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Physical Review A

Vacuum Rabi splitting of energy levels in a strongly coupled system of a two-level atom and dielectric microsphere

В. В. Климов, Vladilen S Letokhov, M. Ducloy

Published Sep 1, 1997• 15 citations
Publisher

Abstract

The effective dipolar Hamiltonian and the vacuum Rabi frequency for a system with a dielectric microsphere and a two-level atom with arbitrary orientation of dipole momentum are found. The energy levels of such a system are determined within the rotating-wave approximation as well as by numerical methods. The solution found is close to that within the pure classical approach.

DOI
10.1103/physreva.56.2308

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The effective dipolar Hamiltonian and the vacuum Rabi frequency for a system with a dielectric microsphere and a two-level atom with arbitrary orientation of dipole momentum are found. The energy levels of such a system are determined within the rotating-wave approximation as well as by numerical methods. The solution found is close to that within the pure classical approach.
A solution is found of the dispersion equation for a classical oscillator in the vicinity of a dielectric microsphere in the case of strong electromagnetic coupling. The solution obtained agrees with the corresponding quantum-mechanical solution obtained within the framework of the dressed states approach and presents a clear signature of the vacuum Rabi splitting of the energy levels in real systems. Based on this solution, a qualitative expression is found for the intensity of the resultant line doublet.
We considered the semiclassical Rabi model: a two-level atom fixed at a position and interacting with a classical, monochromatic electromagnetic field. In resonance and for an arbitrary initial state of the system, we obtained an approximate, analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/21/21/2 (1/41/41/4) of the angular frequency of the field and 111 (101010) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA. We found that the approximate solution is composed of the the solution in the RWA plus terms that oscillate at the angular frequency of the field and at the Rabi frequency. Moreover, these corrections are proportional to the Rabi frequency divided by two times the angular frequency of the field.
A strong resonant interaction of a two-level atom with a dielectric microsphere is studied on the basis of quantum electrodynamics. The initial condition considered is one in which the atom is initially excited and the resonant mode of the microsphere has been excited by a single photon. The spectrum of two emitted photons depends strongly on the method used to excite the microsphere, i.e., on the spatial distribution of the photon energy. The most characteristic feature of the two-photon fluorescence spectrum is a strong energy correlation of the emitted photons. This correlation is expressed in the fact that the energies of the emitted photons are related by the equation of an ellipse (ω+ω 2−2ω vA )2+3(ω 1−ω 2)2= 4Ω Rabi 2 . The relation between the results obtained and the predictions of the theory of dressed states is discussed.
A class of quasi-exact solutions of the Rabi Hamiltonian, which describes a two-level atom interacting with a single-mode radiation field via a dipole interaction without the rotating-wave approximation, are obtained by using a wavefunction ansatz. Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and the field frequency satisfy certain relations. As an example, the lowest exact energy level and the corresponding atom-field entanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes–Cummings and counter-rotating cases of the Rabi Hamiltonian.
We considered the semiclassical Rabi model, that is, a two-level system interacting with a single-mode, classical field. In resonance and for an arbitrary initial state of the system, we obtained a simple, approximate, and analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/2 (1/4) of the angular frequency of the field and 1 (10) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and allows one to describe the evolution of the Bloch vector of the system in terms of a \textit{slow-precessional} and a \textit{fast-nutational} motion similar to that of a symmetric top with one point fixed. Finally, the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA.