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International Journal of Theoretical Physics

Ladder Invariants and Coherent States for Time-Dependent Non-Hermitian Hamiltonians

M. Zenad, F.-Z. Ighezou, Omar Cherbal, Mustapha Maamache

Published Feb 14, 2020• 7 citations
Publisher
DOI
10.1007/s10773-020-04401-8

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In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).
Abstract We introduce a pseudo-squeezed bosonic ladder operator defined as a time-dependent non-Hermitian linear invariant and related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. In fact, they are obtained from the squeezed transformation of the pseudo-bosons annihilation and creation operators. Thus, the pseudo-bosonic squeezed coherent states are just obtained as pseudo-displacement operator method acting on the ground pseudo-squeezed state. To our knowledge, the time-dependent pseudo-squeezed coherent states have not been constructed until now. As an illustration, we study the time-dependent non-Hermitian displaced harmonic oscillator, and the properties of these states are analysed with respect to the localization in position and to uncertainty principle.
We study the time evolution of a class of exactly solvable time-dependent quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators with the help of the invariant Hermitian operator. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the invariant operator. The adiabatic and non-adiabatic Berry phases are calculated with the exact solutions. Moreover, we derive an explicit time-evolution operator which is used to investigate the time-dependent two-photon squeezing states and SU(2) squeezing states. The squeezing properties of the time-dependent SU(1,1) coherent states are also discussed.