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Journal of Modern Optics

Quantum state reconstruction by multichannel unbalanced homodyning

Tomáš Opatrný, D.‐G. Welsch, Sascha Wallentowitz, W. Vogel

Published Nov 1, 1997• 14 citations
Publisher

Abstract

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.

DOI
10.1080/09500349708231891

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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.
We propose the reconstruction of the quantum state of light from the homodyne photocounting statistics of a single, realistic photodetector. Contrary to the development of homodyning over the last decade, our approach is based on unbalanced detection with a weak local oscillator. Representing the quantum state in terms of s-parametrized quasiprobability distributions, the method even allows one to determine the Wigner function provided the quantum efficiency of the detector is sufficiently large. We show that perturbing effects due to classical noise of the local oscillator are small. \textcopyright{} 1996 The American Physical Society.
A theoretical treatment is given which establishes dual-mode balanced-homodyne detection as a practical and well characterized technique for measuring optical field correlations, photon-number correlations, or the full quantum state of a pair of optical modes. The definition of modes used includes temporal wave packets, Gaussian or other beam profiles, or two-frequency fields. The proposed method allows the measurement of two-time correlations on sub-picosecond scales, the disentangling of the statistics of signal light in two spatially overlapping modes, and the measurement of field correlations, such as squeezing, over 100 THz bandwidths. We show how to estimate from the data the statistical errors on the measured correlations and the density matrix arising from finite data sets, and the errors introduced by using finite numbers of phases and relative amplitudes of the two local oscillator fields.
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.
Homodyne detection is the most effective detection scheme employed in quantum optics to characterize quantum states. It is based on mixing at a beam splitter the signal to be measured with a coherent state, called the “local oscillator,” and on evaluating the difference of the photocurrents of two photodiodes measuring the outputs of the beam splitter. If the local oscillator is much more intense than the field to be measured, the homodyne signal is proportional to the signal-field quadratures. If the local oscillator is less intense, the photodiodes can be replaced with photon-number-resolving detectors, which have a smaller dynamics but can measure the light statistics. The resulting new homodyne-like detector acquires a hybrid nature, being it capable of yielding information on both the particle-like (statistics) and wave-like (phase) properties of light signals. The scheme has been tested in the measurement of the quadratures of coherent states, bracket states and phase-averaged coherent states at different intensities of the local oscillator.