• Open Access

Does Nonlinear Metrology Offer Improved Resolution? Answers from Quantum Information Theory

Michael J. W. Hall and Howard M. Wiseman
Phys. Rev. X 2, 041006 – Published 25 October 2012

Abstract

A number of authors have suggested that nonlinear interactions can enhance resolution of phase shifts beyond the usual Heisenberg scaling of 1/n, where n is a measure of resources such as the number of subsystems of the probe state or the mean photon number of the probe state. These suggestions are based on calculations of “local precision” for particular nonlinear schemes. However, we show that there is no simple connection between the local precision and the average estimation error for these schemes, leading to a scaling puzzle. This puzzle is partially resolved by a careful analysis of iterative implementations of the suggested nonlinear schemes. However, it is shown that the suggested nonlinear schemes are still limited to an exponential scaling in n. (This scaling may be compared to the exponential scaling in n which is achievable if multiple passes are allowed, even for linear schemes.) The question of whether nonlinear schemes may have a scaling advantage in the presence of loss is left open. Our results are based on a new bound for average estimation error that depends on (i) an entropic measure of the degree to which the probe state can encode a reference phase value, called the G asymmetry, and (ii) any prior information about the phase shift. This bound is asymptotically stronger than bounds based on the variance of the phase-shift generator. The G asymmetry is also shown to directly bound the average information gained per estimate. Our results hold for any prior distribution of the shift parameter, and generalize to estimates of any shift generated by an operator with discrete eigenvalues.

  • Received 10 May 2012

DOI:https://doi.org/10.1103/PhysRevX.2.041006

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Michael J. W. Hall and Howard M. Wiseman

  • Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia

Popular Summary

One of the wave properties of light is its phase, and phase shifts of light resulted from its interaction with other matter are commonly used to make precision measurements of many quantities, including position, temperature, and gravitational changes. Quantum mechanics has revealed that light also has a particle nature (photons), and this “lumpiness” of light places fundamental bounds on the measurement resolution when a limited number of photons are available. Recently, physicists have begun exploring a new direction for improving the measurement accuracy by using nonlinear interactions between photons to increase the phase shift. But can this work? We study such schemes theoretically, and show that, although it can work in principle, it is surprisingly nontrivial and requires more sophisticated schemes than have yet been implemented.

The strength of our results comes from applying quantum information theory to the problem of estimating the quantity that causes the phase shift. This advance yields stronger bounds, both for the information that can be gained per photon and for the corresponding average accuracy, than have been previously obtained for linear and nonlinear schemes. Unexpectedly, in contrast to existing bounds, our bounds are independent of the degree of nonlinearity of the interaction. Simply replacing a phase shift which is linear in the number of photons by one which is quadratic, for example, gives no measurement advantage per se. However, nonlinear interactions can give, via suitable sequences of pulses having differing photon numbers, a fundamental advantage over the linear case in terms of the scaling of the bounds with the average photon number available.

Our results extend also to atomic interferometry. We believe that they will set directions for future theoretical and experimental investigations of nonlinear metrology, including determining the effect of noise.

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Vol. 2, Iss. 4 — October - December 2012

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