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Physical Review Letters

Interferometric detection of optical phase shifts at the Heisenberg limit

Murray Holland, Keith R. Burnett

Published Aug 30, 1993• 866 citations
Publisher

Abstract

We show that the uncertainty in the relative quantum phase of two fields propagating in the arms of a Mach-Zehnder interferometer can be reduced to the Heisenberg limit by driving the interferometer with two Fock states containing equal numbers of photons. This leads to a minimum detectable phase shift far below that of any interferometer driven by a coherent light source.

DOI
10.1103/physrevlett.71.1355

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We show that the uncertainty in the relative quantum phase of two fields propagating in the arms of a Mach-Zehnder interferometer can be reduced to the Heisenberg limit by driving the interferometer with two Fock states containing equal numbers of photons. This leads to a minimum detectable phase shift far below that of any interferometer driven by a coherent light source.
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 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 discuss an adaptive interferometric approach to quantum estimation of phase shifts. In a first step we find optimized two-mode input states for a generalized Mach-Zehnder interferometer. These two states show a small phase variance at the so-called Heisenberg limit. It turns out that they can be generated by entangling a coherent state and a squeezed vacuum at a beam splitter. Moreover, the interferometric setup allows us to describe an adaptive estimation scheme for arbitrary phase shifts if we assume a fixed mean photon number as a quantum resource.
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.
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