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Journal of Liaoning University

Geometric Phase in a Generalized Λ-type Jaynes-Cummings Model with Imaginary Photon Process

Jiao Zhi-Yong

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

By using the Lewis Riesenfeld invariant theory,we have studied the dynamical and the geometric phases in a generalized time-dependent Λ-type Jaynes-Cummings model with imaginary photon process.We find that the geometric phases in a cycle case have nothing to do with the frequency of the photon field,the coupling coefficient between photons and atoms,and the atom transition frequency.If we use the more accuracy device,the geometric phase in the imaginary photon process may be observed,and the geometric phases in this process have the observable physical effect.

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By using the Lewis Riesenfeld invariant theory,we have studied the dynamical and the geometric phases in a generalized time-dependent Λ-type Jaynes-Cummings model with imaginary photon process.We find that the geometric phases in a cycle case have nothing to do with the frequency of the photon field,the coupling coefficient between photons and atoms,and the atom transition frequency.If we use the more accuracy device,the geometric phase in the imaginary photon process may be observed,and the geometric phases in this process have the observable physical effect.
By using the Lewis–Riesenfeld invariant theory, we have studied the dynamical and the geometric phases in a generalized time-dependent Λ-type Jaynes–Cummings model. We find that, compared with the dynamical phases, the geometric phases in a cycle case are independent of the frequency of the photon field, the coupling coefficient between photons and atoms, and the atom transition frequency. It is pointed out that the geometric phases in a cycle case can be measured under the case of a stronger time-dependent photon field and a stronger coupling photon-atom system. On the other hand, the geometric phases of the model may be measured in the composition of cold atoms.
利用Lewis-Riesenfeld不变量理论，研究了推广的含时二能级虚光子过程的 Jaynes-Cummings模型。发现周期情况下的几何相位与光子场的频率、光子和原子之间的耦合系数以及原子跃迁频率无关。 By using the Lewis-Riesenfeld invariant theory, we have studied the geometric phase in a generalized time-dependent Jaynes-Cummings model with imaginary photon process for two-level atoms interacting with light field. It is found that the geometric phase in a cycle case has nothing to do with the frequency of the photon field, the coupling coefficient between photons and atoms, and the atom transition frequency.
通过使用Lewis-Riesenfeld不变理论，已经研究了具有双模运算符和相位运算符的广义Jaynes-Cummings模型中的几何相位。与动力学相位相比，一个周期情况下的几何相位与双光子场的频率无关，光子和原子之间的耦合系数以及原子跃迁频率。显然，几何相具有纯几何和拓扑特征，这意味着几何相位代表了厄米线性束中的和乐。 By using the Lewis-Riesenfeld invariant theory, the geometric phase in a generalized Jaynes-Cummings model with double mode operators and phase operators has been studied. Compared with the dynamical phase, the geometric phase in a cycle case is independent of the frequency of the double photon field, the coupling coefficient between photons and atoms, and the atom transition frequency. It is apparent that the geometric phase has the pure geometric and topological characteristics, which means that the geometric phase represents the holonomy in the Hermitian linear bundles.
An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.
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.
An extension of the standard one-photon as well as the two-photon Jaynes-Cummings (JC) model has been done to include transient effects due to a modulation of the atom-field coupling coefficient. An explicit time dependence is given for the case of a linear sweep of the coupling parameter. We have shown the effects of such a sweep on the dynamical evolution of a two-level atom undergoing a one- or two-photon process in a single nondecaying mode of the field in a coherent state. The statistical aspects of the field, such as intensity-intensity correlation and squeezing, are also investigated and are found to be both qualitatively as well as quantitatively different from the standard JC models.