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Physical Review A

Approach to a generalized Jaynes-Cummings model and the geometric phase

Zhong Feng Tang

Published Nov 1, 1995• 9 citations
Publisher

Abstract

Adopting the recently introduced inverses of boson creation and annihilation operators, we construct a nonunitary transformation to solve exactly a generalized Jaynes-Cummings (JC) model. The method reveals that the JC model in a new frame can be viewed simply as a system of a spin-1/2 atom interacting with a magnetic field dependent on the photon number. For the initial state with fixed photon number l, the JC model becomes a cyclic evolution, and exhibits a geometric phase ${\ensuremath{\gamma}}_{\mathit{g}}$. We discuss the properties of ${\ensuremath{\gamma}}_{\mathit{g}}$ in the large l limit.

DOI
10.1103/physreva.52.3448

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Adopting the recently introduced inverses of boson creation and annihilation operators, we construct a nonunitary transformation to solve exactly a generalized Jaynes-Cummings (JC) model. The method reveals that the JC model in a new frame can be viewed simply as a system of a spin-1/2 atom interacting with a magnetic field dependent on the photon number. For the initial state with fixed photon number l, the JC model becomes a cyclic evolution, and exhibits a geometric phase \ensuremathγg{\ensuremath{\gamma}}_{\mathit{g}}\ensuremathγg​. We discuss the properties of \ensuremathγg{\ensuremath{\gamma}}_{\mathit{g}}\ensuremathγg​ in the large l limit.
A so far most generalized Jaynes-Cummings (JC) model, which includes not only various ordinary JC models but also a special model describing spin-orbit interaction, is investigated from an algebraic point of view. We find that an alternative representation of su(2) algebra arises naturally from the system. This structure plays an essential role in the generalized system. As one result, the system behaves like a spin-1/2 system moving in a magnetic field. So it is rather easy to obtain the energy levels, eigenstates, and the time evolution operator, or any other quantities of physical interest. Two special cases are considered. Finally, a possible generalization to the three-level system is discussed.
Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.
We introduce a dynamical system that instead of exchanging a single photon as in the atomic system of the usual Jaynes-Cummings model (JCM), it exchanges instead a squeezed coherent photon. Accordingly, the creation and annihilation photon operators in the JCM are replaced by the creation and annihilation operators of squeezed coherent states, respectively. This transformation generates a photon Hamiltonian that modifies the JCM that includes terms usually omitted in the JCM when making the rotating-wave approximation and, in addition, the transformation introduces atom exchange terms absent in the JCM.
Abstract This article is a response to the continued assumption, cited even in reports and reviews of recent experimental breakthroughs and advances in theoretical methods, that the antiJaynes-Cummings (AJC)interaction is an intractable energy non-conserving component of the quantum Rabi model (QRM). We present three key features of QRM dynamics : (a) the AJC interaction component has a conserved excitation number operator and is exactly solvable (b) QRM dynamical space consists of a rotating frame (RF) dominated by an exactly solved Jaynes-Cummings (JC) interaction specied by a conserved JC excitation number operator which generates the U(1) symmetry of RF and a correlated counter-rotating frame (CRF) dominated by an exactly solved antiJaynes-Cummings (AJC) interaction specied by a conserved AJC excitation number operator which generates the U(1) symmetry of CRF (c) for QRM dynamical evolution in RF, the initial atom-eld state je0i is an eigenstate of the effective AJC Hamiltonian HAJC, while the effective JC Hamiltonian HJC drives this initial state je0i into a time evolving entangled state, and, in a corresponding process for QRM dynamical evolution in CRF, the initial atom-eld state jg0i is an eigenstate of the effective JC Hamiltonian, while the effective AJC Hamiltonian drives this initial state jg0i into a time evolving entangled state, thus addressing one of the long-standing challenges of theoretical and experimental QRM dynamics; consistent generalizations of the initial states je0i , jg0i to corresponding n 0 entangled eigenstates j+en i , j 􀀀g ni of the AJC in RF and JC in CRF, respectively, provides general dynamical evolution of QRM characterized by collapses and revivals in the time evolution of the atomic, eld mode, JC and AJC excitation numbers for large initial photon numbers ; the JC and AJC excitation numbers are conserved in the respective frames RF, CRF, but each evolves with time in the alternate frame.
The unitary transformation method is utilized for the closed form solution of the nonlinear generalization of the Jaynes–Cummings (JC) model, which includes arbitrary multiphoton and intensity-dependent interactions between radiation and matter. Specifically, the unitary transformation leading to the diagonalization of the relevant Hamiltonian is established and energy levels and eigenstates are derived. It is also shown that the results for the JC model can be extended to the anti-Jaynes–Cummings (AJC) model through the map between the two models.