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Soviet Journal of Quantum Electronics

Nonlinear and stimulated effects in resonance fluorescence of one and two atoms

Vladislav Cheltsov

Published May 31, 1986• 0 citations
Publisher

Abstract

The Stark representation of the resonator field and the first order of perturbation theory of the fluorescence field are used to study the fluorescence spectrum of one and two two-level atoms in an ideal resonator as a function of the initial excitation of the system. It is shown that in the absence of resonator photons (nc = 0) the spectrum obtained for initial excitation of one atom consists of two lines located symmetrically relative to the central frequency ωα. In all other cases the spectrum consists of the central harmonic ωα and of ωα ± 2Ω satellites (Ω is the Rabi frequency). The intensity of collective higher harmonics ωα ± 4Ω is low and it falls on increase in the resonator field as I/nc2. The nonlinearity of the interaction Hamiltonian results in parametric amplification of the ωα + 2Ω harmonics and in attenuation of the ωα–2Ω harmonics. In the case of two atoms it is possible to achieve stimulated fluorescence at the harmonics ωα ± 2Ω.

DOI
10.1070/qe1986v016n05abeh006611

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•Mar 1, 1982•0 citations
The Stark representation of the resonator field and the first order of perturbation theory of the fluorescence field are used to study the fluorescence spectrum of one and two two-level atoms in an ideal resonator as a function of the initial excitation of the system. It is shown that in the absence of resonator photons (nc = 0) the spectrum obtained for initial excitation of one atom consists of two lines located symmetrically relative to the central frequency ωα. In all other cases the spectrum consists of the central harmonic ωα and of ωα ± 2Ω satellites (Ω is the Rabi frequency). The intensity of collective higher harmonics ωα ± 4Ω is low and it falls on increase in the resonator field as I/nc2. The nonlinearity of the interaction Hamiltonian results in parametric amplification of the ωα + 2Ω harmonics and in attenuation of the ωα–2Ω harmonics. In the case of two atoms it is possible to achieve stimulated fluorescence at the harmonics ωα ± 2Ω.
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.
For pt.I see ibid., vol.12, no.12, p.1939 (1979). The modulated fluorescence in an optical double resonance experiment, i.e. when an atom interacts with two intense monochromatic laser beams each of which is weakly modulated in amplitude at frequency Omega , is studied in detail. The modulation rate corresponding to the steady-state population of the highest atomic level exhibits resonant structures at the modulation frequencies Omega =0, +or- alpha 0 , +or-2 alpha 0 due to the dynamical Stark splitting of the various energy levels; alpha 0 being related to the Rabi frequencies alpha 1 and alpha 2 associated with the two lasers and the two atomic transitions, by alpha 0 2 = mod alpha 1 mod 2 + mod alpha 2 mod 2 . The peak at Omega =+or- alpha 0 is the most intense and exhibits splitting when the applied fields are detuned from exact resonance with the respective atomic frequencies. Resonances are observed in the modulation rate of the intermediate-level population at the modulation frequencies Omega =0, +or-2 alpha 0 . In the presence of detunings of the applied fields, additional resonant structures occur at Omega =+or- alpha 0 which increase in magnitude with higher detunings and split. Explicit expressions for the positions, widths and weights of these resonant components are obtained analytically for intense, resonant fields. Numerical results for general values of the laser field strengths and the detunings are given.
We consider the possible physical processes that may arise in a three-level atom when only two of its levels interact with a strong electromagnetic field and when the atomic transition frequency is nearly equal to once and twice the frequency of the laser field, respectively. There have been found pronounced cooperative effects in the spectrum of the two-level system, which is in resonance with the laser field, arising from the presence of the third level. The excitation spectra describing the transitions from the first excited state into the second excited state and from that to the ground state consist, apart from the two central peaks, of two pairs of sidebands which are induced by the laser field of the neighbouring system. Detailed expressions of the spectral functions for the physical processes of one- and two-photon resonance fluorescence have been derived and discussed in the limit of high photon densities. The excitation spectrum of the low frequency modes has been considered and discussed in detail. It is found that quantum beats in spontaneous emission may appear in the spectra of the one- and two-photon resonance fluorescence arising from the interference between the two atomic transition frequencies and the frequency of the laser field. The importance of the low frequency modes that occur in the processes in question has been pointed out.
We have calculated the spectrum of resonance fluorescence and the second-order correlation function for a system of two two-level atoms interacting via the dipole-dipole interaction and via radiative coupling. The method employed is the "master equation" for the reduced atomic density operator. The scattered spectrum is symmetric about the resonant frequency and is altered significantly by the presence of the dipole-dipole term. For large incident field strengths pairs of sidebands are displaced from resonance by (+OR-)(omega)(,R) and (+OR-)2(omega)(,R), where (omega)(,R) is the Rabi frequency. Additional peaks arise when the dipole-dipole interaction is of the same order as the atom-field interaction. For weak incident fields the spectrum consists of a central peak and two peaks displaced symmetrically by (omega)(,d), the strength of the dipole-dipole interaction. The spectrum when averaged over the atomic separation and orientation consists of the central peak and two pairs of sidebands displaced by (+OR-)(omega)(,R) and (+OR-)2(omega)(,R). The central and first pair of sidebands become weaker and broader as the number density increases, whereas the second pair of sidebands is enhanced and broadened. The first effect is probably observable for suitable atomic number densities. Perhaps the second pair of sidebands could also be observed in a careful experiment. The second-order correlation function of this system predicts incomplete antibunching for strong incident field strengths. At weak field strengths the antibunching depends strongly on the dipole-dipole energy.