© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical Review A

Phase properties of optical linear amplifiers

John A. Vaccaro, David Thomas Pegg

Published Jun 1, 1994• 18 citations

Abstract

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.

DOI: 10.1103/physreva.49.4985

Related papers

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

John A. Vaccaro, David Thomas Pegg

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.

Journal of Modern Optics•Jun 1, 1994•2 citations

Properties of the Optical Quantum Amplifier

R. Loudon, Terence J. Shepherd

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.

Optica Acta International Journal of Optics•Nov 1, 1984•105 citations

Reduction of quantum fluctuation and suppression of the Gordon–Haus effect with phase-sensitive linear amplifiers

Horace P. Yuen

Compared with the use of phase-insensitive linear quantum amplifiers of the same gain G, the use of phase-sensitive linear amplifiers on phase coherent classical light sources in an amplifier-attenuator chain reduces the total quantum noise power by a factor of 4, the homodyne noise variance by 2, and the photon number variance by 2-8 and suppresses the Kerr effect linear phase fluctuation variance as well as the soliton timing-error variance by 2G(2).

Optics Letters•Jan 1, 1992•88 citations

The small-noise approximation in linear amplifiers: Limitations and beyond

A. Chia, Michal Hajdušek, Rosario Fazio +2 more

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). This 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: How many photons must be present in the input to a quantum linear amplifier for the phase noise to be amenable to a small-noise analysis? We show that for a linear amplifier which is signal-to-noise optimised (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down for a coherent-state input 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 an amplifier which does not have its signal-to-noise performance optimised. In order to explore this limit we propose a description of phase diffusion that goes beyond the small-noise approximation which we call the inverse-number expansion. The small-noise approximation can then be understood as the lowest-order contribution to the inverse-number expansion while phase diffusion with only a few-photon input can be captured by the inclusion of higher-order terms.

•Nov 20, 2018•0 citations

Phase diffusion and the small-noise approximation in linear amplifiers: Limitations and beyond

A. Chia, Michal Hajdušek, Rosario Fazio +2 more

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.

arXiv:1811.08105•Quantum•Nov 4, 2019•8 citations

Linear amplification, squeezed states and phase properties of light

John Alfio Vaccaro

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.

•Jan 1, 1990•0 citations

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 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 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). This 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: How many photons must be present in the input to a quantum linear amplifier for the phase noise to be amenable to a small-noise analysis? We show that for a linear amplifier which is signal-to-noise optimised (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down for a coherent-state input 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 an amplifier which does not have its signal-to-noise performance optimised. In order to explore this limit we propose a description of phase diffusion that goes beyond the small-noise approximation which we call the inverse-number expansion. The small-noise approximation can then be understood as the lowest-order contribution to the inverse-number expansion while phase diffusion with only a few-photon input can be captured by the inclusion of higher-order terms.

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.

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.