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Physical Review A

Interferometers and minimum-uncertainty states

Mark S. Hillery, Leonard D. Mlodinow

Published Aug 1, 1993• 97 citations
Publisher

Abstract

Minimum-uncertainty states in variables which are quadratic in mode creation and annihilation operators can be used to increase the accuracy of interferometric measurements. For a general interferometer we show how to find the relevant uncertainty relation. We consider two specific examples, a Mach-Zehnder interferometer and a Mach-Zehnder interferometer in which the first beam splitter has been replaced by a four-wave mixer. For both devices su(2) squeezed minimum-uncertainty states can be used to achieve phase-measurement accuracies of 1/N, where N is the total photon number at the input. We also describe a method of producing approximate versions of these states.

DOI
10.1103/physreva.48.1548

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Minimum-uncertainty states in variables which are quadratic in mode creation and annihilation operators can be used to increase the accuracy of interferometric measurements. For a general interferometer we show how to find the relevant uncertainty relation. We consider two specific examples, a Mach-Zehnder interferometer and a Mach-Zehnder interferometer in which the first beam splitter has been replaced by a four-wave mixer. For both devices su(2) squeezed minimum-uncertainty states can be used to achieve phase-measurement accuracies of 1/N, where N is the total photon number at the input. We also describe a method of producing approximate versions of these states.
As was originally pointed out by Caves, squeezed light can be used to increase the accuracy of interferometric phase measurements. This analysis was carried further by Yurke, McCall, and Klauder, who used group theory to examine the ultimate sensitivity of conventional interferometers and also those constructed from four-wave mixers. Here we consider interferometers composed of mixtures of beam splitters, four-wave mixers, and degenerate parametric amplifiers. We show that each interferometer can be characterized by an uncertainty relation whose minimum uncertainty states can be used to explore its ultimate accuracy. The variables in the uncertainty relation are quadratic in the mode operators, which allows us to connect the study of improved interferometric measurements to recent work on higher-order squeezing.
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The output state of a nonlinear Mach-Zehnder interferometer is shown to be an effective number-phase minimum-uncertainty state with reduced photon-number uncertainty. This interferometer includes an optical Kerr medium in one arm with a coherent-state input. Unusual ``crescent''-shaped squeezing which preserves photon number is revealed in the unitary evolution associated with a self-phase-modulation in the Kerr medium. Photon-number uncertainty 〈\ensuremath{\Delta}n${^}^{2}$〉 can be reduced by interference with a coherent-state reference wave. It can be minimized to 〈n^〉1/3{〉}^{1/3}〉1/3, far below the limit 〈n^〉2/3{〉}^{2/3}〉2/3 achieved by an ordinary squeezed state. The increased phase uncertainty due to self-phase-modulation and the reduced photon-number uncertainty still preserve the minimum-uncertainty product 〈\ensuremath{\Delta}n${^}^{2}$〉〈\ensuremath{\Delta}\ensuremath{\Phi}${^}^{2}$〉\ensuremath{\sim}(1/4). .AE
Photon counting measurements are analyzed for obtaining a classical phase parameter in a linear Mach–Zehnder interferometer (MZI) by the use of phase estimation theories. The detailed analysis is made for four cases: (a) coherent states inserted into the interferometer, (b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port, (c) coherent state inserted into one input port of the interferometer and squeezed-vacuum state into the other input port, and (d) exchanging the first beam splitter of an MZI by a nonlinear system that inserts a NOON (representing a superposition of N particles in the first mode with zero particles in the second mode, and vice versa) state into the interferometer and by using photon counting for parity measurements. The properties of photon counting for obtaining minimal phase uncertainties for the above special cases and for the general case are discussed.
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.
We have studied the sensitivity of four different phase shift measurement schemes with a Mach-Zehnder interferometer. The input light is considered to be in a coherent state and the detectors are assumed to be ideal with the quantum efficiency of unity. It is shown by direct calculation of the operators corresponding to the measurement schemes that the uncertainty of the phase-shift measurement is limited to the classical one 1m\frac{1}{\sqrt{m}}m​1​ (m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.