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arXiv:1306.1279 · Physical Review Letters

Stochastic Heisenberg Limit: Optimal Estimation of a Fluctuating Phase

Dominic William Berry, Michael J. W. Hall, Howard M. Wiseman

Published Sep 10, 2013• 35 citations
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Abstract

The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cram\'er-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ${\ensuremath{\omega}}^{\ensuremath{-}p}$ with $p>1$, the minimum mean-square error in any (single-time) phase estimate scales as ${\mathcal{N}}^{\ensuremath{-}2(p\ensuremath{-}1)/(p+1)}$, where $\mathcal{N}$ is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit $p\ensuremath{\rightarrow}\ensuremath{\infty}$) and provides a stochastic Heisenberg limit for fluctuating phases. For $p=2$ (Brownian motion), this limit can be attained by phase tracking.

DOI
10.1103/physrevlett.111.113601

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The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cram'er-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as \ensuremathω\ensuremath−p{\ensuremath{\omega}}^{\ensuremath{-}p}\ensuremathω\ensuremath−p with p>1p>1p>1, the minimum mean-square error in any (single-time) phase estimate scales as N\ensuremath−2(p\ensuremath−1)/(p+1){\mathcal{N}}^{\ensuremath{-}2(p\ensuremath{-}1)/(p+1)}N\ensuremath−2(p\ensuremath−1)/(p+1), where N\mathcal{N}N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p\ensuremath→\ensuremath∞p\ensuremath{\rightarrow}\ensuremath{\infty}p\ensuremath→\ensuremath∞) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2p=2p=2 (Brownian motion), this limit can be attained by phase tracking.
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.
When measuring a time-varying phase, the standard quantum limit and Heisenberg limit as usually defined, for a constant phase, do not apply. If the phase has Gaussian statistics and a power-law spectrum 1/∣ω∣p1/|\omega|^p1/∣ω∣p with p>1p>1p>1, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of 1/N(p−1)/p1/{\cal N}^{(p-1)/p}1/N(p−1)/p and 1/N2(p−1)/(p+1)1/{\cal N}^{2(p-1)/(p+1)}1/N2(p−1)/(p+1), respectively, where N{\cal N}N is the mean photon flux. We show that this Heisenberg scaling can be achieved via adaptive measurements on squeezed states. We predict the experimental parameters analytically, and test them with numerical simulations. Previous work had considered the special case of p=2p=2p=2. For the coherent state case we derive the achievable mean-square error analytically for even ppp, and compare it to the standard quantum limit as found previously.
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We propose quantum versions of the Bell-Ziv-Zakai lower bounds for the error in multiparameter estimation.As an application, we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power-law spectrum ∼1=jωj p , with p > 1.With no other assumptions, we show that the mean-square error has a lower bound scaling as 1=N 2ðp-1Þ=ðpþ1Þ , where N is the time-averaged mean photon flux.Moreover, we show that this scaling is achievable by sampling and interpolation, for any p > 1.This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.
The ultimate precision of phase estimation is limited by the Heisenberg scaling Δϕ0=K/NΔϕ_0 = K/NΔϕ0​=K/N, where K∼1K\sim1K∼1 is a numerical prefactor and NNN is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity Δϕ0Δϕ_0Δϕ0​ and the width δϕδϕδϕ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that δϕδϕδϕ strongly depends on KKK. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range δϕ=π/2δϕ=π/2δϕ=π/2.
I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cramér-Rao bounds for quantum parameter estimation. From a simple form of the proposed bounds, I derive both a Heisenberg error limit that scales with the average energy and a limit similar to the quantum Cramér-Rao bound that scales with the energy variance. These results are further illustrated by applying the bound to a few examples of optical phase estimation, which show that a quantum Ziv-Zakai bound can be much higher and thus tighter than a quantum Cramér-Rao bound for states with highly non-gaussian photon-number statistics in certain regimes and also stay close to the latter where the latter is expected to be tight.