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Journal of Experimental and Theoretical Physics Letters

Two-photon fluorescence spectrum in an atom + dielectric microsphere system

В. В. Климов, S. Letokhov

Published Aug 1, 1999• 0 citations
Publisher

Abstract

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.

DOI
10.1134/1.568151

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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.
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 resonance interaction of a two-level atom with a continuum of free-space modes modified by the presence of a dielectric microsphere (modified free-space modes — MFSMs) is studied. In the case that quantized MFSMs are initially excited within the contour of one of the resonance modes of the microsphere, the spectrum of emitted photons depends strongly on the excitation method. Under optimal excitation conditions efficient excitation of the atom accompanied by the formation of a Rabi doublet in the fluorescence spectrum occur. As the excitation conditions depart from optimality, the spectrum becomes a triplet. If the departure from optimality of excitation is large, the atom remains essentially unexcited, and the fluorescence spectrum has a singlet character.
The strong resonance interaction of a two-level atom with the continuum of quantized electromagnetic modes falling within the contour of a resonance mode of a dielectric microsphere is considered within the framework of quantum electrodynamics. Analytical solutions are derived. As an initial condition, we consider the case when, at time t = 0, the atom is excited and the resonance modes of the microsphere contain a single-photon wavepacket. It is shown that the properties of the emitted photon pair depend crucially on space-time properties of the photon wavepacket contained in the resonator. When the mean square of the electric field of the photon wavepacket at the initial instant of time at the atom position is close to the vacuum value, the radiation of an atom is similar to a spontaneous one and the emitted photon pair has no correlations. On the contrary, if the mean square of the electric field of the photon wavepacket at the initial instant of time at the atom position is substantially greater than the vacuum value, the radiation of an atom has a stimulated nature and the emitted photon pair has very complicated strong correlations. The relationship between the results obtained and the predictions of the dressed states theory are briefly discussed. The results obtained are of a general character and can be applied to the description of the resonant interaction of an excited atom and an excited resonator of an arbitrary shape.
The excitation spectrum arising from the interaction between two identical atoms (molecules), one of which is excited in the presence of a resonant strong electromagnetic field (pump field), is investigated. The excitation spectrum is found to consist of those describing the symmetric and antisymmetric modes, respectively. The form of both spectra depends on the relation between the distance RRR separating the atoms and the wavelength $\ensuremath{\lambda}$ of the transition from the ground state to the excited state. For $R<\ensuremath{\lambda}$, and when certain conditions prevail, the spectral function for the symmetric modes consists of three Lorentzian lines describing the central peak and the two sidebands whose radiative widths are equal to ${\ensuremath{\gamma}}_{0}$, $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, and $\frac{3{\ensuremath{\gamma}}_{0}}{2}$, which are twice the corresponding ones arising from an isolated single atom interacting with the pump field. For the antisymmetric modes the central peak has a $\ensuremath{\delta}$-function distribution indicating the stability of the mode in question, while, when certain conditions are satisfied, the two sidebands are described by Lorentzian lines each having a radiative width of the order of $\frac{{\ensuremath{\gamma}}_{0}}{2}$, which is equal to the natural linewidth for a photon spontaneously emitted from an isolated atom. For both the symmetric and antisymmetric modes, the dipole-dipole interaction between the atoms brings about small energy shifts. For $R>\ensuremath{\lambda}$, apart from the small energy shifts caused by the dipole-dipole interaction, the spectral functions for symmetric and antisymmetric modes are similar to that for the single atom interacting with the pump field.
The resonance fluorescence spectrum of one and two two-level atoms in an ideal cavity with one resonant mode is investigated as a function of the initial excitation of the system. The analysis is carried out in the Stark representation with respect to the cavity's field and in first-order perturbation theory with respect to the fluorescence field. In the absence of the field of the cavity (n c = 0) and under an initial excitation of one atom only, the spectrum consists of two lines which are symmetrical with respect to ωa. In all other cases the spectrum has a central line at ωa and other lines at ωa ± 2 Ω. The intensity of the higher (ωa ± 4 Ω) harmonics is small and decreases as 1/ 2 c c with increasing n>c. The nonlinearity of the interaction hamiltonian results in the parametric (with respect to the field of the cavity) amplification of the ωa + 2 Ω harmonic and the attenuation of the ωa − 2 Ω harmonic. The stimulated emission of the ωa ± 2 Ω harmonics, vanishing in a strong (n c » 1) field, is shown to exist for two atoms.