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Physical Review A

Two-mode quantum-optical state measurement: Sampling the joint density matrix

Michael G. Raymer, D. F. McAlister, Ulf Leonhardt

Published Sep 1, 1996• 76 citations
Publisher

Abstract

We propose schemes for measuring the joint quantum state of a two-mode optical field using balanced homodyne detection, thereby generalizing earlier single-mode schemes. We distinguish between two cases---the case where the two modes are spatially separable and the case where they are nonseparable. For each case we present a generalization of the direct sampling method of state reconstruction and point out that its advantages over inverse Radon methods are even greater here than in the single-mode case. In the case of two modes that cannot be separated we propose a scheme in which a single balanced homodyne detector is used for data collection. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.2397

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International Conference on Ultrafast Phenomena•Jan 1, 1996•0 citations
We propose schemes for measuring the joint quantum state of a two-mode optical field using balanced homodyne detection, thereby generalizing earlier single-mode schemes. We distinguish between two cases---the case where the two modes are spatially separable and the case where they are nonseparable. For each case we present a generalization of the direct sampling method of state reconstruction and point out that its advantages over inverse Radon methods are even greater here than in the single-mode case. In the case of two modes that cannot be separated we propose a scheme in which a single balanced homodyne detector is used for data collection. \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.
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.
Ultimately the properties of all optical fields are contained in the density matrix that specifies their quantum mechanical state. It has only recently been demonstrated that it is possible to actually reconstruct this density matrix from measured data, and to do so with temporal resolution of a few hundred femtoseconds.[1] This technique, known as Optical Homodyne Tomography (OHT), is unfortunately restricted to the measurement of only single mode radiation. In fact it selects out from the test field only those components which overlap temporally with the local oscillator pulse mode.