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arXiv:quant-ph/0002059 · Physical Review A

Phase measurements at the theoretical limit

Dominic William Berry, Howard M. Wiseman

Published Dec 12, 2000• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.63.013813

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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,{n}^{\ensuremath{-}1.5},n\ensuremath−1.5, where n\ifmmodeˉ\elseˉ\fi\mathrm{n\ifmmode \bar{}\else \={}\fi{}}n\ifmmodeˉ\elseˉ\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.{n}^{\ensuremath{-}2}.n\ensuremath−2. It has been shown that a lower limit to the phase variance that is introduced scales as ln(n\ifmmodeˉ\elseˉ\fi)/n2.\mathrm{ln}(\mathrm{n\ifmmode \bar{}\else \={}\fi{}})/{n}^{2}.ln(n\ifmmodeˉ\elseˉ\fi)/n2. Here we introduce an adaptive technique that attains this theoretical lower limit.
The standard technique for measuring the phase of a single-mode field is heterodyne detection. Such a measurement may have an uncertainty far above the intrinsic quantum phase uncertainty of the state. Recently it has been shown [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] that an adaptive technique introduces far less excess noise. Here we quantify this difference by an exact numerical calculation of the minimum measured phase variance for the various schemes, optimized over states with a fixed mean photon number. We also analytically derive the asymptotics for these variances. For the case of heterodyne detection our results disagree with the power law claimed by D'Ariano and Paris [Phys. Rev. A 49, 3022 (1994)].
In standard single-shot measurements of the phase of an optical mode, the phase and amplitude quadratures are jointly measured, and the latter information discarded. These techniques are consequently suboptimal. Here I suggest an adaptive scheme, whereby the phase is estimated from the results so far and fed back to control the phase of the local oscillator so as to measure the (estimated) phase quadrature only. I show that adaptive phase measurements can approach optimal phase measurements for states with both low and high mean photon numbers.
Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic 1/n1/n1/n decrease in phase uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$ for nnn the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the 1/n1/n1/n limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.
Adaptive techniques make practical many quantum measurements that would otherwise be beyond current laboratory capabilities. For example, they allow discrimination of nonorthogonal states with a probability of error equal to the Helstrom bound, measurement of the phase of a quantum oscillator with accuracy approaching (or in some cases attaining) the Heisenberg limit (HL), and estimation of phase in interferometry with a variance scaling at the HL, using only single qubit measurement and control. Each of these examples has close links with quantum information, in particular, experimental optical quantum information: the first is a basic quantum communication protocol; the second has potential application in linear optical quantum computing; the third uses an adaptive protocol inspired by the quantum phase estimation algorithm. We discuss each of these examples and their implementation in the laboratory, but concentrate upon the last, which was published most recently [Higginset al.,Nature, vol. 450, p. 393, 2007].
In this thesis I consider the general problem of how to make the best possible phase measurements using feedback. Both the optimum input state and optimum feedback are considered for both single-mode dyne measurements and two-mode interferometric measurements. I derive the optimum input states under general dyne measurements when the mean photon number is fixed, both for general states and squeezed states. I propose a new feedback scheme that introduces far less phase uncertainty than mark II feedback, and is very close to the theoretical limit. I also derive results for the phase variance when there is a time delay in the feedback loop, showing that there is a lower limit to the introduced phase variance, and this is approached quite accurately under some conditions. I derive the optimum input states for interferometry, showing that the phase uncertainty scales as 1/N for all the common measures of uncertainty. This is contrasted with the |j0>_z state, which does not scale as 1/N for all measures of phase uncertainty. I introduce an adaptive feedback scheme that is very close to optimum, and can give scaling very close to 1/N for the uncertainty. Lastly I consider the case of continuous measurements, for both the dyne and interferometric cases.
We demonstrate the first experimental Heisenberg-limited scaling of the phase variance in optical phase measurements. Our algorithm replaces complicated entangled states – widely thought to be essential for Heisenberg scaling – with single photons, multiple passes through a phase shift, and adaptive measurement. We show that although a single-photon version of Kitaev’s phase estimation algorithm has standard-quantum-limited variance, our generalized technique is within a small constant factor of the ultimate quantum limit.