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Physical Review A

Sub-Heisenberg phase uncertainties

Luca Pezzè

Published Dec 12, 2013• 23 citations

Abstract

Phase shift estimation with uncertainty below the Heisenberg limit, $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}\ensuremath{\propto}1/{\overline{N}}_{T}$, where ${\overline{N}}_{T}$ is the total average number of particles employed, is a mirage of linear quantum interferometry. Recently, Rivas and Luis, [New J. Phys. 14, 093052 (2012)] proposed a scheme to achieve a phase uncertainty $\ensuremath{\Delta}\ensuremath{\phi}\ensuremath{\propto}1/{\overline{N}}_{T}^{k}$, with $k$ an arbitrary exponent. This sparked an intense debate in the literature which, ultimately, does not exclude the possibility to overcome $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}$ at specific phase values. Our numerical analysis of the Rivas and Luis proposal shows that sub-Heisenberg uncertainties are obtained only when the estimator is strongly biased. No violation of the Heisenberg limit is found after bias correction or when using a bias-free Bayesian analysis.

DOI: 10.1103/physreva.88.060101

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Phase shift estimation with uncertainty below the Heisenberg limit,

\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}\ensuremath{\propto}1/{\overline{N}}_{T}

, where

{\overline{N}}_{T}

is the total average number of particles employed, is a mirage of linear quantum interferometry. Recently, Rivas and Luis, [New J. Phys. 14, 093052 (2012)] proposed a scheme to achieve a phase uncertainty

\ensuremath{\Delta}\ensuremath{\phi}\ensuremath{\propto}1/{\overline{N}}_{T}^{k}

, with

k

an arbitrary exponent. This sparked an intense debate in the literature which, ultimately, does not exclude the possibility to overcome

\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}

at specific phase values. Our numerical analysis of the Rivas and Luis proposal shows that sub-Heisenberg uncertainties are obtained only when the estimator is strongly biased. No violation of the Heisenberg limit is found after bias correction or when using a bias-free Bayesian analysis.

The Heisenberg limit traditionally provides a lower bound on the phase uncertainty scaling as $1/\ensuremath{\langle}N\ensuremath{\rangle}$, where $\ensuremath{\langle}N\ensuremath{\rangle}$ is the mean number of photons in the probe. However, this limit has a number of loopholes which potentially might be exploited to achieve measurements with even greater accuracy. Here we close these loopholes by proving a completely rigorous form of the Heisenberg limit for the average error over all phase shifts, applicable to any estimate of a completely unknown phase shift. Our result gives a completely general, constraint-free, and nonasymptotic statement of the Heisenberg limit. It holds for all phase estimation schemes, including multiple passes, nonlinear phase shifts, multimode probes, and arbitrary measurements.

The ultimate bound to the accuracy of phase estimates is often assumed to be given by the Heisenberg limit. Recent work seemed to indicate that this bound can be violated, yielding measurements with much higher accuracy than was previously expected. The Heisenberg limit can be restored as a rigorous bound to the accuracy provided one considers the accuracy averaged over the possible values of the unknown phase, as we have recently shown [Phys. Rev. A 85, 041802(R) (2012)]. Here we present an expanded proof of this result together with a number of additional results, including the proof of a previously conjectured stronger bound in the asymptotic limit. Other measures of the accuracy are examined, as well as other restrictions on the generator of the phase shifts. We provide expanded numerical results for the minimum error and asymptotic expansions. The significance of the results claiming violation of the Heisenberg limit is assessed, followed by a detailed discussion of the limitations of the Cram\'er-Rao bound.

The phase sensitivity of interferometers is limited by the so-called Heisenberg limit, which states that the optimum phase sensitivity is inversely proportional to the number of interfering particles

N

, a

1∕\sqrt{N}

improvement over the standard quantum limit. We have used simulated annealing, a global optimization strategy, to systematically search for quantum interferometer input states that approach the Heisenberg-limited uncertainty in estimates of the interferometer phase shift. We compare the performance of these states to that of other nonclassical states already known to yield Heisenberg-limited uncertainty.

We investigate the scaling of the phase sensitivity of a nonideal Heisenberg-limited interferometer with the particle number

N

, in the case of the Bayesian detection procedure proposed by Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] for twin boson input modes. Using Monte Carlo simulations for up to

10 000

bosons, we show that the phase error of a nonideal interferometer scales with the Heisenberg limit if the losses are of the order of ${N}^{\ensuremath{-}1}$. Greater losses degrade the scaling which is then in ${N}^{\ensuremath{-}1∕2}$, such as the beam-splitter shot-noise limit, yet the sensitivity stays sub-shot-noise as long as photon correlations are present. These results give the actual limits of Bayesian detection for twin-mode interferometry and prove that it is an experimentally feasible scheme, contrary to what is implied by the coincidence-detection analysis of Kim et al. [Phys. Rev. A 60, 708 (1999)].