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Chinese Physics Letters

Collapse and Revival in the Damped Jaynes-Cummings Model

Weihan Tan, Yan Ke-zhu

Published Dec 1, 1999• 1 citation
Publisher

Abstract

We present the first results for the analytic solution of the dissipative Jaynes-Cummings model and its application to the collapse and revival of atomic inversion. We find that the atomic inversion oscillates more and more slowly with decreasing Rabi frequency due to the cavity losses. At the same time, long drawn-out revivals after collapse are observed.

DOI
10.1088/0256-307x/16/12/014

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We present the first results for the analytic solution of the dissipative Jaynes-Cummings model and its application to the collapse and revival of atomic inversion. We find that the atomic inversion oscillates more and more slowly with decreasing Rabi frequency due to the cavity losses. At the same time, long drawn-out revivals after collapse are observed.
In this paper, we give an analytical treatment to study the behavior of the collapse and the revival of the Rabi oscillations in the Jaynes–Cummings model (JCM). The JCM is an exactly soluble quantum mechanical model, which describes the interaction between a two-level atom and a single cavity mode of the electromagnetic field. If we prepare the atom in the ground state and the cavity mode in a coherent state initially, the JCM causes the collapse and the revival of the Rabi oscillations many times in a complicated pattern in its time-evolution. In this phenomenon, the atomic population inversion is described with an intractable infinite series. (When the electromagnetic field is resonant with the atom, the nth term of this infinite series is given by a trigonometric function for sqrtnt\\sqrt{n} tsqrtnt, where t is a variable of the time.) According to Klimov and Chumakov's method, using the Abel–Plana formula, we rewrite this infinite series as a sum of two integrals. We examine the physical meanings of these two integrals and find that the first one represents the initial collapse (the semi-classical limit) and the second one represents the revival (the quantum correction) in the JCM. Furthermore, we evaluate the first- and second-order perturbations for the time-evolution of the JCM with an initial thermal coherent state for the cavity mode at low temperature, and write down their correction terms as sums of integrals by making use of the Abel–Plana formula.
As a continuation of the previous paper 'Collapse and revival in the damped Jaynes-Cummings model' [Chin. Phys. Lett. 16 (1999) 895], at present the problem is solved under the condition that the atomic spontaneous emission is included. In the case of no damping, c = 0,γ 1 ≠ 0,γ 2 ≠ 0, the half atomic inversion operator ⟨σ z ⟩ possesses an analytic solution, whereas in the general case c ≠ 0, the problem is reduced to the numerical evaluation of first-order nonlinear differential equations. The final results show that with the increase of γ 1 and γ 2 , the kinetic quantities converge rapidly to stationary solutions.
In this paper, we give an analytical treatment to study the behavior of the collapse and the revival of the Rabi oscillations in the Jaynes–Cummings model (JCM). The JCM is an exactly soluble quantum mechanical model, which describes the interaction between a two-level atom and a single cavity mode of the electromagnetic field. If we prepare the atom in the ground state and the cavity mode in a coherent state initially, the JCM causes the collapse and the revival of the Rabi oscillations many times in a complicated pattern in its time-evolution. In this phenomenon, the atomic population inversion is described with an intractable infinite series. (When the electromagnetic field is resonant with the atom, the nth term of this infinite series is given by a trigonometric function for [Formula: see text], where t is a variable of the time.) According to Klimov and Chumakov's method, using the Abel–Plana formula, we rewrite this infinite series as a sum of two integrals. We examine the physical meanings of these two integrals and find that the first one represents the initial collapse (the semi-classical limit) and the second one represents the revival (the quantum correction) in the JCM. Furthermore, we evaluate the first- and second-order perturbations for the time-evolution of the JCM with an initial thermal coherent state for the cavity mode at low temperature, and write down their correction terms as sums of integrals by making use of the Abel–Plana formula.
We study the Jaynes-Cummings model (JCM) which describes the interaction between a two-level atom and a single-mode resonant or quasi-resonant electromagnetic field in the non-dissipative and issipative cases. In particular, we study the atomic population inversion whose behavior presents some aspects of purely quantum origin: collapse and revivals. This observables expectation value can be btained analytically without use of approximations for the non-dissipative case and using first-order perturbation in the dissipative case. We make use of the sum of Poisson and the steepest descents method to obtain analytical formulas for the collapse and for each revival. In this way, we can obtain relevant information like the occurrence time, the maximum amplitude, the approximate frequency, and width. The dissipative JCM models the action of the environment on the quantum system. We show that the characteristic time of the revival diminishes as the interaction with the environment enlarges. We consider the limit in which it becomes experimentally impossible to distinguish the obtained curve from the classical one.