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Physical Review A

Optimal input states and feedback for interferometric phase estimation

Dominic William Berry, Howard M. Wiseman, J. K. Breslin

Published Apr 12, 2001• 96 citations
Publisher

Abstract

We derive optimal N-photon two-mode input states for interferometric phase measurements. Under canonical measurements the phase variance scales as ${N}^{\ensuremath{-}2}$ for these states, as compared to ${N}^{\ensuremath{-}1}$ or ${N}^{\ensuremath{-}1/2}$ for states considered by previous authors. We prove that it is not possible to realize the canonical measurement by counting photons in the outputs of the interferometer, even if an adjustable auxiliary phase shift is allowed in the interferometer. However, we introduce a feedback algorithm based on Bayesian inference to control this auxiliary phase shift. This makes the measurement close to a canonical one, with a phase variance scaling slightly above ${N}^{\ensuremath{-}2}.$ With no feedback, the best result (given that the phase to be measured is completely unknown) is a scaling of ${N}^{\ensuremath{-}1}.$ For optimal input states having up to four photons, our feedback scheme is the best possible one, but for higher photon numbers more complicated schemes perform marginally better.

DOI
10.1103/physreva.63.053804

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We derive optimal N-photon two-mode input states for interferometric phase measurements. Under canonical measurements the phase variance scales as N\ensuremath−2{N}^{\ensuremath{-}2}N\ensuremath−2 for these states, as compared to N\ensuremath−1{N}^{\ensuremath{-}1}N\ensuremath−1 or N\ensuremath−1/2{N}^{\ensuremath{-}1/2}N\ensuremath−1/2 for states considered by previous authors. We prove that it is not possible to realize the canonical measurement by counting photons in the outputs of the interferometer, even if an adjustable auxiliary phase shift is allowed in the interferometer. However, we introduce a feedback algorithm based on Bayesian inference to control this auxiliary phase shift. This makes the measurement close to a canonical one, with a phase variance scaling slightly above N\ensuremath−2.{N}^{\ensuremath{-}2}.N\ensuremath−2. With no feedback, the best result (given that the phase to be measured is completely unknown) is a scaling of N\ensuremath−1.{N}^{\ensuremath{-}1}.N\ensuremath−1. For optimal input states having up to four photons, our feedback scheme is the best possible one, but for higher photon numbers more complicated schemes perform marginally better.
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.
Summary form only given. Interferometry is the basis of many high-precision measurements. The ultimate limit to the precision is due to quantum effects. This limit is most easily explored for a Mach-Zehnder interferometer The outputs of this device can be measured to yield an estimate /spl phi/ of the phase difference /spl phi/ between the two arms of the interferometer. It is well known that this can achieve the standard quantum limit for phase variance when an N-photon number state enters one input port. Several authors have proposed ways of reducing the phase variance to the Heisenberg limit. In this paper we show that there is a physical measurement scheme using photodetectors and feedback that is almost as good as the optimal measurement. The fundamental idea is that, in addition to the unknown phase we wish to measure, /spl phi/, in one arm of the interferometer, we introduce a known phase shift, /spl Phi/, into the other arm of the interferometer. After each photodetection we adjust this introduced phase shift in order to minimize the expected uncertainty of our best phase estimate /spl phi/ after the next photodetection. This feedback process is uniquely defined using Bayesian statistics, and can be solved exactly for moderate photon numbers.
We derive the optimal N-photon two-mode input state for obtaining an estimate straight phi of the phase difference between two arms of an interferometer. For an optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995)], it yields a variance (Deltastraight phi)(2) approximately pi(2)/N2, compared to O(N-1) or O(N-1/2) for states considered by previous authors. Such a measurement cannot be realized by counting photons in the interferometer outputs. However, we introduce an adaptive measurement scheme that can be thus realized, and show that it yields a variance in straight phi very close to that from an optimal measurement.
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.
Single photon count distributions at the outputs of a Mach-Zehnder interferometer are highly dependent on the phase difference between the two arms of the apparatus, so photocount records can be used to estimate an unknown optical phase. Experimental results have shown that phase information can be inferred from the data even at low mean input intensities of ⪅10 photons. Here we consider the optimal Bayesian parameter estimate for the model used by a previous frequentist analysis. In the limit of high input intensity the observation is well-modelled as an intensity measurement of a classical wave in Gaussian measurement noise, however in the low intensity regime this simple model does not apply. It seems that no generally valid closed-form estimator for this problem exists without making the Gaussian approximation. Moreover the 2π radian ambiguity in phase estimates presents a problem in defining cost functions to compare estimator performance, and we consider two reasonable alternatives. We present a numerical study of the experiment along with a novel numerical approach to this problem which out-performs the existing estimators and with a squared error that approaches the Cramer-Rao lower bound previously derived in the literature, in the limit where that bound is valid.
The best performance of a Mach-Zehnder interferometer is achieved with the input state $\ensuremath{\mid}{N}_{T}∕2+1⟩\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩+\ensuremath{\mid}{N}_{T}∕2\ensuremath{-}1⟩\ensuremath{\mid}{N}_{T}∕2+1⟩$, NT{N}_{T}NT​ being the total number of atoms or photons. This gives (i) a phase-shift error confidence ${C}_{68%}=2.67∕{N}_{T}$ with (ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling $\ensuremath{\sim}1∕{N}_{T}$ but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.