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Physical Review A

Quantum theory of correlated-emission lasers: Vacuum state for the mode of the relative phase and the relative amplitude

Ning Lü

Published Jun 1, 1992• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.45.8154

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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.
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 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.
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.
As discussed in the last three chapters, the fundamental source of noise in a laser is spontaneous emission. A simple pictorial model for the origin of the laser linewidth envisions it as being due to the random phase diffusion process arising from the addition of spontaneously emitted photons with random phases to the laser field. In this chapter we show that the quantum noise leading to the laser linewidth can be suppressed below the standard, i.e., Schawlow–Townes limit by preparing the atomic systems in a coherent superposition of states as in the Hanle effect and quantum beat experiments discussed in Chapter 7. In such coherently prepared atoms the spontaneous emission is said to be correlated. Lasers operating via such a phase coherent atomic ensemble are known as correlated spontaneous emission lasers (CEL). An interesting aspect of the CEL is that it is possible to eliminate the spontaneous emission quantum noise in the relative linewidths by correlating the two spontaneous emission noise events. A number of schemes exist in which quantum noise quenching below the standard limit can be achieved. In two-mode schemes a correlation between the spontaneous emisson events in two different modes of the radiation field is established via atomic coherence so that the relative phase between them does not diffuse or fluctuate. In a Hanle laser and a quantum beat laser this is achieved by pumping the atoms coherently such that every spontaneously emitting atom contributes equally to the two modes of the radiation, leading to a reduction and even vanishing of the noise in the phase difference.