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Physical Review A

Semiclassical stochastic description of the two-photon laser

Adi R. Bulsara, William C. Schieve

Published May 1, 1979• 28 citations
Publisher

Abstract

We consider the problem of two-photon transitions between atomic levels having the same parity. The semiclassical theory is developed from Maxwell's equations and the resulting transcendental equations for the pulse envelope solved exactly. They lead to an unstable pulse envelope if the higher-order processes are included. We then treat the laser as a birth/death process, deriving a Markovian (macroscopic) master equation for the probability function, which may be solved exactly in the steady state. By a suitable truncation procedure, this equation predicts a stable steady-state envelope, and is consistent with the microscopic quantum theory when virtual processes are neglected. A comparison is made between the macroscopic (master equation) approach and the microscopic quantum theory at the level of the moment equations. We find that, in the high-gain limit, the fluctuations predicted by the two theories are the same.

DOI
10.1103/physreva.19.2046

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Physical Review A•Jan 1, 1980•38 citations
We consider the problem of two-photon transitions between atomic levels having the same parity. The semiclassical theory is developed from Maxwell's equations and the resulting transcendental equations for the pulse envelope solved exactly. They lead to an unstable pulse envelope if the higher-order processes are included. We then treat the laser as a birth/death process, deriving a Markovian (macroscopic) master equation for the probability function, which may be solved exactly in the steady state. By a suitable truncation procedure, this equation predicts a stable steady-state envelope, and is consistent with the microscopic quantum theory when virtual processes are neglected. A comparison is made between the macroscopic (master equation) approach and the microscopic quantum theory at the level of the moment equations. We find that, in the high-gain limit, the fluctuations predicted by the two theories are the same.
The analogy of a laser with an autocatalytic chemical reaction is used to write down a macroscopic stochastic master equation of the birth-and-death type for the photon probability function in a semiclassical laser formalism. This equation may be solved exactly in the steady state using a generating function. This leads to a prescription for calculating all the higher-order moments of the photon number as well as providing a unique truncation of the hierarchy of moment equations at any given order. Further, the variance is given by a Poisson distribution, in agreement with the strong-signal Scully-Lamb theory. This master equation is also shown to lead to an exact Fokker-Planck equation by using a modified Kramers-Moyal expansion. A comparison between our results and the (microscopic) Scully-Lamb theory shows that the two approaches give the same results at large photon number.
We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model. Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P(n), the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of (n) and log[(n2)/(n)2] versus the pumping parameter show close agreement with the microscopic results of the Casa-grande-Lugiato model. A similar discussion of the fluctuations can also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.
We consider a gas of homogeneously broadened two-level atoms in a Fabry-Perot cavity injected with a coherent resonant electric field. Adopting a macroscopic birth-death description, we construct a stochastic master equation for the output intensity distribution. From this starting point we find the steady-state distribution and show it to be bimodal in the region of the optical bistability. We compare our results for the mean and variance with the microscopic results of Bonifacio, Gronchi, and Lugiato as a test of the stochastic-master-equation approach.
We discuss the propagation of an electromagnetic pulse through an active medium prepared in a state of inversion between two levels of the same parity. Since no electric dipole transition is possible between the chosen atomic levels, we investigate the possibility of amplification of an injected signal having a carrier frequency equal to one-half the atomic-transition frequency. We show that under suitable conditions a nonlinear atomic polarization can be generated which oscillates at the same frequency as the incident electromagnetic pulse. The coupled atom-field evolution is described by the usual self-consistent approach. When atomic relaxation effects are negligible, we derive an equation describing the spatial evolution of the energy of the propagating pulse. From this equation we characterize the threshold condition for power amplification and classify the multiple steady-state solutions of the propagation problem. The evolution of the pulse envelope through the amplifier is analyzed with the help of a hybrid computer simulation. Pulse-envelope modulation and multiple-pulse formation even in the asymptotic limit of long amplifiers are displayed.
We present the theory of a microscopic maser operating on a nondegenerate two-photon transition between two states of same parity. We start from a three-level Hamiltonian, and discuss both the one-photon resonant cascade and the two-photon regime, when the intermediate state is far off resonance. In the semiclassical limit, we derive rate equations for the field intensities and get the corresponding steady-state values. The vacuum is stable in the two-photon regime but becomes unstable for zero detuning (a clear signature of a one-photon process). Solutions displaying true two-photon operation (intermediate state unpopulated) even with zero detuning are displayed. Using the full quantum density-matrix approach, we derive a master equation for the field and discuss both operating regimes, showing that the spontaneous emission may turn the vacuum unstable in the two-photon case. In the off-resonance two-photon regime, there is a strong correlation between the fluctuations of the intensities of the two modes. We find a squeezing factor of 50% for the fluctuations in the difference of intensities.
The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.