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Journal of Modern Optics

Non-diffusive Phase Dynamics from Linear Amplifiers and Attenuators in the Weak-field Regime

John A. Vaccaro, David Thomas Pegg

Published Jun 1, 1994• 2 citations
Publisher

Abstract

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.

DOI
10.1080/09500349414551031

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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.
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.
In this thesis we examine linear amplification of squeezed light, the phase properties of squeezed light and the effect on the phase of light from linear amplification. Previous studies of linear amplifiers of light have shown that (i) when amplifying squeezed light the power gain must be less than 2 to preserve some squeezing in the output light; and (ii) a squeezed output cannot be generated from a non-squeezed input. However, these studies were restricted to a class of linear amplifier that can now be identified as not being squeezed. In this thesis we show how to breach the gain limit with a squeezed linear amplifier, and how to generate squeezed light using a squeezed linear attenuator.
The properties of a linear optical amplifier or attenuator in which the input light is coupled to a collection of non-saturable atoms are considered. The photon-number factorial moments and probability distribution of the amplifier output are derived for arbitrary input statistics. Relations are obtained between the output and input variances in photon number, phase-angle cosine, and electric-field magnitude. The effects of amplification and attenuation on signal-to-noise ratio are obtained for both direct and homodyne detection. Particular attention is paid to the extents to which the non-classical properties of photon antibunching and squeezing are preserved by amplification and attenuation. It is found that both properties can at best survive only twofold intensity amplification.
The noise properties of the two main kinds of linear optical amplifier, phase-insensitive and phase-sensitive, are reviewed, with particular emphasis on their minimum-noise limits. An optical communications link is considered, in which identical sections of attenuating fiber alternate with identical linear amplifiers. The limits on the overall signal-to-noise ratios of such chains are derived. More attention is paid to the signal-to-noise ratio appropriate to phase-sensitive detection, but some results for direct detection are also given.
According to the fundamental laws of quantum optics, noise is necessarily added to the system when one tries to clone or amplify a quantum state. However, it has recently been shown that the quantum noise related to the operation of a linear phase-insensitive amplifier can be avoided when the requirement of a deterministic operation is relaxed. Nondeterministic noiseless linear amplifiers are therefore realizable. Usually nondeterministic amplifiers rely on using single photon sources. We have, in contrast, recently proposed an amplification scheme in which no external energy is added to the signal, but the energy required to amplify the signal originates from the stochastic fluctuations in the field itself. Applying our amplification scheme, we examine the amplifier gain and the success rate as well as the properties of the output states after successful and failed amplification processes. We also optimize the setup to find the maximum success rates in terms of the reflectivities of the beam splitters used in the setup. In addition, we discuss the nonidealities related to the operation of our setup and the relation of our setup with the previous setups.
According to the fundamental laws of quantum optics, noise is necessarily added to the system when one tries to clone or amplify a quantum state. However, it has recently been shown that the quantum noise related to the operation of a linear phase-insensitive amplifier can be avoided when the requirement of a deterministic op-eration is relaxed. Nondeterministic noiseless linear amplifiers are therefore realizable. Usually nondeterministic amplifiers rely on using single photon sources. We have, in contrast, recently proposed an amplification scheme in which no external energy is added to the signal, but the energy required to amplify the signal originates from the stochastic fluctuations in the field itself. Applying our amplification scheme, we examine the amplifier gain and the success rate as well as the properties of the output states after successful and failed amplification processes. We also optimize the setup to find the maximum success rates in terms of the reflectivities of the beam splitters used in the setup. In addition, we discuss the nonidealities related to the operation of our setup and the relation of our setup with the previous setups.