In SPDC, the generation of photons inside a crystal is completely random and it behaves like a thermal source for field interference i.e. the second order (field) spatial coherence of either signal or idler is too small to observe the interference effects. Due to the correlations between signal and idler beams, one can get interference pattern with fourth order correlation which is coincidence counts.

In our publication " Twin Matter Waves for Interferometry Beyond the Classical Limit" we used a process that is the analogue of SPDC for atoms. We mixed the two modes corresponding to signal and idler mode on a beam splitter that is realized using microwave pulses. In the Bloch sphere picture we use it becomes obvious why the number of atoms at the output ports does not depend on the phase that signal and idler beam picked up, i.e. why there is no first order interference. However the fluctuations depend on the mixing angle.

No. Entanglement does not destroy coherence. Coherence is preserved both in the signal and idler beam separately and also mutually between the signal and the idler. To observe interference from one beam (signal or idler) we must put a polarizer to fix the photon polarization and send the beam in an interferometer. Fringes are then observed due to first-order field-field interference. To see field-field interference between the signal and idler we must first convet them to the same polarization and then combine them in one beam using a beam splitter. Because the two photons entering in the beam splitter are indistinguishable, they collapse by second-order photon-photon interference (as it can be tested by the lack of coincidences at the two exits of the beam spliiter) so that both photons are either transmitted or reflected. Inserting an inteferometer in one exit ports of the beam splitter (in the transmitted beam, for example) field-field first order interference can be observed. The crucial point to observe interference is making the two photons indistinguishable: then, a simple reasoning about the "which path" followed by the detected photons can be used to know when fringes should appear. The "which path" reasoning applies both to first-order (field-field) and to second.order (photon-photon) inteference.

Moreover, the arm length is taken such a big to destroy one-photon interference. Therefore an interference pattern in two-arm coincidences becomes a reliable evidence of entanglement.

Thank you for fruitful answers. I read the Dixon’s paper. It seems to me that it is an “entangled” analogue of the original Young’s experiment done with incoherent light, where a single slit is used to filter otherwise incoherent light spatially and double slit is used to create an interference pattern. In Dixon’s case, however, the single slit is placed in the idler beam and double slit in the signal beam. Interference is then observed in coincidences. Can one think about that in that way?

Reply to Enrico: I came to a contradiction after reading the Hong’s paper [J. Opt. Soc. Am. B 15, 1192 (1998)]. He explains the lack of second-order correlations in the signal beam in a purely quantum-mechanical way. He constructs the Bell’s state as |psi> ~ |1,1>s1,i1 + |1,1>s2,i2 and applies the signal field operator to that state in order to evaluate second-order correlations in the signal beam. He then concludes that no interference can be observed in the signal photon counting rate since idler states |1>i1 and |1>i2 are mutually orthogonal. But what if the transverse degree of coherence in the signal beam was high enough to see the “classical” interference? Moreover, what if I didn’t know at all there are any idler photons? I’m sorry for my classical way of thinking. That’s why I’m asking the specialists.

The possibility to observe fringes depends on the experimental setup. The correct way to understand if interference fringes occur in a given experiment is the following: start from the detector (not from the source!) and consider one photon producing a "click" on this detector. Then, if we can find no way (even in principle) to understand from where (along which path from the source) the detected photon is arrived, then we observe fringes otherwise no. This "which path" principle applies irrespective of entanglement and both for the first (field-field) and second order (photon-photon) interference. Try to apply the "which path" criterion to a given experiment with entangled photons: in some cases you find only first-order fringes, in other cases only second-order fringes and in others cases both first and second order fringes with same SPDC source. So the property of entangled light to produce fringes (i.e. its observed "coherence") depends on the apparatus and it is not intrinsic to the light itself. Notice, however, that in all two-photon experiments in the literature the entangled state produced by SPDC source is supposed to be a pure quantum state.

I said previously that entanglement does not destroy coherence, precisely because entangled states are pure states of the pair and, hence, by appropriate manipulations coherence can be recovered even if initially is not apparent in a given experiment. Pure states evolve coherently in quantum mechanics. Decoherece in a two-particle system can appear if we look to only one of the partners, but this kind of decoherence is apparent and coherence can be recoverd if we know what the other partner does. True decoherence is for mixed states arising, for example, by random phases (due to noise) in the interfering fields. In this case, there is no way to recover coherence and the description of the system by density matrix is mandatory (to recover coherence we should have complete knowledge on the quantum state of the environmental thermal bath (reservoir) producing the noise, which is impossible).

Actually，in the experiment the signal or idler photons are interfering with themself，however the path length difference between the two optical path is too big to observe the interference phenomenon. In the SPDC process, the nonlinear process happens in tens of femotoseconds, so every single photon would result in a very broad specral, in the frequency region there are many continues wave wave functions with different frequencies, for each wave function has a very good interference property, if we can use a filter to select wave functions in a narrow frequency region, we can observe the single side interference in an unbalanced interferometer. In most current experiment, people do not care the single side interference and no narrow band interferometer or not narrow enough compared with the optical path length difference, so no interference is observed. In conclusion, I would like to say, the interference property of the single side photon generated in spontaneously nonlinear optics processes, such as SPDC and SFWM processes, is very good. And it is not good to say the filter can improve the interference property of the light; it is help us to observe the interference, but the interference property is there, not caused by the filter. At last, I hope to cooperate with others to fulfill the suggested experiment demonstrating the interference property of the single side photon.

I wish to point you to our article "Engineering of two-photon spatial quantum correlations behind a double slit" WH Peeters, JJ Renema, MP van Exter - Physical Review A, 2009 which adresses precisely the question of the conditions which are necesary to observe young's fringes either in the one (singles) or two-photon (coincidences) signal.