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 $p>1$, 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 $\mathcal{N}$ 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 $p=2$.
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/∣\ensuremathω∣p with p>1, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of 1/N(p\ensuremath−1)/p and 1/N2(p\ensuremath−1)/(p+1), respectively, where 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=2.
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∣p with p>1, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of 1/N(p−1)/p and 1/N2(p−1)/(p+1), respectively, where 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=2. For the coherent state case we derive the achievable mean-square error analytically for even p, and compare it to the standard quantum limit as found previously.
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.
We have previously [Phys. Rev. A 65, 043803 (2002)] analyzed adaptive measurements for estimating the continuously varying phase of a coherent beam, and a broadband squeezed beam. A real squeezed beam must have finite photon flux N and hence can be significantly squeezed only over a limited frequency range. In this paper we analyze adaptive phase measurements of this type for a realistic model of a squeezed beam. We show that, provided it is possible to suitably choose the parameters of the beam, a mean-square phase uncertainty scaling as ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}5∕8}$ is possible, where $\ensuremath{\kappa}$ is the linewidth of the beam resulting from the fluctuating phase. This is an improvement over the ${(\mathcal{N}∕\ensuremath{\kappa})}^{\ensuremath{-}1∕2}$ scaling found previously for coherent beams. In the experimentally realistic case where there is a limit on the maximum squeezing possible, the variance will be reduced below that for coherent beams, though the scaling is unchanged.
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 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.
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 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.
We derive, and experimentally demonstrate, an interferometric scheme for unambiguous phase estimation with precision scaling at the Heisenberg limit that does not require adaptive measurements. That is, with no prior knowledge of the phase, we can obtain an estimate of the phase with a standard deviation that is only a small constant factor larger than the minimum physically allowed value. Our scheme resolves the phase ambiguity that exists when multiple passes through a phase shift, or NOON states, are used to obtain improved phase resolution. Like a recently introduced adaptive technique (Higgins et al 2007 Nature 450 393), our experiment uses multiple applications of the phase shift on single photons. By not requiring adaptive measurements, but rather using a predetermined measurement sequence, the present scheme is both conceptually simpler and significantly easier to implement. Additionally, we demonstrate a simplified adaptive scheme that also surpasses the standard quantum limit for single passes.