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Journal of Modern Optics

Vacuum Rabi splitting of an atom dipole near a dielectric microsphere

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

Published Jun 1, 1997• 18 citations
Publisher

Abstract

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.

DOI
10.1080/09500349708230721

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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.
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.
Studied within the framework of both classical and quantum-mechanical approaches is the resonance interaction between two atomic dipoles one of which is located inside a dielectric microsphere and the other near its outer surface. The dispersion equation of this system is found. The analysis of this equation in a quasistatic approximation has shown that the behavior of the eigenfrequencies depends in a complex manner on the parameters of the system, and that exact resonance can be achieved, provided the parameters are suitably selected. Based on the solutions of the above equation, relationships are found between the energy-transfer rate and the system parameters. It is demonstrated that as the inside atom approaches the surface of the microsphere, resonance excitation transfer from one atom to the other can take place. Similar results are obtained when using the quantum-mechanical density matrix to define the atomic interaction. The results obtained can be of use in implementing selective microscopy with a nanometer-high spatial resolution.
The interaction is considered between a two-level atom and the continuum of electric field modes falling within the profile of one of the resonance modes of a dielectric microsphere (a whispering gallery mode). Subject to minimal assumptions, simple equations are obtained on first principles, which describe the spectral properties of the one-photon continuum and the relaxation processes occurring therein. If the initial state of the atom is excited, a doublet structure is formed in the emitted photon spectrum, provided that the interaction is strong enough. If it is the microsphere that is excited at the initial instant, the excited photon spectrum depends substantially on the excitation method used. When excitation is optimal, the atom is effectively excited, and a Rabi-doublet is then formed in the fluorescence spectrum. When the excitation conditions are other than optimal, the spectrum becomes of triplet character, and when the deviation from the optimal excitation conditions is strong enough, the atom practically fails to get excited, and the fluorescence spectrum is of singlet type. A considerable part of the results obtained is of a general nature and can easily be applied to other cases of strong interaction between atoms and resonators.
The vacuum-field Rabi splitting of an optical mode in a cavity interacting with a system of N excitations (atomic or electronic) represented by harmonic oscillators is obtained quantum mechanically both without and with coupling to dissipation. The optical lineshapes and the scaling of the Rabi splitting with the number of oscillators and its dependence on the damping parameters are given. Corresponding results are given for a classical treatment of the oscillators, and it is shown that to the leading order in the dissipation, the quantum mechanical results for absorption are the same as the classical ones.