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arXiv:1910.02955 · Physica Scripta

Approximate evolution for a system composed by two coupled Jaynes–Cummings Hamiltonians

I. Ramos-Prieto, A Paredes, J. Récamier, H. M. Moya-Cessa

Published Nov 1, 2019• 6 citations
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Abstract

Abstract In this work we construct an approximate time evolution operator for a system composed by two coupled Jaynes–Cummings Hamiltonians. We express the full time evolution operator as a product of exponentials and we analyze the validity of our approximations contrasting our analytical results with those obtained by purely numerical methods.

DOI
10.1088/1402-4896/ab538b

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Abstract In this work we construct an approximate time evolution operator for a system composed by two coupled Jaynes–Cummings Hamiltonians. We express the full time evolution operator as a product of exponentials and we analyze the validity of our approximations contrasting our analytical results with those obtained by purely numerical methods.
A generalization of the time-dependent Jaynes–Cummings Hamiltonian is considered including nonlinear terms in the field and/or an intensity-dependent coupling term. Using algebraic methods we find the corresponding time evolution operator of the problem as a product of exponential operators. Also, we study the effect of the atomic motion in the cavity on its quantum state. We found that, when the atom is vibrating classically with a small amplitude A0 (in comparison with the effective length of the cavity L), i.e. A0 ≪ L, the effects of the atom’s motion are negligible but when A0 ∼ L/4, the state of the atom changes drastically. Our study is devoted to a Jaynes–Cummings Hamiltonian with an additional Kerr medium.
In this work, we start from a phenomenological Hamiltonian built from two known systems: the Hamiltonian of a pumped optomechanical system and the Jaynes-Cummings Hamiltonian. Using algebraic techniques we construct an approximate time evolution operator U^(t) for the forced optomechanical system (as a product of exponentials) and take the JC Hamiltonian as an interaction. We transform the later with U^(t) to obtain a generalized interaction picture Hamiltonian which can be linearized and whose time evolution operator is written in a product form. The analytic results are compared with purely numerical calculations using the full Hamiltonian and the agreement between them is remarkable.
Abstract The Hamiltonian of a system composed of two standard Jaynes‐Cummings cells connected by the overlap of evanescent cavity fields, which allows photons to hop from the first cavity to the second and vice versa, and interacting with an external classical field, is simplified in the framework of unitary canonical transformations. James's method of the effective Hamiltonian is used in the derivation of two effective Hamiltonians with two completely dispersive regimes. Analytical expressions for the time dependent wavefunction and the elements of the reduced atomic operators for the different Hamiltonians are derived. The temporal evolution of atomic quantum entropy is calculated when the fields are initially in their vacuum states with various forms of the atomic initial states. Effects of the photon hopping strength, detuning and the various forms of the atomic initial states on the evolution of the von Neumann entropies are analyzed. The impact of the time‐average technique on the features resulting due to involving the controlling parameters and their influences on the considered functions is discussed. Damping is taken into account through a non‐Hermitian Hamiltonian and results are discussed. General conclusions reached are illustrated by numerical results.