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

A Gaussian measure of quantum phase noise

Wolfgang P. Schleich, Jonathan P. Dowling

Published Feb 1, 1992• 0 citations
OpenAlex

Abstract

We study the width of the semiclassical phase distribution of a quantum state in its dependence on the average number of photons (m) in this state. As a measure of phase noise, we choose the width, delta phi, of the best Gaussian approximation to the dominant peak of this probability curve. For a coherent state, this width decreases with the square root of (m), whereas for a truncated phase state it decreases linearly with increasing (m). For an optimal phase state, delta phi decreases exponentially but so does the area caught underneath the peak: all the probability is stored in the broad wings of the distribution.

Related papers

Exponential decrease in phase uncertainty

Wolfgang P. Schleich, Jonathan P. Dowling, R. J. Horowicz

The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty \ensuremath{\delta}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width \ensuremath{\Delta}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.

Physical Review A•Sep 1, 1991•31 citations

Realistic quantum states of light with minimum phase uncertainty

A. Bandilla, Henryk Paul, H -H Ritze

Starting from a convenient measure for the phase uncertainty delta phi , a rigorous solution to the variational problem in which ( delta phi )2 is minimized under the constraint that the mean photon number N is held fixed, was found. In an approximation valid for large values of N, closed-form expressions for both ( delta phi )2 and the variance of the photon number ( Delta n)2 were obtained. In the final result, the approximation is very similar to an approximation scheme recently discussed by Summy and Pegg Opt. Commun., vol.77, p.75, 1990. The phase uncertainty for the phase optimized states is found to decrease as (N+1)-1 with growing N.

Quantum Optics Journal of the European Optical Society Part B•Oct 1, 1991•57 citations

Phase uncertainties of a squeezed state

Matthias Freyberger, W. P. Schleich

We investigate various measures of phase uncertainty in their dependence on the average number of photons for the case of a phase squeezed state. We use the following three measures: (i) the geometrical uncertainty (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{g}}$${)}^{2}$, (ii) the dispersion (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{d}}$${)}^{2}$, and (iii) the rotational width (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{r}}$${)}^{2}$. Whereas (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{g}}$${)}^{2}$ and (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{r}}$${)}^{2}$ decay quadratically with the average number of photons 〈n〉, the dependence of (\ensuremath{\delta}${\mathit{cphi}}_{\mathit{d}}$${)}^{2}$ is only of the form ln〈n〉/〈n${\mathrm{〉}}^{2}$. We give physical pictures for these scaling laws.

Physical Review A•Jun 1, 1994•27 citations

Ultimate quantum limits on phase measurement

Scott Roger Shepard, Jeffrey H. Shapiro

In a shot-noise-limited (coherent state) phasesensing interferometer, the root-mean-square (rms) phase error is ~ 1 / N av 1 / 2 where Nav is the average number of detected photons. In an optimized squeezed-state interferometer, the rms phase error goes as 1/Nav. We have considered the phase measurement problem from the viewpoint of quantum estimation theory.1 In particular, for the problem of optimum phase-shift estimation from observation of a single mode of the radiation field under an average photon number constraint, we have found both the optimum quantum state and the optimum quantum measurement. The former turns out to have a truncated number-state expansion whose coefficients, out to the truncation point, fall off as 1/n. The latter turns out to be the Susskind-Glogower phase operator. More important, the resulting phase estimation performance improves as 1 / N av 2 .

Annual Meeting Optical Society of America•Jan 1, 1988•5 citations

Quantum Phase Density Distribution of Amplitude Coherent States

Ying Li

Pegg-Barnett Quantum phase density distribution properties of amplitude coherent states are studied.It is find that:1.the Quantum phase density distribution of amplitude coherent states p(θ,m) is influenced by superposition angle φ which is the vacuum state |o〉superposed by coherent state|α〉,average photon number and θ-γ that is the quantum phase θ minus displace angle of coherent state γ.2.the quantum phase density distribution changes with φ and θ-γ periodically;it becomes large with increase of average photon numbers.Under certain conditions,extremums of quantum phase density distribution are restricted by φ and θ-γ.

Journal of Shangluo University•Jan 1, 2008•0 citations

Quantum phase measurements with infinite peak-likelihood and zero phase information

Z. Hradil, Jeffrey H. Shapiro

Maximum likelihood estimation of an unknown c-number phaseshift is considered, along the lines established by Shapiro, Shepard and Wong, Phys. Rev. Lett., vol.62, p.2377, 1989. An input state is exhibited whose reciprocal peak likelihood vanishes, delta phi to 0, while its phase-measurement probability density converges, in distribution, to uniform, p( phi mod Phi ) to d 1/2 pi .

Quantum Optics Journal of the European Optical Society Part B•Feb 1, 1992•22 citations
We study the width of the semiclassical phase distribution of a quantum state in its dependence on the average number of photons (m) in this state. As a measure of phase noise, we choose the width, delta phi, of the best Gaussian approximation to the dominant peak of this probability curve. For a coherent state, this width decreases with the square root of (m), whereas for a truncated phase state it decreases linearly with increasing (m). For an optimal phase state, delta phi decreases exponentially but so does the area caught underneath the peak: all the probability is stored in the broad wings of the distribution.
The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty \ensuremath{\delta}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width \ensuremath{\Delta}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.
Starting from a convenient measure for the phase uncertainty delta phi , a rigorous solution to the variational problem in which ( delta phi )2 is minimized under the constraint that the mean photon number N is held fixed, was found. In an approximation valid for large values of N, closed-form expressions for both ( delta phi )2 and the variance of the photon number ( Delta n)2 were obtained. In the final result, the approximation is very similar to an approximation scheme recently discussed by Summy and Pegg Opt. Commun., vol.77, p.75, 1990. The phase uncertainty for the phase optimized states is found to decrease as (N+1)-1 with growing N.
We investigate various measures of phase uncertainty in their dependence on the average number of photons for the case of a phase squeezed state. We use the following three measures: (i) the geometrical uncertainty (\ensuremath{\delta}cphig{\mathit{cphi}}_{\mathit{g}}cphig​)2{)}^{2})2, (ii) the dispersion (\ensuremath{\delta}cphid{\mathit{cphi}}_{\mathit{d}}cphid​)2{)}^{2})2, and (iii) the rotational width (\ensuremath{\delta}cphir{\mathit{cphi}}_{\mathit{r}}cphir​)2{)}^{2})2. Whereas (\ensuremath{\delta}cphig{\mathit{cphi}}_{\mathit{g}}cphig​)2{)}^{2})2 and (\ensuremath{\delta}cphir{\mathit{cphi}}_{\mathit{r}}cphir​)2{)}^{2})2 decay quadratically with the average number of photons 〈n〉, the dependence of (\ensuremath{\delta}cphid{\mathit{cphi}}_{\mathit{d}}cphid​)2{)}^{2})2 is only of the form ln〈n〉/〈n〉2{\mathrm{〉}}^{2}〉2. We give physical pictures for these scaling laws.
In a shot-noise-limited (coherent state) phasesensing interferometer, the root-mean-square (rms) phase error is ~ 1 / N av 1 / 2 where Nav is the average number of detected photons. In an optimized squeezed-state interferometer, the rms phase error goes as 1/Nav. We have considered the phase measurement problem from the viewpoint of quantum estimation theory.1 In particular, for the problem of optimum phase-shift estimation from observation of a single mode of the radiation field under an average photon number constraint, we have found both the optimum quantum state and the optimum quantum measurement. The former turns out to have a truncated number-state expansion whose coefficients, out to the truncation point, fall off as 1/n. The latter turns out to be the Susskind-Glogower phase operator. More important, the resulting phase estimation performance improves as 1 / N av 2 .
Pegg-Barnett Quantum phase density distribution properties of amplitude coherent states are studied.It is find that:1.the Quantum phase density distribution of amplitude coherent states p(θ,m) is influenced by superposition angle φ which is the vacuum state |o〉superposed by coherent state|α〉,average photon number and θ-γ that is the quantum phase θ minus displace angle of coherent state γ.2.the quantum phase density distribution changes with φ and θ-γ periodically;it becomes large with increase of average photon numbers.Under certain conditions,extremums of quantum phase density distribution are restricted by φ and θ-γ.