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Physical Review A

Test of Bell's inequalities with harmonic oscillators

Shahen Hacyan

Published Apr 1, 1997• 1 citation
Publisher

Abstract

We present another version of the Bell inequality experiment employing four harmonic oscillators proposed by Yurke and Stoler [Phys. Rev. A 51, 3437 (1995)]. In the present gedanken experiment, the interaction time between two oscillators plays the same role as the polarization angle in the standard version of the experiment employing two photons. We obtain a correlation between two causally disconnected pairs of oscillators which violates Bell's inequalities.

DOI
10.1103/physreva.55.r2492

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We present another version of the Bell inequality experiment employing four harmonic oscillators proposed by Yurke and Stoler [Phys. Rev. A 51, 3437 (1995)]. In the present gedanken experiment, the interaction time between two oscillators plays the same role as the polarization angle in the standard version of the experiment employing two photons. We obtain a correlation between two causally disconnected pairs of oscillators which violates Bell's inequalities.
Most experimental tests of Bell inequalities have involved measurements of correlations between two photons. One of the first definitive experiments involved the linear polarization correlation between two photons in an atomic cascade of 200 Hg. That experiment is discussed in depth in this paper. The form of the initial state density matrix was a critical part of that experiment; and the experimental and theoretical determination of the density matrix, including finite solid-angle effects is discussed. However, all Bell inequality experiments to date have two widely recognized loopholes, detector efficiency and locality. This paper concludes with a very brief description of an experiment that will close both of these loopholes. In contrast to existing experiments with photons, this new test involves spin correlations between the nuclei of two 199 Hg atoms; it is an exact experimental realization of Bohm's version of the Einstein-Podolsky-Rosen gedankenexperiment.
We show that Bell's-inequality experiments can be carried out using four harmonic oscillators, pairs of which interact with each other at suitable instants of time to allow for the exchange of energy. Between interactions, the oscillators are shifted in phase relative to each other in a controllable manner. Energy becomes distributed among the four harmonic oscillators in a manner that is incompatible with a local realism picture in which information is exchanged only when the oscillators interact with each other. Entanglement is dynamically generated from an initial product state.
Correlations of linear polarizations of pairs of photons have been measured with time-varying analyzers. The analyzer in each leg of the apparatus is an acousto-optical switch followed by two linear polarizers. The switches operate at incommensurate frequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps between two orientations in a time short compared with the photon transit time. The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.
Experiments testing Bell's inequalities generally uses sources of entangled photons emitted in opposite direction, sent through polarizer of specific different orientation and finally correlated. But the correlation result would be the same if photons was emitted by only one source of 360° rotating polarization, separated in two flux by a semi-transparent mirror, and finally correlated as in the standard experiment. Therefore, it would demonstrate that the result is not due to any entangled state of the photons.
We propose a practical scheme to use photons from causally disconnected cosmic sources to set the detectors in an experimental test of Bell's inequality. In current experiments, with settings determined by quantum random number generators, only a small amount of correlation between detector settings and local hidden variables, established less than a millisecond before each experiment, would suffice to mimic the predictions of quantum mechanics. By setting the detectors using pairs of quasars or patches of the cosmic microwave background, observed violations of Bell's inequality would require any such coordination to have existed for billions of years-an improvement of 20 orders of magnitude.