# Time entanglement in photon pairs

I'm studying nonlinear optics and in some processes, for example, Spontaneous Parametric Down Conversion or Spontaneous Four Wave Mixing, photon pairs are created and they say that the photons are energy and time entangled. I understood the energy entangled part, which comes easily from energy conservation, but I don't understand why there must be also time entanglement, the only reason I found is experimental, they perform coincidence measure and it can be seen that there is a zero delay between the photons' arrival. Is there a more theoretical explanation for the time entanglement? There was a similar question here, but I find the explanation with Feynman diagrams not satisfactory.

• I would think timing is the reason perfect correlation is so hard to accomplish. Timings not just about simultaneous emission but also equal distance traveled. – Bill Alsept Mar 28 '18 at 3:12
• @BillAlsept synchronizing the photons is not difficult. Not sure what you mean by perfect correlation. – ostrichCamel Mar 28 '18 at 15:51

Energy is conserved in the crystal in SPDC, so (global) energy conservation requires simultaneity of three events: (1) annihilation of the pump photon, (2) creation of SPDC photon #1, and (3) creation of SPDC photon #2. Without a projective measurement of the time at which the SPDC event occurred, emission at times $t_1$ and $t_2$ are indistinguishable, when $\lvert{t_1-t_2}\rvert<t_c,$ the coherence time of the laser. For indistinguishable events, the amplitudes add, so we expect the single-pair term of the full SPDC state at time $t_0$ to be proportional to $\int_{t_0-t_c}^{t_0} dt \lvert t,t\rangle$ (there are other terms as well, corresponding to emission of no pairs as well as emission of more than one pair). Here $\lvert t_s,t_i\rangle$ is the two-photon state corresponding to emission of photons #1 and 2 at times $t_s$ and $t_i.$
Regarding the experimental evidence you mentioned, note that simply observing simultaneous detections does not prove that the state is time-entangled. At best this shows that the state is classically correlated in time. For example, it does not rule out the possibility that (the single-pair part of) the state is described by a density matrix proportional to $\int dt \lvert t,t\rangle\langle t,t \rvert$, which is not entangled. There are other experiments that demonstrate time entanglement.