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(2) interactions. Furthermore, we have shown how the low-gain limit of the polarization-entangled Gaussian state is well approximated by vacuum plus biphoton components [15]. Because our down-converter operates in this low-gain regime, it is appropriate to describe its nonvacuum output component in terms of the probability amplitudes of a biphoton state. Because the biphoton description is more intuitive than the rigorous Gaussian-state characterization, we have chosen to use the former throughout this paper.
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The biphoton state from Sec. II is the leading (first-order) nonvacuum term when the Gaussian state produced by SPDC is expanded in the number-state basis. That ∼1% of our coincidences are due to double-pair occurrences represents the contribution of the second-order term in this number-state expansion. In Ref. [18] we used the Gaussian-state treatment of SPDC to analyze a quantum-interference measurement similar to that of Fig. 2. The same analysis procedure can be used for the Bell's inequality configuration. When the coincidence gate is short enough that the probability of a singles count in that time window is low, then the Gaussian-state results reduce to those obtained from the biphoton description.
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