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arXiv:quant-ph/0603189 · Physical Review A

Adaptive phase measurements for narrowband squeezed beams

Dominic William Berry, Howard M. Wiseman

Published Jun 23, 2006• 34 citations

Abstract

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 $\mathcal{N}$ 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.

DOI: 10.1103/physreva.73.063824

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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

\mathcal{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

\sqrt{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.

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

.

It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as ${n}^{\ensuremath{-}1.5},$ where $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as ${n}^{\ensuremath{-}2}.$ It has been shown that a lower limit to the phase variance that is introduced scales as $\mathrm{ln}(\mathrm{n\ifmmode \bar{}\else \={}\fi{}})/{n}^{2}.$ Here we introduce an adaptive technique that attains this theoretical lower limit.

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/|\omega|^p

with

p>1

, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of

1/{\cal N}^{(p-1)/p}

and

1/{\cal N}^{2(p-1)/(p+1)}

, respectively, where

{\cal 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.