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Annual Meeting Optical Society of America

Violations of Bell’s Inequality in an eight-port receiver

Brian J. Oliver, Carlos R. Stroud

Published Jan 1, 1988• 0 citations
Publisher

Abstract

It is shown that a system of correlated electromagnetic field modes in an eight-port balanced homodyne detector is useful for testing Bell's inequality. This system is different from all previously considered tests of Einstein locality that use photons1-3 in that the correlated field modes present in the eight-port are all of the same polarization, and no polarizing filters are used. The variables that replace the angular orientation of the polarizing filters in the other systems are the optical path lengths between the beam splitters in the eight-port. A variety of different states of the free electromagnetic field may be tested for violations of locality using this system. It is shown that combinations of a number state and any other free- field state as inputs to the eight-port can demonstrate violations of the inequality, thus indicating the failure of locality for a specific set of input states.

DOI
10.1364/oam.1988.mm3

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It is shown that a system of correlated electromagnetic field modes in an eight-port balanced homodyne detector is useful for testing Bell's inequality. This system is different from all previously considered tests of Einstein locality that use photons1-3 in that the correlated field modes present in the eight-port are all of the same polarization, and no polarizing filters are used. The variables that replace the angular orientation of the polarizing filters in the other systems are the optical path lengths between the beam splitters in the eight-port. A variety of different states of the free electromagnetic field may be tested for violations of locality using this system. It is shown that combinations of a number state and any other free- field state as inputs to the eight-port can demonstrate violations of the inequality, thus indicating the failure of locality for a specific set of input states.
The balanced homodyne detector has made possible the detection of squeezed states of light. The essential feature of this detector is that it allows measurement of an arbitrary linear combination of the canonically conjugate observables of the electromagnetic field. We show that it is precisely this feature of the detector which provides a means for studying EPR nonlocality and Bell's inequality. A simple physical statement of this is that a homodyne detector is to its signal field as a Stern-Gerlach magnet is to a spin. Thus a pair of homodyne detectors can be used in the same manner as the magnets in Bohm’s gedanken experiment for studying the EPR question. The quantum mechanical state which demonstrates a violation of Bell's inequality in this system has correlations between the modes entering the two homodyne detectors. It should be noted that this correlation is not between the polarizations of the modes, as in all past EPR experiments using light, rather it is between the occupation numbers of the two modes.
We provide a detailed analysis of the recently proposed setup for a loophole-free test of Bell inequality violation using conditionally generated non-Gaussian states of light and balanced homodyning. In the proposed scheme, a two-mode squeezed vacuum state is de-Gaussified by subtracting a single photon from each mode with the use of an unbalanced beam splitter and a standard low-efficiency single-photon detector. We thoroughly discuss the tolerance of the achievable Bell violation in the various experimentally relevant parameters such as the detector efficiencies, the electronic noise, the mixedness of the initial Gaussian state, and the probability of false triggers. We also consider several alternative schemes involving squeezed states, linear optical elements, conditional photon subtraction, and homodyne detection.
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.
A Bell inequality is a fundamental test to rule out local hidden variable model descriptions of correlations between two physically separated systems. There have been a number of experiments in which a Bell inequality has been violated using discrete-variable systems. We demonstrate a violation of Bell's inequality using continuous variable quadrature measurements. By creating a four-mode entangled state with homodyne detection, we recorded a clear violation with a Bell value of B=2.31±0.02. This opens new possibilities for using continuous variable states for device independent quantum protocols.
Nonlocal effects associated with quantum-mechanical states have been theoretically interesting because they emphasize the differences between classical and quantum mechanics. We describe a system in which a single input photon causes a nonlocal influence on two spatially separated homodyne detectors, as evidenced by a violation of Bell's inequalities. Unlike previous configurations for violating Bell's inequalities, the correlation is not imposed on a pair of particles that spatially separate before being detected. Rather, the correlation arises from the interaction of the local oscillators at each homodyne detector with the spatially extended field of the input photon. The above effect is only one of a class of situations in which nonlocal correlations can be imposed on separated homodyne detectors by quantum mechanical fields. By deriving the conditions for violating Bell's inequality with phase-sensitive detectors, new tests can be developed that relate to squeezed light and other nonclassical fields.