Wave packet evolution in non-Hermitian quantum systems

Abstract

The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\hslash \to 0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase space metric---a phenomenon having no analog in Hermitian theories.

Example added, references updated, 4 pages, 2 figures

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• Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states
• Conceptual inconsistencies in finite-dimensional quantum and classical mechanics
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• Phase space formulation of density operator for non-Hermitian Hamiltonians and its application in quantum theory of decay
• Bloch oscillations in a Bose-Hubbard chain with single-particle losses
• Density operator approach to turbulent flows in plasma and atmospheric fluids
• Complex extension of Wigner's theorem
• Quantum-to-semiclassical Husimi dynamics of non-Hermitian localization transitions
• $\mathcal{P}\mathcal{T}$-symmetry in compact phase space for a linear Hamiltonian