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arXiv:1710.08548 · Physical Review A

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, Dominic William Berry

Published Apr 23, 2018• 10 citations
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Abstract

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.

DOI
10.1103/physreva.97.042334

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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 \ensuremathκp\ensuremath−1/∣\ensuremathω∣p{\ensuremath{\kappa}}^{p\ensuremath{-}1}/{|\ensuremath{\omega}|}^{p}\ensuremathκp\ensuremath−1/∣\ensuremathω∣p for large \ensuremathω\ensuremath{\omega}\ensuremathω, 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(\ensuremathπ/p)]\ensuremath−1(4N/\ensuremathκ)\ensuremath−(p\ensuremath−1)/p{[psin(\ensuremath{\pi}/p)]}^{\ensuremath{-}1}{(4\mathcal{N}/\ensuremath{\kappa})}^{\ensuremath{-}(p\ensuremath{-}1)/p}[psin(\ensuremathπ/p)]\ensuremath−1(4N/\ensuremathκ)\ensuremath−(p\ensuremath−1)/p. Next, we consider whether the bound can be achieved by an adaptive homodyne measurement in the limit N/\ensuremathκ\ensuremath≫1\mathcal{N}/\ensuremath{\kappa}\ensuremath{\gg}1N/\ensuremathκ\ensuremath≫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 N/\ensuremathκ\mathcal{N}/\ensuremath{\kappa}N/\ensuremathκ 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.
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 $1/{|\ensuremath{\omega}|}^{p}$ with p>1p>1p>1, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of $1/{\mathcal{N}}^{(p\ensuremath{-}1)/p}$ and $1/{\mathcal{N}}^{2(p\ensuremath{-}1)/(p+1)}$, respectively, where N\mathcal{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.
Quantum parameter estimation has many applications, from gravitational wave detection to quantum key distribution. The most commonly used technique for this type of estimation is quantum filtering, using only past observations. We present the first experimental demonstration of quantum smoothing, a time-symmetric technique that uses past and future observations, for quantum parameter estimation. We consider both adaptive and nonadaptive quantum smoothing, and show that both are better than their filtered counterparts. For the problem of estimating a stochastically varying phase shift on a coherent beam, our theory predicts that adaptive quantum smoothing (the best scheme) gives an estimate with a mean-square error up to 2sqrt[2] times smaller than nonadaptive filtering (the standard quantum limit). The experimentally measured improvement is 2.24+/-0.14.
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>1p>1p>1, the minimum mean-square error in any (single-time) phase estimate scales as ${\mathcal{N}}^{\ensuremath{-}2(p\ensuremath{-}1)/(p+1)}$, where N\mathcal{N}N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit $p\ensuremath{\rightarrow}\ensuremath{\infty}$) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2p=2p=2 (Brownian motion), this limit can be attained by phase tracking.
Adaptive homodyne estimation of a continuously evolving optical phase using time-symmetric quantum smoothing has been demonstrated experimentally to provide superior accuracy in the phase estimate compared to adaptive or non-adaptive estimation using filtering alone. Here, we illustrate how the mean-square error in the adaptive phase estimate may be further reduced below the standard quantum limit for the stochastic noise process considered by using a Rauch-Tung-Striebel smoother as the estimator, along with an optimal Kalman filter in the feedback loop. Further, the estimation using smoothing can be made robust to uncertainties in the underlying parameters of the noise process modulating the system phase to be estimated. This has been done using a robust fixed-interval smoother designed for uncertain systems satisfying a certain integral quadratic constraint.
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.