# Will a two-photon state be detected at the same spot?

Let us say I excite a particular mode $$\omega_l$$ of the electromagnetic field by means of parametric downconversion such that both of them are identical in all aspects: polarisation, direction and of course, frequency. So, I have the state: $$|\psi\rangle = |2\rangle$$

Let this state impinge on a detector array whose active area is bigger than the photons’ spatial stretch. Within this region let the probability of detection be uniform (top hat profile).

If the detectors are ideal (zero dark counts), can two independent detectors fire simultaneously? Or, given the identical nature of the photons will each instance always trigger the same detector in the array?

In other words, is the detection of a single photon within our state $$|\psi\rangle$$ an independent event?

• Is two-photon absorption allowed by the selection rules? Oct 15, 2021 at 13:54
• I’m thinking of a solid state based detector (like a photodiode) which has broad bands. So yes, two-photon absorption should be possible. Oct 15, 2021 at 14:04
• I don't mean just absorbing two photons, but absorbing them in a single process. Oct 15, 2021 at 14:26
• I meant the same. Assume both single and two photon absorption from a single process is possible. Oct 15, 2021 at 14:58

From a purely theoretical perspective, you are considering a two-photon state of the form $$|2\rangle\equiv\frac{1}{\sqrt2}(a^\dagger_\psi)^2|0\rangle$$ where $$a_\psi^\dagger=\sum_k c_k a_k^\dagger|0\rangle\equiv\sum_k c_k |k\rangle$$ creates a single-photon state that is a superposition of a bunch of spatial mode (think of $$\psi$$ as the wavepacket of the photon).

So now you are asking: suppose you can detect independently single-photon absorption at different modes. I'd model this as asking what are the probabilities $$\langle 1_i 1_j|2_\psi\rangle\equiv \langle 0|a_i a_j \frac{1}{\sqrt2}(a_\psi^\dagger)^2|0\rangle = \frac{1}{\sqrt2}\sum_{k\ell} c_k c_\ell \langle a_i a_j a_k^\dagger a_\ell^\dagger\rangle = \frac{1}{\sqrt2}\sum_{k\ell} c_kc_\ell (\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk}) = \sqrt2 c_i c_j.$$ It follows that the probability of detecting a coincidence on two different modes $$i,j$$ is $$p(i,j)=2 |c_i c_j|^2$$, where $$|c_i|^2$$ is the probability of finding $$a_\psi^\dagger$$ in the $$i$$-th mode.

Note that, on paper, there isn't really any fundamental difference between the scenario you consider, and a two-photon state delocalised on two (or more) separate spatial modes. You could generate with whatever method you like a two-photon state in a single mode, call it $$|2_i\rangle$$, and then pass it through a beamsplitter, and do coincidence measurements on the two output spatial modes. Mathematically, the scenario is identical to the one you consider (except that you can deal with easily distinguishable spatial modes, rather than pixels distinguishing positions on a single-photon spatial distribution).

In other words, on paper, sure, this is possible. I'm not aware of this having been directly observed experimentally though (I mean the scenario where you resolve pixels within a wavepacket). The thing is that single-photon detectors generally cannot resolve different spatial modes within a single-photon beam, and vice versa, pixel-resolving devices (such as CCD cameras) generally only detect intensities. The technology to do pixel-resolving single-photon detectors is relatively recent, and still under active development. See e.g. https://arxiv.org/abs/2007.16037 for some discussion of recent developments in this field. I'm also not really expert in this field, so take the above with a grain of salt.

• This is also what I thought, that the photon detection is an independent process. In your answer by modes i&j I’m guessing you’re meaning localised wavepackets. I don’t think pixel resolving detectors. If my photon mode is large enough that two detectors can fit in, we can measure this. Maybe I’ll set this up in my lab and do an experiment. Should not be that hard. Oct 20, 2021 at 7:27
• @SuperfastJellyfish $i$ and $j$ can represent either. If you are considering a single wavepacket, then it's different pixels. Pixel resolving detectors are a thing (CCD cameras being the trivial example). Having both pixel resolution and single-photon resolution is what's hard, but it also can be done (not very easily as far as I'm aware). Otherwise, doing the experiment with $i,j$ separate spatial modes isn't hard sure (well, at least as far as generating a two-photon Fock state "isn't hard")
– glS
Oct 20, 2021 at 8:19

Two photon that are produced as a pair due to the down conversion from a single pump photon would be entangled (correlated) in their transverse momentum. This is a property used in many application. What it means is that you can detect the two photons with different detector in coincidence. In other words, they can fire simultaneously due to the detection of the respective photons in the pair.

• I know that we can detect them in different places provided we spatially separate them. But I’m asking if we don’t spatially separate them, can they be detected by two different detectors (provided that they’re within the spatial mode). Oct 15, 2021 at 14:57