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arXiv:quant-ph/9802023

Homodyne measurement of exponential phase moments for quantum-phase reconstruction

Mohammed Dakna, Gerd Breitenbach, J. Mlynek, Tomáš Opatrný, Stephan Schiller, D.‐G. Welsch

Published Feb 9, 1998• 0 citations
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Abstract

We directly sample the exponential moments of the canonical phase for various quantum states from the homodyne output. The method enables us to study the phase properties experimentally, without making the detour via reconstructing the density matrix or the Wigner function and calculating the phase statistics from them. In particular, combing the measurement with a measurement of the photon-number variance, we verify fundamental number-phase uncertainty.

DOI
10.48550/arxiv.quant-ph/9802023

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We directly sample the exponential moments of the canonical phase for various quantum states from the homodyne output. The method enables us to study the phase properties experimentally, without making the detour via reconstructing the density matrix or the Wigner function and calculating the phase statistics from them. In particular, combing the measurement with a measurement of the photon-number variance, we verify fundamental number-phase uncertainty.
We investigate exponential phase moments of the s-parametrized quasidistributions (smoothed Wigner functions). We show that the knowledge of these moments as functions of s provides, together with photon-number statistics, a complete description of the quantum state. We demonstrate that the exponential phase moments can be directly sampled from the data recorded in balanced homodyne detection and we present simple expressions for the sampling kernels. The phase moments are Fourier coefficients of phase distributions obtained from the quasidistributions via integration over the radial variable in polar coordinates. We performed Monte Carlo simulations of the homodyne detection and we demonstrate the feasibility of direct sampling of the moments and subsequent reconstruction of the phase distribution.
We show that the exponential phase moments associated with the s-ordered generalized Wigner function can be sampled from homodyne data for s 0. The tomographic sampling functions are evaluated for non-unit homodyne quantum efficiency, and the statistical reliability of the reconstruction is tested by Monte Carlo simulated experiments. The reconstruction of the s-ordered phase distribution from sampled moments is discussed, and the reduction of statistical fluctuations by adaptive techniques is also analysed.
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.
We consider double balanced homodyne detection schemes with and without (local oscillator) phase-randomized detection. We discuss the reconstruction of the photon statistics from phase-randomized measurements. We show how the Wigner function of a photon-number state can be synthesized from phase-randomized double homodyne measurements of two properly prepared field states. Moreover we propose a new procedure to determine the whole density matrix from the Q function. Finally we express the Q function in terms of normally ordered moments and discuss their reconstruction.