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Chinese Journal of Quantum Electronics

The Solution of Full-Quantum Equation on Two-Photon Optical Bistability and Two-Photon Laser

Z Zhang

Published Jan 1, 1984• 0 citations
OpenAlex

Abstract

In this paper the two-photon optical bistability and the two-photon laser are treated unifiedly by Haken's full-quantum theory. The Langevin equation and the Fokker-Planck equation of the light field are derived by eliminating the atom variables adiabatioally. The stationary probability distribution of the light field is obtained under the detailed balance condition. It is shown that the two stationary states of the transmitted light field for a given value of the external parameter (inoident light fiield or pumping parameter) are not equally stable. In fact, the system can exist in metastable states and it is this that leads to a hysteresis.

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In this paper the two-photon optical bistability and the two-photon laser are treated unifiedly by Haken's full-quantum theory. The Langevin equation and the Fokker-Planck equation of the light field are derived by eliminating the atom variables adiabatioally. The stationary probability distribution of the light field is obtained under the detailed balance condition. It is shown that the two stationary states of the transmitted light field for a given value of the external parameter (inoident light fiield or pumping parameter) are not equally stable. In fact, the system can exist in metastable states and it is this that leads to a hysteresis.
This paper presents a unified theory of nonequilibrium transitions which occur in radiation interacting via an nnn-photon transition with atomic systems inside optical cavities. Incoherent pumping of the atoms and coherent driving of the cavity are included. Examples of such systems are the nnn-photon laser, the nnn-photon laser with injected signal, and nnn-photon optical bistability. The state equation and stability conditions for these phenomena are derived. Fluctuations are included via a Fokker-Planck equation in the constant-diffusion approximaton for which potential solutions may be obtained in the steady state. When these solutions are used, moments of the photon distribution can be calculated for the above systems in the specific cases of one- and two-photon transitions.
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A quantum treatment of a coherently driven dispersive cavity is given based on a cubic nonlinearity in the polarisability of the internal medium. This system displays bistability and hysteresis in the semiclassical solutions. Quantum fluctuations are included via a Fokker-Planck equation in a generalised P representation. The transmitted light shows a transition from a single-peaked spectrum to a double-peaked spectrum above the threshold of the lower branch. Fluctuations in the field are reduced on the upper branch and both photon bunching and photon antibunching are predicted, for different operating points. An exact solution obtained for the steady-state generalised P function shows decidedly non-equilibrium behaviour, e.g. the lack of a Maxwell construction.
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The systematic method of Fokker-Planck equation (FPE) for the generalized Wigner distribution is used to study the multi-photon transition process of a driven optical system. The general multi-photon FPE, the state equation and the spectrum of transmitted light in good cavity limit are obtained- The two-photon optical bistability spectrum of transmitted light is calculated and compared with single-photon case. It is shown that for multi-photon case, the atom -atom correlation is also negligible in the good cavity limit.