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arXiv:1005.5038 · Physical Review A

Heisenberg-limited interferometry with pair coherent states and parity measurements

Christopher C. Gerry, Jihane Mimih

Published Jul 26, 2010• 72 citations
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Abstract

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.

DOI
10.1103/physreva.82.013831

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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.
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.
We theoretically investigate the quantum enhanced metrology using two-mode squeezed twin-Fock states and parity detection. Our results indicate that, for a given initial squeezing parameter, compared with the two-mode squeezed vacuum state, both phase sensitivity and resolution can be enhanced when the two-mode squeezed twin-Fock state is considered as an input state of a Mach–Zehnder interferometer. Within a constraint on the total photon number, although the two-mode squeezed vacuum state gives the better phase sensitivity when the phase shift φ to be estimated approaches to zero, the phase sensitivity offered by these non-Gaussian entangled Gaussian states is relatively stable with respect to the phase shift itself. When the phase shift slightly deviates from φ = 0, the phase sensitivity can be still enhanced by the two-mode squeezed twin-Fock state over a broad range of the total mean photon number where the phase uncertainty is still below the quantum standard noise limit. Finally, we numerically prove that the quantum Cramér–Rao bound can be approached with the parity detection.
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.
We review the use of parity measurements to obtain Heisenberg-limited phase shift measurements from interferometry performed with maximally path entangled two mode states containing N photons, contrasting this measure with the method of coincidence detection which requires a number of such detections that scales with N. For large N, coincidence measurements will become impractical. In contrast, the parity method we propose is independent of N. In addition, with the parity approach, one can use an entanglement of a coherent state and the vacuum to approach the Heisensberg limit of phase resolution without the requirement of first producing a Fock state of large photon number. We also show in a very simple way, using Maximally entangled states and parity measurements that it is possible to go beyond the Heisenberg limit if the phase shifts are generated by a nonlinear Kerr-like interaction.
Parity measurement-based quantum optical interferometry and metrology is known to be sensitive to the joint photon number distribution, , of the two-mode quantum states inside the interferometer. In cases where the statistics display a clustering of the populations around the borders (i.e. along the axes ) with a plateau over most of the interior, Heisenberg-limited (or super-sensitive) measurements of phase shifts are enabled. This follows from the heuristic number-phase uncertainty relation because of the uncertainty in the location of the photons being ΔN ∼ . As an example of the connection between the photon statistics and Heisenberg-limited phase uncertainty, we study parity measurement-based interferometry of the two-mode Barut–Girardello coherent states for SU(1,1)SU(1,1) obtained from the coupling together of two representations of SU(1,1) each associated with single-mode boson realizations with Bargmann indices as discussed by Gerry and Benmoussa [Phys. Rev. A 2000, 62, 033812]. These states generally have distributions with clustered populations along the borders. We demonstrate that these states do, indeed, lead to super-sensitive measurements and also lead to super-resolved interference fringes.
The canonical Mach-Zehnder interferometer fed with a coherent state and a squeezed-vacuum state of equal intensities is theoretically predicted to achieve Heisenberg scaling in phase sensitivity. However, this ultimate performance is unattainable using direct photon-number-difference detection due to a divergence arising precisely at the optimal equal-intensity regime. In this work, we introduce a dual-squeezing approach that overcomes this fundamental limitation. Our scheme employs an additional single-mode squeezer before detection, forming a paired configuration with the input squeezer used to generate the squeezed-vacuum state. We analytically demonstrate that the resulting dual-squeezing Mach-Zehnder interferometer enables Heisenberg-limited phase sensitivity with di rect photon-number-difference detection, while remaining robust against detection noise. Our work provides a feasible and robust route toward quantum-limited interferometric phase measurements