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Physical Review A

Unified approach to multiphoton lasers and multiphoton bistability

M. D. Reid, K. J. McNeil, Daniel F. Walls

Published Oct 1, 1981• 58 citations
Publisher

Abstract

This paper presents a unified theory of nonequilibrium transitions which occur in radiation interacting via an $n$-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 $n$-photon laser, the $n$-photon laser with injected signal, and $n$-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.

DOI
10.1103/physreva.24.2029

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Acta Physica Sinica•Jan 1, 1988•1 citation
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.
A two-photon laser is modeled as NNN two-level atoms interacting via a two-photon transition with a single resonant-cavity field mode. The quadratic form of this interaction makes such a system a candidate to exhibit reduced quantum fluctuations (squeezing) in one quadrature of the output field. A Fokker-Planck equation containing all leading quantum noise terms is derived following the method of Haken. A linearized fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing may be present in two-photon optical bistability. Our calculations confirm the results of Lugiato and Strini, who adopted a factorization of moment equations.
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.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
A fully quantum-mechanical theory of bistable inversionless lasers is developed by generalizing the Scully-Lamb laser theory to this kind of laser systems. By converting the quantum master equation of the system into a Fokker-Planck equation, we obtain the steady-state properties and the fluctuation properties of the laser field for both coherently and incoherently injected atomic systems. It is shown that both the intensity and phase fluctuations of the laser system with coherent atomic injection are reduced corresponding to their counterpart of the incoherent atomic injection. The phase noise of bistable inversionless lasers can be quenched down to the vacuum-noise level for both atomic injections.
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.