Note that although the idea "vacuum/zero-point energy" can sometimes be useful, it is a *fiction*, cannot be relied upon, and can be particularly misleading in the context of nonlinear interactions (see e.g.:

Article Testing quantum mechanics using third-order correlations

Notably, if you use the coherent-state basis positive-P representation of an OPA process, the basis states already contain the vacuum, and so no "vacuum noise"-like terms appear; although there are instead nonlinear noise terms, which allow for the spontaneous emission.

If you represent the quantum uncertainty by the simple addition of statistical noise to your field amplitudes, you have in effect a hidden-variable theory (see e.g. the "stochastic electrodynamics" alternative to QED).

There is, however, a difference between spontaneously generated signal/idler amplitudes, and coherently generated ones: (a) Spontaneous one are matched in the signal and idler, and depend only on the pump field amplitude and phase, (b) coherent ones depend on the pump and signal (for idler) and pump and idler (for signal). However, if the existing states of the pump, signal, and idler are well defined coherent states, so will any coherently generated field be. I do not therefore see that such generation will be any less "entangled" than spontaneous generation; although (depending on specified signal/idler phases), it might not be an entanglement identical to a spontaneously generated fields.

Thank you Paul for your answer. Is there then a metric to quantify the degree to which a pair of photons are "entangled" and could one estimate this for the cases of SPDC and OPA? In the case of OPA, the pre existing 'seed' photon which gets amplified, would be a thermal photon. So in terms of any possible degree of entanglement, would this be any different to that case where the seed photon is from a coherent man-made oscillator?

Hi Neil,

No, I cannot think of any reason why the origin of a (specific) seed photon would affect the degree of entanglement.

However, note that what you are imagining to be a "seed photon" might not be in a pure basis state, it could be a mixture of some kind. It might be (indeed, it seems likely) that a thermal photons would not be an exact "thermal photon", and would be more likely to a mixture of different photon numbers, than a photon generated by a specialized single photon source [1].

[1] NB: if you assume a laser beam is in a coherent state, remember that these are never a single photon state; but a coherent combination of all photon numbers.

Hi Paul,

thank you for that, yes agreed.

I'm happy that SPDC signal and idler are entangled, so now 'all' that is needed now is some mathematical proof to show that OPA signal and idler photons are entangled or not.