Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Modern Optics

Effect of losses on phase noise

Brian Y. Han, Mark S. Hillery, Adil Benmoussa, Christopher C. Gerry

Published Jan 1, 2003• 0 citations
Publisher

Abstract

We study the effect of losses on the phase noise of single-mode field states. The losses are described by the standard loss master equation, and it is used to find an upper bound for the increase in the phase noise as a function of time. We compare the time dependence of the phase noise of an initial coherent state to that of a state that initially has very small phase noise. Both states have the same initial mean photon number. While the small-phase noise state is more susceptible to losses, the difference between its behaviour and that of the coherent state is not great.

DOI
10.1080/0950034021000013246

Related papers

Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum

Takafumi Ono, Holger Friedrich Hofmann

Two-path interferometry with coherent states and a squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information $F$ of the state. In the limit of high squeezing, the ratio $(F--N)/{N}^{2}$ of Fisher information above shot noise to the square of the average photon number $N$ depends only on the average number of photons lost, ${n}_{\mathrm{loss}}$, and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by $(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})$. The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.

arXiv:0910.3727•Physical Review A•Mar 11, 2010•102 citations

Lower bound on phase noise of classical states

Mark S. Hillery

An uncertainty relation for the number and phase of a single-mode field state is derived. It is then used to find a lower bound on the phase noise of a classical state. Any state that violates this condition is nonclassical. An example of such a nonclassical state is presented. Because a nonclassical state can have less phase noise than a classical state with the same average photon number, nonclassical states can play a role in the measurement of small phase shifts.

arXiv:2309.08450•Sep 15, 2023•0 citations

Phase properties of optical linear amplifiers

John A. Vaccaro, David Thomas Pegg

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.

Physical Review A•Jun 1, 1994•18 citations

Quantum Phase Diffusion in Two-Mode Bose-Einstein Condensates with Particle Losses

Jianning Liu, Jiahao Xu, Kejela Wodajo Ebawak +3 more

We study the effect of particle losses on dynamical phase diffusion in a two-mode Bose-Einstein condensate.Starting with an equal-population coherent state, the single-particle coherence is solved analytically by the Monte Carlo wave function method.We show that decoherence slows down the loss of single-particle coherence.Moreover, we find that the bigger phase diffusion is, the more sensitive to the noise it becomes.Finally, we discuss the effect of the losses on the spin squeezing and show that the squeezing angle is robust to the decoherence even though the value of the squeezing is greatly affected.

Acta Physica Polonica A•Oct 1, 2022•0 citations

Difference-phase squeezing from amplitude squeezing by means of a beamsplitter

Mark S. Hillery, Mingliang Zou, Vladimír Bužek

A classical analysis of two equal intensity coherent fields incident on a beamsplitter shows that the difference-phase noise of the output depends only on the noise in the difference of the amplitudes of the input fields and not on their phase noise. This suggests that in the quantum mechanical case squeezing in the amplitudes of the input beams can lead to squeezing in the phase difference of the output beams. We show that this is true. We also find the phase properties of the output when the input consists of two number states with an equal number of photons. The difference phase distribution consists of two narrow peaks, at and . States with small phase difference noise should be useful in the measurement of phase shifts.

Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Oct 1, 1996•20 citations

Nonlinear phase measurement using a coherent state in noisy environments

Bao-Hua Yang, Huajie Zou, Weijie Gu

Nonlinear phase measurement plays an important role in many dynamic processes. In this paper, we propose a single-mode measurement scheme for second-order nonlinear phase shifts that utilizes coherent states and homodyne measurement. The sensitivity at operating point φ = 0 is 1 / 4 N 3 / 2 with N photons on average and approaches the quantum Cramér–Rao bound. In addition, sensitivity with a small value of φ is discussed. For practical purposes, we analyze the effects of photon losses, thermal photon noise, and phase diffusion on sensitivities with φ = 0 as well as a small value of φ . The sensitivity still scales as 1 / N 3 / 2 in the presence of photon losses and thermal photon noise but degrades to the scaling of 1 / N with phase diffusion.

Journal of the Optical Society of America B•Nov 18, 2021•0 citations
We study the effect of losses on the phase noise of single-mode field states. The losses are described by the standard loss master equation, and it is used to find an upper bound for the increase in the phase noise as a function of time. We compare the time dependence of the phase noise of an initial coherent state to that of a state that initially has very small phase noise. Both states have the same initial mean photon number. While the small-phase noise state is more susceptible to losses, the difference between its behaviour and that of the coherent state is not great.
Two-path interferometry with coherent states and a squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information FFF of the state. In the limit of high squeezing, the ratio (F−−N)/N2(F--N)/{N}^{2}(F−−N)/N2 of Fisher information above shot noise to the square of the average photon number NNN depends only on the average number of photons lost, nloss{n}_{\mathrm{loss}}nloss​, and the fraction of squeezed vacuum photons $\ensuremath{\mu}$. For $\ensuremath{\mu}=1/2$, the effect of losses is given by (F−−N)/N2=1/(1+2nloss)(F--N)/{N}^{2}=1/(1+2{n}_{\mathrm{loss}})(F−−N)/N2=1/(1+2nloss​). The possibility of increasing the robustness against losses by lowering the squeezing fraction $\ensuremath{\mu}$ is considered, and an optimized result is derived. However, the improvements are rather small, with a maximal improvement by a factor of 2 at high losses.
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.
We study the effect of particle losses on dynamical phase diffusion in a two-mode Bose-Einstein condensate.Starting with an equal-population coherent state, the single-particle coherence is solved analytically by the Monte Carlo wave function method.We show that decoherence slows down the loss of single-particle coherence.Moreover, we find that the bigger phase diffusion is, the more sensitive to the noise it becomes.Finally, we discuss the effect of the losses on the spin squeezing and show that the squeezing angle is robust to the decoherence even though the value of the squeezing is greatly affected.
A classical analysis of two equal intensity coherent fields incident on a beamsplitter shows that the difference-phase noise of the output depends only on the noise in the difference of the amplitudes of the input fields and not on their phase noise. This suggests that in the quantum mechanical case squeezing in the amplitudes of the input beams can lead to squeezing in the phase difference of the output beams. We show that this is true. We also find the phase properties of the output when the input consists of two number states with an equal number of photons. The difference phase distribution consists of two narrow peaks, at and . States with small phase difference noise should be useful in the measurement of phase shifts.
Nonlinear phase measurement plays an important role in many dynamic processes. In this paper, we propose a single-mode measurement scheme for second-order nonlinear phase shifts that utilizes coherent states and homodyne measurement. The sensitivity at operating point φ = 0 is 1 / 4 N 3 / 2 with N photons on average and approaches the quantum Cramér–Rao bound. In addition, sensitivity with a small value of φ is discussed. For practical purposes, we analyze the effects of photon losses, thermal photon noise, and phase diffusion on sensitivities with φ = 0 as well as a small value of φ . The sensitivity still scales as 1 / N 3 / 2 in the presence of photon losses and thermal photon noise but degrades to the scaling of 1 / N with phase diffusion.