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Physical Review A

Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements

Richard A. Campos, Christopher C. Gerry, Adil Benmoussa

Published Aug 25, 2003• 217 citations
Publisher

Abstract

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 $N.$

DOI
10.1103/physreva.68.023810

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Holland and Burnett [Phys. Rev. Lett. 71, 1355 (1993)] have argued that twin Fock states of equal photon number NNN injected at both input ports of a Mach-Zehnder interferometer lead to phase measurements with accuracies approaching the Heisenberg limit \ensuremathΔ\ensuremathφHL=1/(2N).\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).\ensuremathΔ\ensuremathφHL​=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.N.N.
Phase measurement using a lossless Mach–Zehnder interferometer with certain entangled N-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.
We discuss the prospects of performing Heisenberg-limited quantum sensing and metrology using a Mach-Zehnder interferometer with input states that are superpositions of twin-Fock states and where photon number parity measurements are made on one of the output beams of the interferometer. This study is motivated by the experimental challenge of producing twin-Fock states on opposite sides of a beam splitter. We focus on the use of the so-called pair coherent states for this purpose and discuss a possible mechanism for generating them. We also discuss the prospect of using other superstitions of twin-Fock states, for the purpose of interferometry.
It is known that the phase measurement using lossless Mach-Zehnder interferometer with certain entangled N-photon states can achieve a phase sensitivity of the order of 1/N, the Heisenberg limit. However, the output observable used is different for different input states to achieve the Heisenberg limit. In this paper, we show that it is possible to achieve this limit just by parity measurement for all the commonly proposed entangled states. Taking account of path absorption effect, the phase shifts with parity measurement for these states are obtained and indicate that the N00N state still remains to be the best candidate to perform phase detection if the transmittance of the medium is not too small and the number of photons is not very large.
The phase sensitivity of the coincidence detection in one output port of a Mach-Zehnder interferometer is analysed for twin Fock state inputs. Firstly, the ideal detectors with quantum efficiency of unity are assumed for the detection of the output photons. The sensitivity is found out to be independent of the photon number of input light, which means that the Heisenberg limit cannot be reached in the coincidence detection even with ideal detectors. Secondly, the practical detectors with quantum efficiencies less than unity are discussed.
Quantum phase measurement with multiphoton twin-Fock states has been shown to be optimal for detecting equal numbers of photons at the output ports of a Mach–Zehnder interferometer (i.e., the so-called single-fringe detection), since the phase sensitivity can saturate the quantum Cramér–Rao lower bound at certain values of phase shift. Here we report a further step to achieve a global phase estimation at the Heisenberg limit by detecting the particle-number difference (i.e., the J ^ z measurement). We show the role of experimental imperfections on the ultimate estimation precision with the six-photon twin-Fock state of light. Our results show that both the precision and the sensing region of the J ^ z measurement are better than those of the single-fringe detection, due to combined contributions of the measurement outcomes. We numerically simulate the phase estimation protocol using an asymptotically unbiased maximum likelihood estimator.