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Physical Review Letters

Steady-state solutions to the master equation for polarization-dependent two-photon absorption

Daniel L. Schauer

Published May 3, 1993• 7 citations
Publisher

Abstract

Steady-state solutions of double-beam polarization-dependent two-photon absorption (PDTPA) are examined. A new basis is used to solve the master equation for PDTPA. It is found that these steady-state solutions are composed of nontrivial states of the two initial beams. This is in disagreement with the predictions of two-photon absorption ignoring the vector nature of the electromagnetic field.

DOI
10.1103/physrevlett.70.2710

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Steady-state solutions of double-beam polarization-dependent two-photon absorption (PDTPA) are examined. A new basis is used to solve the master equation for PDTPA. It is found that these steady-state solutions are composed of nontrivial states of the two initial beams. This is in disagreement with the predictions of two-photon absorption ignoring the vector nature of the electromagnetic field.
The physics of the two-photon absorption process is investigated for a three-state system. The density-matrix equations for the two-photon interaction are solved in the steady-state limit assuming that the pump laser radiation is monochromatic. Collisional broadening, saturation, and Stark shifting of the two-photon resonance are investigated in detail by numerical solution of the steady-state density-matrix equations. Analytical expressions for the saturation intensity and the Stark shift are derived for the case where the single-photon transitions between the intermediate state and the initial and final states are far from resonance with the pump laser. For this case, it is found that the direction of the Stark shift is dependent on the relative magnitudes of the dipole-moment matrix elements for the single-photon transitions that couple the intermediate state with the initial and final states. Saturation and Stark shifting are also investigated for the case where the single-photon transitions between the intermediate state and the initial and final states are close to resonance with the pump laser.
The fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function p(n,m)p(n,m)p(n,m) to find nnn photons in the light field and mmm atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.
Starting from the multi-level master equation a non-Markoffian rate equation is derived. Explicit expressions for the transition rates between each pair of atomic levels in this equation are derived at it is shown that these may be interpreted as the sum of one-photon, two-photon,..., etc. transition rates connecting the given pair of levels. The non-Markoffian master equation is exact and so can describe long- or short-time behaviour under monochromatic or broad-band excitation. The exact steady-state solutions are obtained by solving a Markoffian rate equation.
Transmission characteristics of a nonlinear Fabry-Perot interferometer with two independent optical beams are considered under steady-state conditions. The sum frequency of two beams is taken to be near a two-photon resonance of the nonlinear medium. Two beams are coupled through the intensity-dependent susceptibility and influence propagation of each other in a mutual way. A set of two coupled-state equations is derived and it incorporates saturated absorption and dispersion as well as the effect of optical Stark shift. The system is found to have as many as seven stationary states, and at most, three are stable among them. The degenerate case of a single beam in two-photon resonance with the medium is also considered with particular emphasis on the effect of the optical Stark shift on optical bistability.