Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps

Abstract

Here we present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of a quasi-Hermitian Hamiltonian. The time-dependent Dyson map is derived through a constructed Schrödinger-like equation governed by the non-Hermitian Hamiltonian itself; despite its time-dependence our scheme ensures the time-independence of the metric operator, a necessary condition for the observability of the quasi-Hermitian Hamiltonian. As an illustrative example we consider a driven Harmonic oscillator described by a time-dependent non-Hermitian Hamiltonian. After computing the Dyson map and demonstrating the time-independence of the associated metric operator, we successfully derive an eigenvalue equation for this time-dependent Hamiltonian which enable us to analyze the PT -symmetry breaking process.

Exact solutions for time-dependent non-Hermitian oscillators: classical and quantum pictures

“…The structure of H sw resembles the expression of the Hamiltonian for the Swanson oscillator [43,47,53,54]. Nevertheless, subtle but relevant differences must be noted since the coefficients α j (t), β j (t), j = 1, 2, and θ(t), are time-dependent functions introduced to design the profile of H sw in Eq.…”

“…In this form, the point transformation provides a straightforward mechanism to determine such a factor, which is nothing but the transformation of the unitary time-evolution phase of the stationary oscillator. Alternative approaches for non-Hermitian Hamiltonians have been previously studied in [53].…”

Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation

“…It is precisely this omission that we aim to remedy here. This technical complexity also probably explains why only a relatively few realistic applications of several alternative implementations of the NHIP ideas themselves may yet be found in the current literature [30,[72][73][74][75][76][77][78][79][80][81].…”

Enhancement of photon creation through the pseudo-Hermitian dynamical Casimir effect

“…Such an indefiniteness of the unitarity of time evolution was overcome by constructing a Hilbert space based on a time independent metric operator where the non-Hermitian Hamiltonian enjoys self-adjointness. When a TD metric operator is considered for treating TD non-Hermitian Hamiltonians, different proposals have been presented for reestablishing the self-adjointness to these TD Hamiltonians [64][65][66][67]. Many interesting contributions have been presented regarding TD PT -symmetric Hamiltonians [68][69][70][71][72][73].…”