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Physical Review A

Adaptive single-shot phase measurements: A semiclassical approach

Howard M. Wiseman, Rowan Killip

Published Jul 1, 1997• 75 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.56.944

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Adaptive single-shot phase measurements: The full quantum theory

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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.

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Heterodyne and adaptive phase measurements on states of fixed mean photon number

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The standard technique for measuring the phase of a single-mode field is heterodyne detection. Such a measurement may have an uncertainty far above the intrinsic quantum phase uncertainty of the state. Recently it has been shown [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] that an adaptive technique introduces far less excess noise. Here we quantify this difference by an exact numerical calculation of the minimum measured phase variance for the various schemes, optimized over states with a fixed mean photon number. We also analytically derive the asymptotics for these variances. For the case of heterodyne detection our results disagree with the power law claimed by D'Ariano and Paris [Phys. Rev. A 49, 3022 (1994)].

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Experimental Demonstrations

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For intermediate values of the photon number the adaptive variances should ideally be even lower [10], but the performance of our experiment is limited by finite feedback bandwidth. For signals with large mean photon number, the adaptive scheme is inferior to heterodyne because of excess technical noise in the feedback loop. As the intrinsic phase uncertainty of coherent states becomes large for very low photon numbers, the relative differences among the expected variances for adaptive, heterodyne, and ideal estimation become small. Accordingly, we have been unable to beat the heterodyne limit for phase estimation variances with adaptive homodyne measurement for mean photon numbers N & 8. (Note that all theoretical curves in Fig. 1(a) correspond FIG. 1 (color). Experimental results from the adaptive and heterodyne measurements. (a) Adaptive (blue circles) and heterodyne (red crosses) phase estimate variance vs pulse photon number. The blue dash-dotted line is a second order curve through the adaptive data, to guide the eye. The thin lines are the theoretical curves for heterodyne detection with (solid) and without (dotted) corrections for detector electronic noise. The thick solid line denotes the fundamental quantum uncertainty limit, given our overall photodetection efficiency. (b) Phase-estimator distributions for adaptive (blue circles) and heterodyne (red crosses) measurements, for pulses with mean photon number 2:5. (c) Polar plot showing the variance of adaptive phase estimates (blue dots) for different signal phases (mean photon number 50). The solid blue line is a linear fit to the data. The double red lines indicate the 1 scatter for our heterodyne data, averaged over initial phase.

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Single-Shot Non-Gaussian Measurements for Optical Phase Estimation

M. T. DiMario, F. E. Becerra

Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.

arXiv:2008.08185•Physical Review Letters•Sep 18, 2020•11 citations
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{N}^{\ensuremath{-}1}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 \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/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{N}^{\ensuremath{-}3/2}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{N}^{\ensuremath{-}2}N\ensuremath−2. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.
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 technique for measuring the phase of a single-mode field is heterodyne detection. Such a measurement may have an uncertainty far above the intrinsic quantum phase uncertainty of the state. Recently it has been shown [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] that an adaptive technique introduces far less excess noise. Here we quantify this difference by an exact numerical calculation of the minimum measured phase variance for the various schemes, optimized over states with a fixed mean photon number. We also analytically derive the asymptotics for these variances. For the case of heterodyne detection our results disagree with the power law claimed by D'Ariano and Paris [Phys. Rev. A 49, 3022 (1994)].
For intermediate values of the photon number the adaptive variances should ideally be even lower [10], but the performance of our experiment is limited by finite feedback bandwidth. For signals with large mean photon number, the adaptive scheme is inferior to heterodyne because of excess technical noise in the feedback loop. As the intrinsic phase uncertainty of coherent states becomes large for very low photon numbers, the relative differences among the expected variances for adaptive, heterodyne, and ideal estimation become small. Accordingly, we have been unable to beat the heterodyne limit for phase estimation variances with adaptive homodyne measurement for mean photon numbers N & 8. (Note that all theoretical curves in Fig. 1(a) correspond FIG. 1 (color). Experimental results from the adaptive and heterodyne measurements. (a) Adaptive (blue circles) and heterodyne (red crosses) phase estimate variance vs pulse photon number. The blue dash-dotted line is a second order curve through the adaptive data, to guide the eye. The thin lines are the theoretical curves for heterodyne detection with (solid) and without (dotted) corrections for detector electronic noise. The thick solid line denotes the fundamental quantum uncertainty limit, given our overall photodetection efficiency. (b) Phase-estimator distributions for adaptive (blue circles) and heterodyne (red crosses) measurements, for pulses with mean photon number 2:5. (c) Polar plot showing the variance of adaptive phase estimates (blue dots) for different signal phases (mean photon number 50). The solid blue line is a linear fit to the data. The double red lines indicate the 1 scatter for our heterodyne data, averaged over initial phase.
In standard single-shot measurements of the phase of an optical mode, the phase and amplitude quadratures are jointly measured, and the latter information discarded. These techniques are consequently suboptimal. Here I suggest an adaptive scheme, whereby the phase is estimated from the results so far and fed back to control the phase of the local oscillator so as to measure the (estimated) phase quadrature only. I show that adaptive phase measurements can approach optimal phase measurements for states with both low and high mean photon numbers.
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 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 ${N}^{\ensuremath{-}2/3},$ 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.
Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.