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Physical Review A

Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons

Taesoo Kim, Olivier Pfister, Murray Holland, Jaewoo Noh, J. L. Hall

Published May 1, 1998• 159 citations

Abstract

The feasibility of a Heisenberg-limited phase measurement using a Mach-Zehnder interferometer fed with twin photon correlated light is investigated theoretically. To take advantage of the Heisenberg limit, $1/N$, for the phase sensitivity, one wants the number of correlated photons, $N$, to be high. This favors the use of parametric oscillation rather than the weaker but better correlated source given by parametric down-conversion. In real systems, decorrelation arising from photon absorption, mode mismatch, and nonideal detectors must be considered. In this paper we address the problem of detection when correlated photons are used as the input. We study the influence of photon statistics and of imperfect quantum correlation of the input light, and show that it is still possible to break the classical $1/\sqrt{N}$ phase sensitivity limit in nonideal experimental conditions. All the results are valid in the general case of quantum correlated bosons.

DOI: 10.1103/physreva.57.4004

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The feasibility of a Heisenberg-limited phase measurement using a Mach-Zehnder interferometer fed with twin photon correlated light is investigated theoretically. To take advantage of the Heisenberg limit,

1/N

, for the phase sensitivity, one wants the number of correlated photons,

N

, to be high. This favors the use of parametric oscillation rather than the weaker but better correlated source given by parametric down-conversion. In real systems, decorrelation arising from photon absorption, mode mismatch, and nonideal detectors must be considered. In this paper we address the problem of detection when correlated photons are used as the input. We study the influence of photon statistics and of imperfect quantum correlation of the input light, and show that it is still possible to break the classical

1/\sqrt{N}

phase sensitivity limit in nonideal experimental conditions. All the results are valid in the general case of quantum correlated bosons.

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)].

We propose a Heisenberg-limited quantum interferometer whose input is twin optical beams from which one or more photons have been indistinguishably subtracted. Such an interferometer can yield Heisenberg-limited performance while at the same time giving a direct fringe reading, unlike for the twin-beam input of the Holland-Burnett interferometer. We propose a feasible experimental realization using a photon-number correlated source, such as nondegenerate parametric down-conversion, and perform realistic analyses of performance in the presence of loss and detector inefficiency.

An interferometric detection scheme to measure optical phase shift with sensitivity beyond the shot noise limit is proposed. The theoretical calculation shows that using the quantum correlated twin beams produced from an optical parametric amplifier as the input fields of a Mach-Zehnder interferometer, the minimum detectable phase shift will exceed the shot noise limit N-1/2 and approach the Heisenberg limit N-1. The parametric dependences of the minimum detectable phase shift on the nonlinear interaction, input photon number, and detection efficiency are shown.

Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] have argued that twin Fock states of equal photon number

N

injected at both input ports of a Mach-Zehnder interferometer lead to phase measurements with accuracies approaching the Heisenberg limit $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).$ However, the method of phase detection suggested by those authors, obtaining the difference of the photocurrents at the output ports of the interferometer, is not sensitive to the phase difference between the two interferometer paths; in fact, the photocurrent vanishes. In this paper we show that the use of parity measurements on just one of the output modes not only is sensitive to the phase difference but that the sensitivity approaches the Heisenberg limit for large

N.

Mach-Zehnder interferometry based on mixing the coherent and the squeezed vacuum states of light has Heisenberg limited capabilities for phase estimation. This is also, because the quantum Cramer-Rao bound on sensitivity of phase estimation with the above interferometric scheme reaches the Heisenberg limit when the inputs are mixed in hear equal proportions. We show that a detection strategy based on the measurement of parity of photon number in one of the output modes of the interferometer saturates the quantum Cramer-Rao bound of the interferometric scheme, and therefore- as a consequence- hits the Heisenberg limit when the inputs are mixed in equal intensities.