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Chinese Physics Letters

Damped Jaynes-Cummings Model Including Spontaneous Emission

Tan Weihan

Published Feb 1, 2001• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0256-307x/18/2/322

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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.
We have in this thesis looked at two very different problems related to quantum optics. In Part I we have studied the Jaynes-Cummings model with cavity damping and a thermal background. To give an introduction to the field we started Chapter 2 by repeating how one in general derives the master equations for a system coupled to a reservoir. We then repeat the Jaynes-Cummings model and show extensions of the Jaynes-Cummings model where cavity-damping and a thermal background are taken into account. We proceed, in Chapter 3, by finding an analytical solution to the Jaynes-Cummings model with cavity damping in a thermal background when there is no detuning in the system, i.e. we have resonance between the energy of the photons in the cavity and the energy difference in the excited and ground state of the atom. We derive analytical expressions for P+(t), the probability of finding the atom in an excited state after some transition time t given that the atom was prepared in an excited state, and P++(t), the probability of finding the atom in an excited state after some transition time t when a previous atom was measured to be in the excited state. Both atoms where initially prepared in the excited state. We also, in Section 3.7 make use of the Poisson summation formula to rewrite the analytical solution P+(t) in terms of a Fourier series, from which we can predict the location of the revivals in the system. In Section 3.8 we derive stationary results, i.e. when the transition time t→∞, for the probabilities P+(t) and P++(t) and the mean number of photons ‹ n(t) › in the cavity. From our analytical results one is, for instance, led to believe that the probability P+(t) approaches 0 as the transitiontime increases, when it in reality approaches the value nb/(1 + 2nb). The results of our study of the extended Jaynes-Cummings model is presented in the article “Macroscopic Interference Effects in Resonant Cavities” and is published in Physica Scripta [5]. The article is reprinted as Article 1 at the end of this thesis. In Chapter 4 we present the result of a numerical solution of the Jaynes-Cummings model with cavity damping in a thermal background. Here we consider both the resonant case and also the case where detuning is present in the system. When we compare our numerical simulation with our analytical solutions we find that the correlation between the two are close to 1, at least when experimental values of Walther et al. [22] and Brune et al. [15, 16] are used. We also present some figures that show how the revival times and probabilities previously discussed is affected by detuning. In Section 4.2 we find an analytical formula for the revival time where detuning is embedded. In Chapter 4 we also, shortly, discuss how the detuning changes the effective number of photons in the cavity and what criteria should be present in order to separate the revival times.In Part II we study some aspects of quantum mechanical phase. We start, in Section 5.1, by repeating the definition and some results of the Susskind-Glogower phase operator. We then turn our attention to the Pegg-Barnett phase operator [26, 27, 28] in Section 5.4. We describe how to evaluate expectation-values and variances of the Pegg-Barnett phase. In Section 5.5 we establish an upper and lower bound on the phase based on the Pegg-Barnett phase description.In Section 5.9 we repeat some details of a new formalism of the phase which we have conveniently called the NFM-formalism after its inventors Noh, Foug`eres and Mandel [29, 30, 31]. We then describe how to find expectation-values using the Pegg-Barnett phase operator definition applied to the NFM-experiment described in Section 5.9. Our work on the quantum mechanical phase operator is presented in the paper “On the Quantum Phase Operator for Coherent States” and is published in Physica Scripta [6]. This paper is reprinted as Article II at the end of this thesis.