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Physical Review A

Phase-diffusion coefficients and phase-diffusion rate

Ning Lü

Published Nov 1, 1990• 9 citations
Publisher

Abstract

We point out that the phase-diffusion rate in an amplifier or attenuator is not proportional to the phase-diffusion coefficient in the Glauber P nor the antinormal-ordering Q distribution. It is only proportional to the phase-diffusion coefficient in the Wigner (W) distribution. We also find that, for a phase-insensitive oscillator, the steady-state phase-diffusion coefficients in the P, W, and Q distributions must be identical to each other.

DOI
10.1103/physreva.42.5641

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We point out that the phase-diffusion rate in an amplifier or attenuator is not proportional to the phase-diffusion coefficient in the Glauber P nor the antinormal-ordering Q distribution. It is only proportional to the phase-diffusion coefficient in the Wigner (W) distribution. We also find that, for a phase-insensitive oscillator, the steady-state phase-diffusion coefficients in the P, W, and Q distributions must be identical to each other.
We examine the effects of linear amplification and attenuation on the quantum-mechanical phase properties of light for fields with mean photon numbers of at least the order of 10. The phase probability density is found to satisfy a diffusion equation for both phase-insensitive and phase-sensitive amplifiers and attenuators. The solution is a convolution of the initial phase probability density with an infinite series of expanding Gaussians which clearly illustrates the diffusion of the phase. In particular, we find that for phase-insensitive amplification the phase of the field undergoes time-dependent uniform diffusion. In the limit of large amplification the diffusion ceases and the phase variance of the amplified light is given by the input phase variance plus an extra term which is equal to the phase variance of a coherent state of the same intensity as the initial field. We show that the reduced phase variance of phase-optimized states (relative to coherent states of the same intensity) is lost for power gains greater than the photon-cloning value of 2. In contrast, phase-sensitive amplifiers give rise to time-dependent nonuniform phase diffusion. The amount of phase diffusion depends on the relative phase angle between the light and the amplifier. If the peak of a relatively narrow phase probability density is near a minimum in the phase diffusion coefficient, then the phase noise added by the amplifier will be less than that found for a phase-insensitive amplifier. Further squeezing of the amplifier reduces the added phase noise proportionally. We find that it is possible, using phase-sensitive amplifiers, to amplify phase-optimized states by power gains considerably larger than 2 and still retain a reduced phase variance.
Previous studies of the phase properties of optical linear amplifiers and attenuators have been restricted to situations where the effect on the phase of single-mode light is found to be diffusive. In this paper we give an analysis of phase-sensitive linear amplification and attenuation in the weak-field regime that clearly reveals a non-diffusive effect on the phase. Indeed we show that a weak field which has initially a uniform phase probability density can inherit non-random phase properties from both phase-sensitive attenuators and amplifiers. In contrast, we find that phase-insensitive amplifiers and attenuators have merely a diffusive effect on the phase of the light.
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.
The phase of an optical field inside a linear amplifier is widely known to diffuse with a diffusion coefficient that is inversely proportional to the photon number. The same process occurs in lasers which limits its intrinsic linewidth and makes the phase uncertainty difficult to calculate. The most commonly used simplification is to assume a narrow photon-number distribution for the optical field (which we call the small-noise approximation). For coherent light, this condition is determined by the average photon number. The small-noise approximation relies on (i) the input to have a good signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be maintained throughout the amplification process. Here we ask: For a coherent input, how many photons must be present in the input to a quantum linear amplifier for the phase noise at the output to be amenable to a small-noise analysis? We address these questions by showing how the phase uncertainty can be obtained without recourse to the small-noise approximation. It is shown that for an ideal linear amplifier (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down with only a few photons on average. Interestingly, when the input strength is increased to tens of photons, the small-noise approximation can be seen to perform much better and the process of phase diffusion permits a small-noise analysis. This demarcates the limit of the small-noise assumption in linear amplifiers as such an assumption is less true for a nonideal amplifier.