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.
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.
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 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.
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)].
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.
We consider a two-mode bosonic state with fixed photon number n∈N , whose upper and lower modes pick up a phase ϕ and −ϕ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating ϕ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider ϕ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider n=1 and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.
Phase fluctuations between optical inputs limit programmable photonic processors because their output powers depend on coherent interference. We study the phase-drift penalty that arises when sine and cosine quadratures are measured sequentially rather than simultaneously. The analysis is motivated by measurements from an eight-mode programmable photonic processor, including 35 free-running recordings of 300 s acquired at approximately 125 samples per second per channel. These recordings provide an empirical route for estimating the phase-increment variance at a selected reconfiguration interval.
The estimate is defined at the time of the second measurement. For fixed quadrature order, perturbation of the atan2 reconstruction gives eC→S=−δτsin2φ0+O(δτ2) and eS→C=−δτcos2φ0+O(δτ2). Writing Qτ=Var(δτ), uniform phase averaging gives the first-order drift mean-square error 3Qτ/8. A phase-predicted ordering rule measures the locally less informative quadrature first and the more informative quadrature second. Its uniform first-order penalty is (3/8−1/π)Qτ, which is 84.9 percent below the fixed-order value.
We also derive an increment-aware estimator from a local state-space model. Marginalizing the unknown phase increment increases the variance of a stale phase observation by Qτ, reducing its Fisher information from I to I/(1+IQτ). For ideal balanced Poisson detection, the Fisher information of each quadrature equals its detected signal-photon number. This yields dimensionless architecture boundaries in spatial information and phase-increment variance. Nonlinear Monte Carlo simulations validate the perturbative laws, quantify robustness to prediction error, and compare simultaneous, fixed-order, increment-aware, and adaptive receivers under a common noise model.