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Physical Review Letters

Violation of Bell's Inequality by a Generalized Einstein-Podolsky-Rosen State Using Homodyne Detection

A. Kuzmich, Ian A. Walmsley, L. Mandeļ

Published Aug 14, 2000• 102 citations
Publisher

Abstract

Using homodyning with weak coherent fields and photon counting, we have observed violations of Bell-type inequalities by the generalized Einstein-Podolsky-Rosen state produced in a pulsed nondegenerate optical parametric amplifier, as predicted by Grangier et al. [Phys. Rev. A 38, 3132 (1988)]. The maximum observed visibility of the interference pattern was (89+/-4)%. This interference can be regarded as a manifestation of nonlocality in the sense described by Banaszek and Wodkiewicz [Phys. Rev. A 58, 4345 (1998)]. We have investigated the interference both theoretically and experimentally and have measured the influence of dispersion and phase matching.

DOI
10.1103/physrevlett.85.1349

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Using homodyning with weak coherent fields and photon counting, we have observed violations of Bell-type inequalities by the generalized Einstein-Podolsky-Rosen state produced in a pulsed nondegenerate optical parametric amplifier, as predicted by Grangier et al. [Phys. Rev. A 38, 3132 (1988)]. The maximum observed visibility of the interference pattern was (89+/-4)%. This interference can be regarded as a manifestation of nonlocality in the sense described by Banaszek and Wodkiewicz [Phys. Rev. A 58, 4345 (1998)]. We have investigated the interference both theoretically and experimentally and have measured the influence of dispersion and phase matching.
An optical Einstein-Podolsky-Rosen (EPR) state is produced in the spontaneous parametric down-conversion of an ultrashort classical pump pulse. It is shown both theoretically and experimentally that this results in the violation of a Bell-type inequality of the kind proposed by Grangier et al. The experiment is based on measuring interference between the light in the EPR state and weak light pulses in a coherent state. The maximum observed visibility of the interference pattern was $(89\ifmmode\pm\else\textpm\fi{}4)%.$ This interference can be regarded as a manifestation of nonlocality in the sense described by Banaszek and Wodkiewicz, even though the EPR state has a positive Wigner function. We develop the theory of two-channel homodyne detection with ultrafast pulses and then apply it to the EPR state generated in a type-II collinear geometry. Particular attention is paid to the investigation of how dispersion and phase matching influence the violation of local realism.
A single-photon incident on a beam splitter produces an entangled field state, and in principle could be used to violate a Bell inequality, but such an experiment (without postselection) is beyond the reach of current experiments. Here we consider the somewhat simpler task of demonstrating Einstein-Podolsky-Rosen (EPR) steering with a single photon (also without postselection). We demonstrate that Alice's choice of measurement on her portion of the entangled state can affect Bob's portion of the entangled state in his laboratory, in a sense rigorously defined by us and Doherty [Phys. Rev. Lett. 98, 140402 (2007)]. Previous work by Lvovsky and coworkers [Phys. Rev. Lett. 92, 047903 (2004)] has addressed this phenomenon (which they called remote preparation) experimentally using homodyne measurements on a single photon. Here we show that, unfortunately, their experimental parameters do not meet the bounds necessary for a rigorous demonstration of EPR steering with a single photon. However, we also show that modest improvements in the experimental parameters, and the addition of photon counting to the arsenal of Alice's measurements, would be sufficient to allow such a demonstration.
Over the past three decades, numerous demonstrations of the violation of Bell’s inequalities have been reported in optical experiments based upon observations of polarization correlations for photon pairs. Without exception, these measurements as well as those in other systems have involved discrete variables for which the Bell inequalities are applicable. By contrast, an experiment to demonstrate the original proposal by Einstein, Podolsky, and Rosen (EPR)1 for a system of observables with a continuous spectrum has not previously been realized. From many possible theoretical avenues, we have followed the suggestion of Reid and Drummond2 to achieve a demonstration of the EPR paradox for continuous variables by employing a subthreshold optical parametric oscillator (OPO) operated in a frequency-degenerate but polarization-nondegenerate mode.
We discuss violation of Bell inequalities by the regularized Einstein-Podolsky-Rosen (EPR) state, which can be produced in a quantum optical parametric down-conversion process. We propose an experimental photodetection scheme to probe nonlocal quantum correlations exhibited by this state. Furthermore, we show that the correlation functions measured in two versions of the experiment are given directly by the Wigner function and the Q function of the EPR state. Thus, the measurement of these two quasidistribution functions yields a novel scheme for testing quantum nonlocality.
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.
For measurements describable by a discrete set of possibilities, numerous demonstrations of the violation of Bell's inequalities have been reported in optical physics based on observations of polarization correlations for photon pairs. By contrast for observables with a continuous spectrum, there appears to be no general inequalities corresponding to those of Bell and no experimental realization of the original Einstein-Podolsky-Rosen paradox. Following Reid and Drummond,1 we are attempting to demonstrate the EPR paradox in the context of the measurement of continuous quantum variables with a subthreshold optical parametric oscillator operated in a frequency degenerate but polarization nondegenerate mode. In this case, the EPR paradox is formulated in terms of two sets of quadrature phase amplitudes for the spatially separated signal and idler fields. As the parametric gain is increased toward threshold, the quadrature amplitudes of the signal and idler beams become increasingly faithful quantum copies of one another. Our measurements of quadrature amplitudes are carried out using independent balanced homodyne detectors at two sites. When operated for the detection of squeezing at one site, noise reductions of 4 dB relative to the vacuum state limit are observed, which is most encouraging since noise reductions >3 dB in the difference photocurrent from the two sites are sufficient for inference at a distance of a signal observable from a measurement of the corresponding idler variable.