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

Abstract

The non-Hermitian quadratic oscillator studied by Swanson is one of the popular $PT$-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describes the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the $PT$-symmetry is unbroken, i.e., the eigenvalues are purely real.

extended version, accepted for publication in J. Phys. A

References in corpus (3)

• Complexified Dynamical Systems
• Wave packet evolution in non-Hermitian quantum systems
• Complex trajectory method in time-dependent WKB

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• New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization
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• Scattering of accelerated wave packets
• Exceptional points and quantum phase transition in a fermionic extension of the Swanson oscillator
• Exploring Branched Hamiltonians for a Class of Nonlinear Systems
• Enhanced frequency and temperature estimation by a $\mathcal{P}\mathcal{T}$-symmetric quantum oscillator
• A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians
• Pseudo-Hermitian extensions of the harmonic and isotonic oscillators
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• Revisited Swanson's Hamiltonian