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Annual Meeting Optical Society of America

Bloch vector model of Raman coupled system

Christopher C. Gerry

Published Jan 1, 1991• 0 citations
Publisher

Abstract

I study the coherent interaction of a three-level atom in the lambda configuration with pump, Stoke, and anti-Stokes laser fields. The pump laser field drives transitions from the two non-degenerate lower levels to the excited state which is highly detuned for either transition. This excited state is also coupled to each of the lower states by Stokes or anti-Stokes transitions so that this system may represent a model of four-wave mixing. Assuming large detunings, the excited state is adiabatically eliminated from obtaining an effective two-level system with Raman coupling between the levels. Optical Bloch equations are obtained for this effective two-level system, and phenomenological damping terms are added for collisional dephasing of the Raman transitions. The equations are solved for the steady-state as well as by the method of adiabatic following. From these solutions we obtain the polarization.

DOI
10.1364/oam.1991.thmm44

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•Jan 1, 1995•0 citations
I study the coherent interaction of a three-level atom in the lambda configuration with pump, Stoke, and anti-Stokes laser fields. The pump laser field drives transitions from the two non-degenerate lower levels to the excited state which is highly detuned for either transition. This excited state is also coupled to each of the lower states by Stokes or anti-Stokes transitions so that this system may represent a model of four-wave mixing. Assuming large detunings, the excited state is adiabatically eliminated from obtaining an effective two-level system with Raman coupling between the levels. Optical Bloch equations are obtained for this effective two-level system, and phenomenological damping terms are added for collisional dephasing of the Raman transitions. The equations are solved for the steady-state as well as by the method of adiabatic following. From these solutions we obtain the polarization.
The population of a three-level system in the lambda configuration interacting with three classical fields is studied. The system is essentially a Raman coupled system with pump, Stokes and anti-Stokes fields. One channel between the two ground states involves the pump and Stokes field while the other involves the pump and anti-Stokes field. Since exact solutions are not generally available the author studies the dynamics numerically. But when the excited state is highly detuned, it may be adiabatically removed so that the system becomes effectively a two-level system but with an effective Rabi frequency that depends on the relative phases of the driving fields. For some choices of the phases this frequency can be made to vanish, an effect of the interference between the two channels. For the special case of equal and opposite detunings, an exact solution of the full three-level equations is presented. The effects of ionization of the excited state are also briefly considered.
A quantum theory of degenerate four-wave mixing is presented in which the atomic medium consists of stationary four-level atoms, each with three degenerate excited states, and the interacting light beams are allowed to be in different states of polarization. This vector-field theory differs from the scalar-field theory of Reid and Walls [Phys. Rev. A 31, 1622 (1985)] in that there are new atomic variables in the Langevin equations which are related to the induced coherence between the upper atomic states. It is found, for the assumed set of atomic levels, that this seemingly different mechanism of four-wave-mixing gain does not circumvent the degrading effect of spontaneous emission on squeezing obtainable via degenerate four-wave mixing. The theory is applied to both forward and backward degenerate four-wave mixing with nearly collinear geometry, and specialized to the case in which the polarization states of the two pump modes are mutually orthogonal. It is found that for both forward and backward configurations, the range of pump intensity for which squeezing can be achieved in the vector-field case is larger than that in the scalar-field case, and the maximum amount of squeezing obtainable at a particular pump detuning is comparable in both cases.
A general theory of transient stimulated Raman scattering and the phase-matching diagrams are developed based on an interaction Hamiltonian which includes all the possible competing interaction channels. In a Raman system pumped by multiple crossing beams with multiple Stokes and anti-Stokes components, these channels are defined by the interactions among fields at different frequencies and/or k vectors of the pump pairs and Stokes and anti-Stokes fields. An ensemble-averaged equation of motion for the active phonon mode is derived within the Markovian approximation in which the phonon-phonon coupling (pressure effect) is reduced to an energy relaxation and a phase fluctuation. A complete set of transient coupled equations including phase modulation, Stark shift, medium (anharmonic) saturation, and all the possible four-wave-mixing channels is presented. In the transient regime, it is found that the effects of the Stark shift on the Raman efficiency are mainly governed by the input pump energy in contrast to the steady-state theory where the pump intensity plays the main role. Both analytic and numerical methods are developed in describing the nonlinear effects due to the energy feedback mechanism in the strong pump-depletion regime. Twelve possible four-wave-mixing channels are analyzed for nonlinear phase diagrams characterized by the angles among pump pairs, and Stokes and anti-Stokes fields. Finally, numerical examples are shown for the effects of the coherence lengths of high-order Stokes fields and pump pairs on the generation of low-order Stokes fields.
We consider a two-channel \ensuremath{\Lambda}-type three-level atom interacting simultaneously with two quantized cavity fields and a classically described cw laser field, all of different frequencies. The effective interactions are of the Raman type, in which the two cavity fields are regarded as Stokes and anti-Stokes modes, and the classical field acts as the pump mode, so that both the pump-Stokes and the pump--anti-Stokes transitions are included. We present the analytic solution of the system evolution operator. We also study the time dependence of the atomic inversion and the statistical properties of the cavity fields. We show that strongly correlated photons can be generated in our model.
The quantum theory of Stokes--anti-Stokes processes in a degenerate system in a cavity is developed. The Raman transition is pumped by a coherent field. Effects of many atoms, cavity losses, and thermal photons are included in the theory. An exact solution for the model is obtained. An explicit expression for the reduced density matrix of the cavity field is given. The spectrum of the cavity field is obtained and is interpreted in terms of the eigenstates of the effective Hamiltonian describing Stokes--anti-Stokes processes. The spectrum develops sidebands due to strong Raman coupling.
Recently experiments on investigation of four wave mixing in sodium vapor cell have been performed by group of Dr P. Hemmer. It has been shown in these experiments that generation of a phase-conjugate wave takes place at large detuning of a pumping wave from the resonant transition in a double Lambda system of atomic levels only (so called case of Raman interaction). Moreover, the experiments demonstrated high efficiency of such Raman interaction for generation of the conjugate wave with low intensities of the exciting waves that indicates realization of new type of Raman laser. The aim of our work is theoretical analysis and numerical simulation of the experiment on the basis of exact solution of self-consistent system of Maxwell-Bloch equations. The atom-field interaction scheme is shown in the report. A peculiarity of this scheme is that the same pumping wave interacts with both transitions 1 ---> 3 and 2 ---> 4 simultaneously.