A time-dependent pseudo-Hermitian Hamiltonian for a cavity mode with pure imaginary frequency

Abstract

In this work we first present a general treatment for a time-dependent (TD) pseudo-Hermitian quadratic Hamiltonian, considering an equally TD and non-Hermitian Dyson map. Then we particularize our study to a TD pseudo-Hermitian Hamiltonian describing a cavity mode with pure imaginary frequency. Our goal is to produce squeezed states of the radiation field with an infinite degree of squeezing at a finite time interval. This mechanism is prevented by Hermitian Hamiltonians which allow an infinite degree of squeezing only asymptotically in time. Beyond the specifics of the technical treatment of the problem, we also present a discussion of its experimental implementation in the domain of the radiation–matter interaction. Such a discussion, is crucial to put into perspective the relevance of the quantum mechanics of pseudo-Hermitian Hamiltonians, whose effects can go beyond those coming from Hermitian Hamiltonians. Basically, we conclude that pseudo-Hermitian Hamiltonians enable much higher pumping rates than those of Hermitian processes.