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Journal of Optics B Quantum and Semiclassical Optics

Phase-difference measurements, distributions and the phase-difference minimum uncertainty states

Gunnar G. Bjork, Jonas Söderholm

Published Jan 1, 1999• 13 citations
Publisher

Abstract

We discuss phase-difference measurements of two bosonic modes and propose an experimental realization of the phase-difference operator by Luis and S?nchez-Soto (1993 Phys. Rev. A 48 4702). We show that a measurement of a two-mode state will collapse any state into a phase-difference eigenstate. We also show that the phase-difference number-difference commutator can be expressed in the phase-difference distribution. Finally, we discuss the associated minimum uncertainty states.

DOI
10.1088/1464-4266/1/3/303

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We discuss phase-difference measurements of two bosonic modes and propose an experimental realization of the phase-difference operator by Luis and S?nchez-Soto (1993 Phys. Rev. A 48 4702). We show that a measurement of a two-mode state will collapse any state into a phase-difference eigenstate. We also show that the phase-difference number-difference commutator can be expressed in the phase-difference distribution. Finally, we discuss the associated minimum uncertainty states.
From the experimentalist’ spoint of view, it is natural to think of aphase measurement as a measurement that involves a reference time. As a part of the system, the reference time generating “clock” must be taken into account in the quantum treatment of the measuring process. This could be argued to be the reason why there does not exist any observable corresponding to the absolute phase in the unrestricted Hilbert space.Luis and Sánchez-Soto [Phys. Rev. A 48, 4702 (1993)] have investigated the case where the “clock” is another harmonic oscillator and defined a Hermitian phase-difference operator, which is valid in the infinite Hilbert space.In the present work we propose and experimentally demonstrate a setup for direct measurement of the phase-difference. We derive the phase-difference distributions for some two-mode states and compare them with the experimental results.
It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as ${n}^{\ensuremath{-}1.5},$ where $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as ${n}^{\ensuremath{-}2}.$ It has been shown that a lower limit to the phase variance that is introduced scales as $\mathrm{ln}(\mathrm{n\ifmmode \bar{}\else \={}\fi{}})/{n}^{2}.$ Here we introduce an adaptive technique that attains this theoretical lower limit.
For the two-mode photon number-difference operator D = a†a -b†b and the Noh, Fougers and Mandel (NFM) operational phase operator [Formula: see text], we introduce the concept of number-difference-phase squeezing. We then find a new minimum-uncertainty state for number-difference-phase squeezing, which turns out to be a special combination of generalized SU(1,1) coherent state. As a by-product, a new orthonormal and complete representation in two-mode Fock space made up of R†n|m, m> is found.
We examine the operator \ensuremath{\Phi}${\mathrm{^}}_{12}$ proposed by Luis and S\'anchez-Soto [Phys. Rev. A 48, 4702 (1993)] as a possible description of the phase difference between two electromagnetic fields. We show that there is a difficulty in directly representing the phase difference between arbitrary fields by this operator, but conclude that the incorporation of the operator into an alternative definition of phase difference constitutes a valuable alternative approach to the quantum description of phase.
Experiments have been performed to determine the Wigner distribution and the density matrix (and for pure states the wave function) of a light mode, by using tomographic inversion of a set of measured probability distributions for quadrature amplitudes.From these measurements the quantum distributions of optical phase and photon number have been obtained.The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balanced homodyne detection.We refer to this new method as optical homodyne tomography.Given the measured density matrix, one can experimentally infer any of the various quantum distributions of optical phase, in particular the Pegg-Barnett (or, equivalently, Shapiro-Shepard) phase distribution, the marginal Wigner distribution, and the Vogel-Schleich operational phase distribntion.We have used this approach to make measurements of the number-plase uncertainty relation for coherent-state fields.The coherent states do not attain the minimum value for the number-phase uncertainty product, as set by the expectation value of the commutator of the number and phase operators; this is true theoretically and experimentally.