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Physical Review A

Nonlinear theory of a correlated emission laser

Khalid Zaheer, Muhammad Suhail Zubairy

Published Nov 1, 1988• 32 citations
Publisher

Abstract

A nonlinear theory of correlated emission lasers is presented. It is shown that if three-level atoms in the V configuration are injected in a coherent superposition of the upper two states in a doubly resonant cavity, the diffusion coefficient for the relative phase vanishes under certain conditions.
DOI
10.1103/physreva.38.5227

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Physical Review A•May 1, 1989•69 citations
We develop a quantum theory of a correlated-emission laser (CEL) with injected atomic coherence for arbitrary initial atomic conditions and arbitrary atomic decay rates. The effect of initial atomic coherence on the CEL is studied. It is shown that, with a proper form of the initial atomic coherence and equal atomic decay rates between the two upper levels of V-type three-level lasing atoms, the diffusion coefficients for the relative phase and the relative amplitude of two laser modes can vanish under certain conditions. These conditions, plus equal cavity-loss rates, lead to the vanishing of the normally ordered variances (quantum-noise quenching) of Hermitian operators corresponding to the relative phase and the relative amplitude.
We present a comprehensive quantum theory of correlated-emission lasers (CEL's) in a combination-mode approach, which is valid for both the quantum-beat laser and the CEL with injected atomic coherence. We first show that, under certain conditions, the linear master equation of the CEL's breaks down into two uncoupled equations in the combination modes, which are linear combinations of two CEL modes. One combination mode that is above threshold represents the average phase and the average amplitude of two CEL modes (called the sum mode), whereas the other that is below threshold represents the relative phase and the relative amplitude of the two CEL modes (called the relative mode). When the relative mode is in its vacuum state, the normally ordered variances of two Hermitian operators corresponding, respectively, to the relative phase and the relative amplitude vanish. Through deriving a nonlinear master equation for the CEL with injected atomic coherence in the combination modes, we show how a set of proper initial atomic conditions can keep the relative mode in its vacuum state. We obtain the Glauber P functions, the mean photon numbers, and the photon-number variances in both the combination modes and the CEL modes, and find the mode pullings of the CEL's and the natural linewidth of the sum mode. We also show a correspondence between the vacuum state of the relative mode and the vanishing of the relative-phase diffusion coefficient.
Quantum mechanical theory of a non-degenerate correlated emission laser in a four-level atomic system in the double-Λ configuration is presented. It is shown that the mode coupling gains in this model have a phase-sensitive nature so that the spontaneous emission events in the two modes are highly correlated in such a way that the relative phase diffusion coefficient can vanish under certain conditions. Also, because of mode coupling and for a certain range of atomic population in the upper atomic levels, lasing without inversion and correlated emission take place simultaneously. Stability conditions and the steady-state modal ratio are derived and it is shown that this ratio remains stable and for this stationary value of the modal ratio both the modes amplify with equal gains.
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.