We present an experimental demonstration of quantum smoothing, a non causal version of quantum parameter estimation that has applications from gravitational wave detection to quantum key distribution. An existing technique for this type of parameter estimation is quantum filtering, which uses past observations only. The technique presented here is time-symmetric that uses past and future observations to obtain a more precise estimate. We couple this innovative estimation technique with an adaptive detection scheme to demonstrate an estimation of the optical phase of light measured to be 2.24 ± 0.14 times better than the standard quantum limit. This approaches our theoretically derived prediction of a mean square error improvement of 2√2 times over the standard quantum limit [1].
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.
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) 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
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, 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.
Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.
We consider the task of estimating the randomly fluctuating phase of a continuous-wave beam of light. Using the theory of quantum parameter estimation, we show that this can be done more accurately when feedback is used (adaptive phase estimation) than by any scheme not involving feedback (nonadaptive phase estimation) in which the beam is measured as it arrives at the detector. Such schemes not involving feedback include all those based on heterodyne detection or instantaneous canonical phase measurements. We also demonstrate that the superior accuracy of adaptive phase estimation is present in a regime conducive to observing it experimentally.