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Physical Review A

Comparative study of various quasiprobability distributions in different models of correlated-emission lasers

Ning Lu, Shi-Yao Zhu, G. S. Agarwal

Published Jul 1, 1989• 32 citations
Publisher

Abstract

We make a comparative study of various quasiprobability distributions in phase-sensitive quantum-optical systems. Starting from a general, linear master equation for the field, which emerges in different models of correlated-emission lasers, we derive the Fokker-Planck equations in the Glauber-Sudarshan P, the antinormal ordering Q, the Wigner W, the complex P, and the positive P representations and find the steady-state solutions for the five distributions. Simple relations between the complex and positive P functions are discovered for the first time. Various moments calculated by using these distributions are found to be identical, as expected. An application of these distributions to the two-photon correlated-emission laser shows that the intracavity field can be near-perfectly squeezed in the phase quadrature and the maximum quadrature squeezing is reached when the mean laser amplitude vanishes.

DOI
10.1103/physreva.40.258

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We make a comparative study of various quasiprobability distributions in phase-sensitive quantum-optical systems. Starting from a general, linear master equation for the field, which emerges in different models of correlated-emission lasers, we derive the Fokker-Planck equations in the Glauber-Sudarshan P, the antinormal ordering Q, the Wigner W, the complex P, and the positive P representations and find the steady-state solutions for the five distributions. Simple relations between the complex and positive P functions are discovered for the first time. Various moments calculated by using these distributions are found to be identical, as expected. An application of these distributions to the two-photon correlated-emission laser shows that the intracavity field can be near-perfectly squeezed in the phase quadrature and the maximum quadrature squeezing is reached when the mean laser amplitude vanishes.
The linear theory of the correlated-emission laser (CEL) is studied using the quantum-mechanical definition of the phase of the electromagnetic field recently introduced by Barnett and Pegg [J. Mod. Opt. 36, 17 (1989); Phys. Rev. A 39, 1665 (1989)]. From the master equation for the density matrix of the field, a Fokker-Planck equation for the phase probability function is derived, and the phase properties of the CEL discussed. The Adler equation and the equation of motion for the variance of the phase difference are derived. The Fokker-Planck equation satisfied by the phase probability function is compared with the corresponding equations for the P, Q, and Wigner quasiprobability distributions, specialized to the case when amplitude fluctuations are negligible. We show that the phase probability function and the Wigner function satisfy the same equation, and conclude that, of the three quasiprobability functions studied, only the Wigner function is reliable for calculating directly the phase properties. The commonly used P function overestimates the laser linewidth by a factor of 2.
The Lindblad master equation for the damped quantum harmonic oscillator is transformed into Fokker–Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal-ordering Q representation and the Wigner W representation. It will be proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker–Planck equations in the steady state, it will be shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients.
We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit. As a by-product we also study the ordinary two-photon laser and find, e.g., photon-number variance and laser linewidth.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
We show that the most complete and accurate description of fluctuations in quantum-optical systems is that obtained using the generalized Wigner distribution for the macroscopic observables of the system. The smallness of the inverse of the saturation photon number entitles us to neglect the terms with derivatives of order higher than second order in the time-evolution equation for the quasiprobability distribution. This treatment allows us to also describe correctly nonclassical effects such as photon antibunching or "squeezing," which are maltreated or even destroyed if one neglects the atom-atom correlations. We derive and analyze the Fokker-Planck equation for the Wigner function in the case of the usual laser and compare the results with those following from other approaches. In the region very high above threshold we find a small antibunching effect, which was not discovered in previous treatments due to the neglect of atom-atom correlations.