Predictions of Violations of Bell’s Inequality in an S-port Homodyne Detector

Since Bell’s work [ I] was published a number of experiments [2-l 11 have been undertaken to test Bell’s inequality. Some of these experiments made use of correlated spin systems [ IO,1 1 ] such as the ones described by Bohm [ 121 and Bell [ 11. However, the system used in the majority of these experiments [ 2-91 consisted of a pair of photons correlated via their polarizations. The correlated twophoton state is essentially the same as the spin singlet state. Such a state can be produced in the two photons emitted successively in a cascade atomic decay. The devices used to analyze the pair of photons are polarizing filters or polarizing beam splitters followed by photodetectors. These devices are precisely analogous to the Stern-Gerlach magnets used in Bohm’s and Bell’s gedanken experiments in that they facilitate the measurement of an arbitrary linear combination of the noncommuting observables associated with each photon polarizaton. Recently Ou. Hong and Mandel [ 13 ] have discovered a new way to produce the photon-polarization singlet state by using the output of a parametric down converter. When this output is prepared using a polarizing beam splitter the desired singlet state is produced. Once the correlated state is created the experiment proceeds the same as the other previous ones that used the cascade atomic decay. However, in another recent paper Ou and Mandel [ 141 have also shown that one can measure the same arbitrary linear combination of noncommuting observables associated with a photon polarization by translating the position of a photodetector behind a polarizer oriented at 45”, rather than leaving the detector fixed and rotating the polarizer. In this paper we consider a system of correlated electromagnetic field modes of the same polarization. Specifically the pair of modes emerging from an ordinary beam splitter. The degree of correlation between these modes is strongly dependent on the state of the field incident on the beam splitter. The correlated field modes are then analyzed with independently adjustable homodyne detectors. The homodyne detectors facilitate the measurement of the arbitrary linear combination of noncommuting observables associated with each of the correlated field modes. These devices replace the polarizing filters used in other previously considered systems. A theoretical analysis of this system leads to the derivation of a Bell inequality that holds for all possible input states. Then, by choosing different states of the field for the input to the correlating beam splitter, one can determine which states violate or satisfy the inequality. The Bell inequality provides a means of testing the predictions of quantum mechanics against the predictions of a local hidden variable theory. We will first calculate the quantum predictions for the g-port homodyne detector system. To do so consider the 8port homodyne detector illustrated in fig. 1. The beam splitter labeled A is used to produce a corre-