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A Two-state System With Evolving Process Beyond Jaynes-Cummings Model

Qin Gan

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The two-level system interacting with a single-mode boson is studied using the method of coherent approximation. We calculated the occupation probability P i(t) of the system, and compare it with the results of Jaynes-Cummings model under different situations. The results show that this method can be used to deal with the problems when J-C model is not valid. CA(coherent approximation) method can provide better approximate results when the differential physical effect of P i(t) is considered. CA is a non-variational and non-perturbational method that provides a systematic scheme to improve the approximation of the system.

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The two-level system interacting with a single-mode boson is studied using the method of coherent approximation. We calculated the occupation probability P i(t) of the system, and compare it with the results of Jaynes-Cummings model under different situations. The results show that this method can be used to deal with the problems when J-C model is not valid. CA(coherent approximation) method can provide better approximate results when the differential physical effect of P i(t) is considered. CA is a non-variational and non-perturbational method that provides a systematic scheme to improve the approximation of the system.
By a model of a two-level particle coupled with boson field, we made it clear that an evolution problem can be solved beyond the rotating wave approximation. We have applied the coherent approximation method, which had been proved to be effective in dealing with stationary state problems of polaron, to the evolution problem of the system mentioned above. The results obtained showed that the coherent approximation method is effective to treat the evolution problem and, in general cases, the non-rotating wave terms in Hamiltonian should not be ignored. Our results may provide a deep physical insight for further experiments to test the effects of non-rotating wave terms.
We propose a new method to discuss the evolution of physical systems, which has an analytic form. And we systematically introduce this new method by the example of Jaynes-Cummings model without rotation wave approximation. Simultaneously, while we repeat previous work by our method, we also calculate it by adding the time growth factor to the initial state unfolded in the steady state which is based on Fock state and is obtained by solving the time-independent Schrodinger equation (in this article, the traditional method refers to this method). By comparing these two results, we find the drawback of our method and improve it. Finally, we show that our improved method need the smaller Fock space than the traditional method for physical systems with two-mode cavity field and put forward expectations for the follow-up study.
The Jaynes-Cummings model (JCM) is studied in the absence of the rotating-wave approximation (RWA) by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansion was performed only to the third order, we carry out in this paper a complete expansion to demonstrate exactly the dynamics of the JCM without the RWA. Our study gives a systematic method to solve the non-RWA problem, which would be useful in various physical systems, e.g., in a system with an ultracold trapped ion experiencing the running waves of lasers.
We revisit the mathematical formulation of the famous Jaynes–Cummings–Paul Hamiltonian, which describes the interaction of a two-level atom with a single mode of an electromagnetic cavity reservoir. We rigorously show that under the condition of ultrastrong coupling between the atom and cavity in which the transition frequency is comparable to the coupling frequency, the bosonic field operators undergo non-sinusoidal time variations. As a result, the well-known solutions to the Jaynes–Cummings–Paul model are no longer valid, even when the rotating wave approximation is not used. We show how a correct mathematical solution could be found instead.
The exactly analytic solution of the coherent sbate Jaynes-Cummings (JC) model is derived for the detunings using the interaction picture. The atomic level occupation probabilities are investig-ated numerically as function of time and detuning parameter and initial mean photon number. For detuning parameter, the quantum collapse and revival phenomenon of the atomic level occupation probabilities is found. Interference between revival signals is discussed. The time evolution of the coherent states of field is studied numerically. It is discovered that the evolution of the state of field appears to be quasi-reversible for a short period of time. This is an interesting and important new phenomenon of JC model. Detailed discussion is presented.
The problem of the interaction of a single two-level atom with a single mode of the quantized electromagnetic field in a coherent state in an ideal resonator in the resonance case is considered. The evolution in time of the atomic inversion, represented by the Jaynes-Cummings sum, is studied. On the basis of the application of the theorem on the approximation of a trigonometric sum by a shorter one (ATS), a new efficient method for approximating the Jaynes-Cummings sum is constructed. New asymptotic formulas for the atomic inversion are found, which approximate it on various time intervals, defined by relation between the atom-field coupling constant and the average photon number in the resonator field before the interaction of the field with the atom. The asymptotics that we obtain give the possibility to predetermine the details of the process of the inversion in the Jaynes-Cummings model depending on the field characteristics.