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arXiv:quant-ph/0010059 · Journal of Modern Optics

The effects of time delays in adaptive phase measurements

Dominic William Berry, Howard M. Wiseman

Published Apr 1, 2001• 4 citations
arXivHTMLPDF

Abstract

It is not possible to make measurements of the phase of an optical mode using linear optics without introducing an extra phase uncertainty. This extra phase variance is quite large for heterodyne measurements, however it is possible to reduce it to the theoretical limit of log [nbar]/(4[nbar]2) using adaptive measurements. These measurements are quite sensitive to experimental inaccuracies, especially time delays and inefficient detectors. Here it is shown that the minimum introduced phase variance when there is a time delay of log τ is τ/(8[nbar]). This result is verified numerically, showing that the phase variance introduced approaches this limit for most of the adaptive schemes using the best final phase estimate. The main exception is the adaptive mark II scheme with simplified feedback, which is extremely sensitive to time delays. The extra phase variance due to time delays is considered for the mark I case with simplified feedback, verifying the τ/2 result obtained by Wiseman and Killip both by a more rigorous analytic technique and numerically.

DOI
10.1080/09500340010011682

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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.

CRC Press eBooks•Apr 27, 2007•0 citations
It is not possible to make measurements of the phase of an optical mode using linear optics without introducing an extra phase uncertainty. This extra phase variance is quite large for heterodyne measurements, however it is possible to reduce it to the theoretical limit of log [nbar]/(4[nbar]2) using adaptive measurements. These measurements are quite sensitive to experimental inaccuracies, especially time delays and inefficient detectors. Here it is shown that the minimum introduced phase variance when there is a time delay of log τ is τ/(8[nbar]). This result is verified numerically, showing that the phase variance introduced approaches this limit for most of the adaptive schemes using the best final phase estimate. The main exception is the adaptive mark II scheme with simplified feedback, which is extremely sensitive to time delays. The extra phase variance due to time delays is considered for the mark I case with simplified feedback, verifying the τ/2 result obtained by Wiseman and Killip both by a more rigorous analytic technique and numerically.
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.
It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as ${n}^{\ensuremath{-}1.5},$ where $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as ${n}^{\ensuremath{-}2}.$ It has been shown that a lower limit to the phase variance that is introduced scales as $\mathrm{ln}(\mathrm{n\ifmmode \bar{}\else \={}\fi{}})/{n}^{2}.$ Here we introduce an adaptive technique that attains this theoretical lower limit.
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)].
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.
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.