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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Heisenberg-limited quantum sensing and metrology with superpositions of twin-Fock states

Christopher C. Gerry, Jihane Mimih

Published Feb 10, 2011• 1 citation
Publisher

Abstract

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.

DOI
10.1117/12.873201

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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.
After reviewing parity-measurement-based interferometry with twin Fock states, which allows for supersensitivity (Heisenberg limited) and super-resolution, we consider interferometry with two different superpositions of twin Fock states, namely, two-mode squeezed vacuum states and pair coherent states. This study is motivated by the experimental challenge of producing twin Fock states on opposite sides of a beam splitter. We find that input two-mode squeezed states, while allowing for Heisenberg-limited sensitivity, do not yield super-resolutions, whereas both are possible with input pair coherent states.
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 $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{HL}}=1/(2N).$ 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.N.N.
Photon number states are assigned a parity of +1 if their photon number is even and a parity of −1 if odd. The parity operator, which is minus one to the power of the photon number operator, is a Hermitian operator and thus a quantum mechanical observable although it has no classical analogue, the concept being meaningless in the context of classical light waves. In this paper we review work on the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00N states, and states obtained by injecting twin Fock states into a beam splitter. With such states and with the performance of parity measurements on one of the output beams of the interferometer, one can breach the standard quantum limit, or shot-noise limit, of sensitivity down to the Heisenberg limit, the greatest degree of phase sensitivity allowed by quantum mechanics for linear phase shifts. Heisenberg limit sensitivities are expected to eventually play an important role in attempts to detect gravitational waves in interferometric detection systems such as LIGO and VIRGO.
Nonlinear interferometers are Mach-Zehnder interferometers with Kerr media in either one or both arms. We refer to these devices, respectively, as the asymmetric and symmetric nonlinear interferometers. In the asymmetric case, with one input mode in the vacuum, it is possible to generate maximally entangled photonic states or superpositions of such states. We consider the device as a resource of entangled states for applications to Heisenberg-limited interferometry. Interferometry with the maximally entangled states cannot be performed by simply subtracting the output photocounts as in standard interferometry. Instead, one must perform parity measurements on only one of the output beams. We show that the symmetric nonlinear interferometer, with one input mode in the vacuum, may be used to perform such parity measurements. The same device is shown to produce, with an input coherent state and upon projective measurements, even or odd coherent states, examples of the Schr\"odinger-cat states.
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.