Let us analyze the situation sketched in Fig. 1. The pump laser passes through two nonlinear crystals, labeled crystal 1 and crystal 2. Twin photons can be produced in each one of the crystals. Signal and idler photons produced in crystal 1 are directed to detectors A and B respectively, so that coincidence between signal and idler channels can be measured. Suppose that degenerate photon pairs produced in crystal 2, can also be directed to the same detectors. This condition is simply fulfilled by tilting crystal 2 relatively to the vertical axis.
The situation described above is suitable for quantum interference. Note that the time of emission of photon pairs cannot be specified, since it is a spontaneous emission and the coherence length of the pump laser is larger than the distance between crystals. In this case, coincidence counts produced by photon pairs originated in crystal 1 are indistinguishable from those of crystal 2. Interference fringes in the coincidence counting rate can be observed, as long as the phase difference between these two probabilities is varied. As each crystal works like an extended source, no interference is observed for the individual intensities.
This interference process can be described in a simplified form with the use of a monomode quantum approach. It is enough to explain the main properties of the coincidence patterns, including the effective wavelength. However, for taking into account for the degree of coherence and its consequences in the visibility of the fringes, a multi-mode theory would be necessary. In this work, we will restrict ourselves to the simpler case.
The quantum state of the field produced by both crystals is given by:
The electric field operators for signal and idler modes at the detection planes are given by:
where fjx is the phase of the field at the emission point, with j = 1,2 and x = s,i and k is the wave number for both signal and idler modes. rjx is the distance between crystal j and the detector at the x side.
The coincidence counting rate can be easily calculated:
From the phase matching conditions we have that f1i + f1s = f1p and f2i + f2s = f2p, where f1p and f2p are the pump laser phases at crystals 1 and 2 respectively. We see that one condition for observing interference is that f1p-f2p = const. That is to say the coherence length of the pump laser must be larger than the distance between crystals.
Eq. 3 can be put in the form:
With the aid of Eqs. 3 and 4 it is clearly seen that the interference fringes are sensible to phases that depend on the paths from crystals 1 and 2 to detectors. It is worth noting that phase di can be varied independently from ds. Displacing signal or idler detector one can vary each one of these phases. Consider the case where di = ads. In this case, the variable phase in Eq. 4 can be written as:
The parameter aa