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

arXiv:quant-ph/0111013 · Physical Review A

Adaptive quantum measurements of a continuously varying phase

Dominic William Berry, Howard M. Wiseman

Published Mar 14, 2002• 79 citations
arXivHTMLPDF

Abstract

We analyze the problem of quantum-limited estimation of a stochastically varying phase of a continuous beam (rather than a pulse) of the electromagnetic field. We consider both nonadaptive and adaptive measurements, and both dyne detection (using a local oscillator) and interferometric detection. We take the phase variation to be $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}=\sqrt{\ensuremath{\kappa}}\ensuremath{\xi}(t),$ where $\ensuremath{\xi}(t)$ is \ensuremath{\delta}-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is $N=P/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\kappa},$ the number of photons per coherence time. For the case of dyne detection, both continuous-wave (cw) coherent beams and cw (broadband) squeezed beams are considered. For a coherent beam a simple feedback scheme gives good results, with a phase variance $\ensuremath{\simeq}{N}^{\ensuremath{-}1/2}/2.$ This is $\sqrt{2}$ times smaller than that achievable by nonadaptive (heterodyne) detection. For a squeezed beam a more accurate feedback scheme gives a variance scaling as ${N}^{\ensuremath{-}2/3},$ compared to ${N}^{\ensuremath{-}1/2}$ for heterodyne detection. For the case of interferometry only a coherent input into one port is considered. The locally optimal feedback scheme is identified, and it is shown to give a variance scaling as ${N}^{\ensuremath{-}1/2}.$ It offers a significant improvement over nonadaptive interferometry only for N of order unity.

DOI
10.1103/physreva.65.043803

Related papers

Adaptive phase estimation is more accurate than nonadaptive phase estimation for continuous beams of light

Damian T. Pope, Howard M. Wiseman, Nathan K. Langford

We consider the task of estimating the randomly fluctuating phase of a continuous-wave beam of light. Using the theory of quantum parameter estimation, we show that this can be done more accurately when feedback is used (adaptive phase estimation) than by any scheme not involving feedback (nonadaptive phase estimation) in which the beam is measured as it arrives at the detector. Such schemes not involving feedback include all those based on heterodyne detection or instantaneous canonical phase measurements. We also demonstrate that the superior accuracy of adaptive phase estimation is present in a regime conducive to observing it experimentally.

arXiv:quant-ph/0405066•Physical Review A•Oct 19, 2004•28 citations

Adaptive phase measurements

Dominic William Berry

In this thesis I consider the general problem of how to make the best possible phase measurements using feedback. Both the optimum input state and optimum feedback are considered for both single-mode dyne measurements and two-mode interferometric measurements. I derive the optimum input states under general dyne measurements when the mean photon number is fixed, both for general states and squeezed states. I propose a new feedback scheme that introduces far less phase uncertainty than mark II feedback, and is very close to the theoretical limit. I also derive results for the phase variance when there is a time delay in the feedback loop, showing that there is a lower limit to the introduced phase variance, and this is approached quite accurately under some conditions. I derive the optimum input states for interferometry, showing that the phase uncertainty scales as 1/N for all the common measures of uncertainty. This is contrasted with the |j0>_z state, which does not scale as 1/N for all measures of phase uncertainty. I introduce an adaptive feedback scheme that is very close to optimum, and can give scaling very close to 1/N for the uncertainty. Lastly I consider the case of continuous measurements, for both the dyne and interferometric cases.

arXiv:quant-ph/0202136•Jan 1, 2001•6 citations

Adaptive phase measurements for narrowband squeezed beams

Dominic William Berry, Howard M. Wiseman

We have previously [Phys. Rev. A 65, 043803 (2002)] analyzed adaptive measurements for estimating the continuously varying phase of a coherent beam, and a broadband squeezed beam. A real squeezed beam must have finite photon flux $\mathcal{N}$ and hence can be significantly squeezed only over a limited frequency range. In this paper we analyze adaptive phase measurements of this type for a realistic model of a squeezed beam. We show that, provided it is possible to suitably choose the parameters of the beam, a mean-square phase uncertainty scaling as ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}5∕8}$ is possible, where $\ensuremath{\kappa}$ is the linewidth of the beam resulting from the fluctuating phase. This is an improvement over the ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}1∕2}$ scaling found previously for coherent beams. In the experimentally realistic case where there is a limit on the maximum squeezing possible, the variance will be reduced below that for coherent beams, though the scaling is unchanged.

arXiv:quant-ph/0603189•Physical Review A•Jun 23, 2006•34 citations

Adaptive single-shot phase measurements: The full quantum theory

Howard M. Wiseman, Rowan Killip

The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.

arXiv:quant-ph/9710056•Physical Review A•Mar 1, 1998•71 citations

Adaptive single-shot phase measurements: A semiclassical approach

Howard M. Wiseman, Rowan Killip

The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as ${N}^{\ensuremath{-}1}$, where $N$ is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus $\ensuremath{\pi}/2$, so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time $t$ is a functional of the photocurrent from time $0$ up to time $t$ in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as ${N}^{\ensuremath{-}3/2}$. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as ${N}^{\ensuremath{-}2}$. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.

Physical Review A•Jul 1, 1997•75 citations

Adaptive estimation of a time-varying phase with coherent states: Smoothing can give an unbounded improvement over filtering

Kiarn T. Laverick, Howard M. Wiseman, Hossein T. Dinani +1 more

The problem of measuring a time-varying phase, even when the statistics of the variation is known, is considerably harder than that of measuring a constant phase. In particular, the usual bounds on accuracy, such as the $1/(4\overline{n})$ standard quantum limit with coherent states, do not apply. Here, by restricting to coherent states, we are able to analytically obtain the achievable accuracy, the equivalent of the standard quantum limit, for a wide class of phase variation. In particular, we consider the case where the phase has Gaussian statistics and a power-law spectrum equal to ${\ensuremath{\kappa}}^{p\ensuremath{-}1}/{|\ensuremath{\omega}|}^{p}$ for large $\ensuremath{\omega}$, for some $p>1$. For coherent states with mean photon flux $\mathcal{N}$, we give the quantum Cram\'er-Rao bound on the mean-square phase error as ${[psin(\ensuremath{\pi}/p)]}^{\ensuremath{-}1}{(4\mathcal{N}/\ensuremath{\kappa})}^{\ensuremath{-}(p\ensuremath{-}1)/p}$. Next, we consider whether the bound can be achieved by an adaptive homodyne measurement in the limit $\mathcal{N}/\ensuremath{\kappa}\ensuremath{\gg}1$, which allows the photocurrent to be linearized. Applying the optimal filtering for the resultant linear Gaussian system, we find the same scaling with $\mathcal{N}$, but with a prefactor larger by a factor of $p$. By contrast, if we employ optimal smoothing we can exactly obtain the quantum Cram\'er-Rao bound. That is, contrary to previously considered ($p=2$) cases of phase estimation, here the improvement offered by smoothing over filtering is not limited to a factor of 2 but rather can be unbounded by a factor of $p$. We also study numerically the performance of these estimators for an adaptive measurement in the limit where $\mathcal{N}/\ensuremath{\kappa}$ is not large and find a more complicated picture.

arXiv:1710.08548•Physical Review A•Apr 23, 2018•10 citations
We analyze the problem of quantum-limited estimation of a stochastically varying phase of a continuous beam (rather than a pulse) of the electromagnetic field. We consider both nonadaptive and adaptive measurements, and both dyne detection (using a local oscillator) and interferometric detection. We take the phase variation to be \ensuremathφ\ifmmode˙\else˙\fi=\ensuremathκ\ensuremathξ(t),\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}=\sqrt{\ensuremath{\kappa}}\ensuremath{\xi}(t),\ensuremathφ\ifmmode˙\else˙\fi​=\ensuremathκ​\ensuremathξ(t), where \ensuremathξ(t)\ensuremath{\xi}(t)\ensuremathξ(t) is \ensuremath{\delta}-correlated Gaussian noise. For a beam of power P, the important dimensionless parameter is N=P/\ensuremath\Elzxh\ensuremathω\ensuremathκ,N=P/\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\kappa},N=P/\ensuremath\Elzxh\ensuremathω\ensuremathκ, the number of photons per coherence time. For the case of dyne detection, both continuous-wave (cw) coherent beams and cw (broadband) squeezed beams are considered. For a coherent beam a simple feedback scheme gives good results, with a phase variance \ensuremath≃N\ensuremath−1/2/2.\ensuremath{\simeq}{N}^{\ensuremath{-}1/2}/2.\ensuremath≃N\ensuremath−1/2/2. This is 2\sqrt{2}2​ times smaller than that achievable by nonadaptive (heterodyne) detection. For a squeezed beam a more accurate feedback scheme gives a variance scaling as N\ensuremath−2/3,{N}^{\ensuremath{-}2/3},N\ensuremath−2/3, compared to N\ensuremath−1/2{N}^{\ensuremath{-}1/2}N\ensuremath−1/2 for heterodyne detection. For the case of interferometry only a coherent input into one port is considered. The locally optimal feedback scheme is identified, and it is shown to give a variance scaling as N\ensuremath−1/2.{N}^{\ensuremath{-}1/2}.N\ensuremath−1/2. It offers a significant improvement over nonadaptive interferometry only for N of order unity.
We consider the task of estimating the randomly fluctuating phase of a continuous-wave beam of light. Using the theory of quantum parameter estimation, we show that this can be done more accurately when feedback is used (adaptive phase estimation) than by any scheme not involving feedback (nonadaptive phase estimation) in which the beam is measured as it arrives at the detector. Such schemes not involving feedback include all those based on heterodyne detection or instantaneous canonical phase measurements. We also demonstrate that the superior accuracy of adaptive phase estimation is present in a regime conducive to observing it experimentally.
In this thesis I consider the general problem of how to make the best possible phase measurements using feedback. Both the optimum input state and optimum feedback are considered for both single-mode dyne measurements and two-mode interferometric measurements. I derive the optimum input states under general dyne measurements when the mean photon number is fixed, both for general states and squeezed states. I propose a new feedback scheme that introduces far less phase uncertainty than mark II feedback, and is very close to the theoretical limit. I also derive results for the phase variance when there is a time delay in the feedback loop, showing that there is a lower limit to the introduced phase variance, and this is approached quite accurately under some conditions. I derive the optimum input states for interferometry, showing that the phase uncertainty scales as 1/N for all the common measures of uncertainty. This is contrasted with the |j0>_z state, which does not scale as 1/N for all measures of phase uncertainty. I introduce an adaptive feedback scheme that is very close to optimum, and can give scaling very close to 1/N for the uncertainty. Lastly I consider the case of continuous measurements, for both the dyne and interferometric cases.
We have previously [Phys. Rev. A 65, 043803 (2002)] analyzed adaptive measurements for estimating the continuously varying phase of a coherent beam, and a broadband squeezed beam. A real squeezed beam must have finite photon flux N\mathcal{N}N and hence can be significantly squeezed only over a limited frequency range. In this paper we analyze adaptive phase measurements of this type for a realistic model of a squeezed beam. We show that, provided it is possible to suitably choose the parameters of the beam, a mean-square phase uncertainty scaling as ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}5∕8}$ is possible, where $\ensuremath{\kappa}$ is the linewidth of the beam resulting from the fluctuating phase. This is an improvement over the ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}1∕2}$ scaling found previously for coherent beams. In the experimentally realistic case where there is a limit on the maximum squeezing possible, the variance will be reduced below that for coherent beams, though the scaling is unchanged.
The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.
The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as ${N}^{\ensuremath{-}1}$, where NNN is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus $\ensuremath{\pi}/2$, so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time ttt is a functional of the photocurrent from time 000 up to time ttt in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as ${N}^{\ensuremath{-}3/2}$. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as ${N}^{\ensuremath{-}2}$. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.
The problem of measuring a time-varying phase, even when the statistics of the variation is known, is considerably harder than that of measuring a constant phase. In particular, the usual bounds on accuracy, such as the 1/(4n‾)1/(4\overline{n})1/(4n) standard quantum limit with coherent states, do not apply. Here, by restricting to coherent states, we are able to analytically obtain the achievable accuracy, the equivalent of the standard quantum limit, for a wide class of phase variation. In particular, we consider the case where the phase has Gaussian statistics and a power-law spectrum equal to ${\ensuremath{\kappa}}^{p\ensuremath{-}1}/{|\ensuremath{\omega}|}^{p}$ for large $\ensuremath{\omega}$, for some p>1p>1p>1. For coherent states with mean photon flux N\mathcal{N}N, we give the quantum Cram\'er-Rao bound on the mean-square phase error as ${[psin(\ensuremath{\pi}/p)]}^{\ensuremath{-}1}{(4\mathcal{N}/\ensuremath{\kappa})}^{\ensuremath{-}(p\ensuremath{-}1)/p}$. Next, we consider whether the bound can be achieved by an adaptive homodyne measurement in the limit $\mathcal{N}/\ensuremath{\kappa}\ensuremath{\gg}1$, which allows the photocurrent to be linearized. Applying the optimal filtering for the resultant linear Gaussian system, we find the same scaling with N\mathcal{N}N, but with a prefactor larger by a factor of ppp. By contrast, if we employ optimal smoothing we can exactly obtain the quantum Cram\'er-Rao bound. That is, contrary to previously considered (p=2p=2p=2) cases of phase estimation, here the improvement offered by smoothing over filtering is not limited to a factor of 2 but rather can be unbounded by a factor of ppp. We also study numerically the performance of these estimators for an adaptive measurement in the limit where $\mathcal{N}/\ensuremath{\kappa}$ is not large and find a more complicated picture.