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arXiv:quant-ph/9808004 · Journal of Optics B Quantum and Semiclassical Optics

An analytically solvable time-dependent Jaynes-Cummings model

Ananda Dasgupta

Published Jan 1, 1999• 16 citations
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Abstract

Using the underlying SU (2) algebra of the Jaynes-Cummings model, we construct a time-dependent interaction term that allows analytical solution for even off-resonance conditions. Exact solutions for the time evolution of any state have been found. The effect of detuning on the Rabi oscillations and the collapse and revival of inversion is indicated. It is also shown that at resonance, the time-dependent Jaynes-Cummings model is analytically solvable for an arbitrary interaction term.

DOI
10.1088/1464-4266/1/1/003

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Using the underlying SU (2) algebra of the Jaynes-Cummings model, we construct a time-dependent interaction term that allows analytical solution for even off-resonance conditions. Exact solutions for the time evolution of any state have been found. The effect of detuning on the Rabi oscillations and the collapse and revival of inversion is indicated. It is also shown that at resonance, the time-dependent Jaynes-Cummings model is analytically solvable for an arbitrary interaction term.
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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.
The interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonalizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By keeping only certain terms, we can achieve an excellent approximation to the exact dynamics within specified parameter ranges.