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Physical Review A

Local sampling of phase-space distributions by cascaded optical homodyning

Z. Kis, T. Kiss, József Janszky, Петер Адам, Sascha Wallentowitz, W. Vogel

Published Jan 1, 1999• 30 citations
Publisher

Abstract

We propose the determination of phase-space distributions of optical fields by cascaded optical homodyning, where phase-randomized balanced homodyning is used for measuring the output photon-number statistics of an unbalanced homodyne detection scheme. The phase-space point of interest is controlled by the complex amplitude of the local oscillator and a universal sampling function is sufficient for mapping the measured quadrature statistics onto the phase-space distribution.

DOI
10.1103/physreva.59.r39

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We propose the determination of phase-space distributions of optical fields by cascaded optical homodyning, where phase-randomized balanced homodyning is used for measuring the output photon-number statistics of an unbalanced homodyne detection scheme. The phase-space point of interest is controlled by the complex amplitude of the local oscillator and a universal sampling function is sufficient for mapping the measured quadrature statistics onto the phase-space distribution.
It is shown how the quantum phase-space distribution of a continuous-wave (cw) optical beam can be obtained independently at each point in phase space by a combination of unbalanced homodyne and balanced-heterodyne techniques. The unbalanced homodyning allows for the local sampling of phase space, whereas the heterodyne part, although introducing a loss of detection efficiency, provides a highly efficient reduction of 1/ f excess noise that is required for the sampling of cw fields. The method complements well-known techniques for light pulses by providing a robust method for cw optical beams.
The technique of dc-balanced, pulsed homodyne detection for the purpose of determining optical-field statistics on short time scales is analyzed theoretically. Such measurements provide photon-number and phase distributions associated with a repetitive signal light field in a short time window. Time- and space-varying signal and local-oscillator pulses are treated, thus generalizing earlier treatments of photoelectron difference statistics in homodyne detection. Experimental issues, such as the effects of imperfect detector balancing on (time-integrated) dc detection and the consequences of background noise caused by non-mode-matched parts of the multimode signal field, are analyzed. The Wigner, or joint, distribution for the two field-quadrature amplitudes during the sampling time window can be directly determined by tomographic inversion of the measured photoelectron distributions. It is pointed out that homodyne detection provides a new method for the simultaneous measurement of temporal and spectral information. Although the theory is generally formulated, with both signal and local-oscillator fields being quantized, emphasis is placed on the limit of a strong, coherent local-oscillator field, making semiclassical interpretation possible.
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.
Multichannel unbalanced homodyning is proposed for measuring the density matrix of a single-mode optical field in the photon-number representation. Combining the signal beam and the local oscillator by a hightransmittance beam splitter, the interfering field is detected using multichannel photocounting. The density matrix is determined by direct statistical sampling of the coincident events recorded for various values of the local-oscillator phase. The usefulness of the method is demonstrated by computer simulations of measurements including error estimations.