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Physical Review A

Density-matrix reconstruction by unbalanced homodyning

Tomáš Opatrný, D.‐G. Welsch

Published Feb 1, 1997• 50 citations
Publisher

Abstract

Recently a method for measuring quasidistributions of single-mode optical fields in unbalanced homo- dyne detection has been proposed [S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996); K. Bana- szek and K. W\'odkiewicz, Phys. Rev. Lett. 76, 4344 (1996)]. We show that the scheme can be used for direct sampling of the signal-mode density matrix in the Fock basis. In this case it is sufficient to vary only the phase of the local oscillator, keeping its amplitude constant.

DOI
10.1103/physreva.55.1462

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Recently a method for measuring quasidistributions of single-mode optical fields in unbalanced homo- dyne detection has been proposed [S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996); K. Bana- szek and K. W'odkiewicz, Phys. Rev. Lett. 76, 4344 (1996)]. We show that the scheme can be used for direct sampling of the signal-mode density matrix in the Fock basis. In this case it is sufficient to vary only the phase of the local oscillator, keeping its amplitude constant.
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 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.
We develop a multimode theory of direct homodyne measurements of quantum optical quasidistribution functions. We demonstrate that unbalanced homodyning with appropriately shaped auxiliary coherent fields allows one to sample point-by-point different phase-space representations of the electromagnetic field. Our analysis includes practical factors that are likely to affect the outcome of a realistic experiment, such as nonunit detection efficiency, imperfect mode matching, and dark counts. We apply the developed theory to discuss feasibility of observing a loophole-free violation of Bell's inequalities by measuring joint two-mode quasidistribution functions by photon counting under locality conditions. We determine the range of parameters of the experimental setup that enable violation of Bell's inequalities for two states exhibiting entanglement in the Fock basis: a one-photon Fock state divided by a 50:50 beam splitter, and a two-mode squeezed vacuum state produced in the process of nondegenerate parametric down-conversion.
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.