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

Abstract

Abstract. The non-Hermitian quadratic oscillator known as Swanson oscillator is one of the popular P T -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 phasespace 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 P T -symmetry is unbroken, i.e., the eigenvalues are purely real.

Quasiclassical analysis of Bloch oscillations in non-Hermitian tight-binding lattices

“…It has been recently argued in [9] that a quasiclassical description of Bloch oscillations in the non-Hermitian case is of little use. Here, however, we show that the classical dynamics recently developed in [21][22][23][24] as a counterpart of non-Hermitian quantum dynamics, is capable of describing the main features of Bloch oscillations as well in the non-Hermitian case, as long as only a single Bloch band is involved in the dynamics (a constraint that also holds in the Hermitian case).…”

“…In the following we shall show, however, that while it may indeed be difficult to proceed directly from equation (25) for position and momentum, the semiclassical limit of the quantum dynamics generated by non-Hermitian Hamiltonians, as derived in [22] is capable of accurately describing basic features of the quantum dynamics. Using a Gaussian wavepacket approximation in the spirit of Heller [31], it has been shown that the classical dynamics associated to a non-Hermitian Hamiltonian = -H H H i R I , depending on the real valued canonical coordinates p q , are given by [22,23] (…”

“…Note that we constrain the discussion to one-dimensional systems here, and use the scaled covariance matrix Σ instead of its inverse G that is used in [22,23].…”

Tests of quantum gravity-induced non-locality: Hamiltonian formulation of a non-local harmonic oscillator

“…where H + = x 4 /2 and H − = 2ixp + 1, which are Hermitian and anti-Hermitian respectively. It is interesting to note that, as H − = i{x, p}, by neglecting the anharmonic part of the potential H + , we are left with (a particular case of) the so-called Swanson Hamiltonian [27][28][29].…”

$\mathcal{PT}$ -symmetric quantum oscillator in an optical cavity

“…For example, the classical limit of the PT -symmetric quantum dynamics, described by generalized canonical equations, shows an interesting geometric structure [20,21] which provides a nontrivial extension of the Ehrenfest theorem. As shown in a series of papers by E.M. Graefe et al [20][21][22][23], the generalized canonical equations involve a metric gradient flow, in addition to the usual Hamiltonian (conservative) dynamics. The gradient flow is coupled to an evolution equation for the metric, which in turn depends on the phase space coordinates.…”