
• Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations
• A squeezed review on coherent states and nonclassicality for non-Hermitian systems with minimal length
• An invariant class of wave packets for the Wigner transform
• Quantum-jump vs stochastic Schrödinger dynamics for Gaussian states with quadratic Hamiltonians and linear Lindbladians
• The norm of the non-self-adjoint harmonic oscillator semigroup
• Husimi dynamics generated by non-Hermitian Hamiltonians
• Scattering of accelerated wave packets