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For example, can a single 1 eV photon excite 0.5 eV transitions in both of two atoms that are widely separated (events outside each other's light-cones)?

In order to excite a transition, the incoming photon energy must be greater than or equal to the energy difference between the two electronic energy levels of an atom. If it's greater, then the excess energy can potentially go into re-emission of a lower-energy photon. The lower energy photon could go on to ionise another atom that might be nearby. This would be a causal process from the first atom to the second. It requires the ionization event of the second atom to be in the future light-cone of the event on the first atom. But what happens if that is not the case?

I think the question boils down to this. Can the collapse of a wavefunction encompass two seemingly simultaneous distinct events? Or, even more extreme, can the collapse of a wavefunction encompass two non causally-connected events occuring at different time-instants? Both of these possibilities seem unlikely and would require some pretty spooky action at a distance.

[edit per @Jagerber48 comments] Perhaps an easier scenario to deal with is a single 1 eV photon trapped in a cavity, with a 1/2 eV excitable atom at each end.

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    $\begingroup$ Just a quick note - you refer to one atom being in the other’s future lightcone, but that’s a descriptor used to talk about events, not objects or spatial positions. We can say that one event is in the future lightcone of another event. Is that what you mean? $\endgroup$
    – J. Murray
    Jan 30, 2022 at 21:22
  • $\begingroup$ @J.Murray Yes, thanks very much. I'll clarify. $\endgroup$
    – Roger Wood
    Jan 30, 2022 at 23:30
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    $\begingroup$ For what it's worth, the absorption of a photon by two atoms at once has already been demonstrated in the lab, though in a cavity. $\endgroup$
    – knzhou
    Mar 4, 2022 at 18:53
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    $\begingroup$ @knzhou Good find, but note that this is a theory paper. $\endgroup$ Mar 4, 2022 at 19:56
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    $\begingroup$ @Wolpertinger Whoops! My grip on reality must be weakening these days... $\endgroup$
    – knzhou
    Mar 4, 2022 at 20:05

4 Answers 4

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As knzhou mentiones, and Wolpertinger expands on, a system of atoms in a cavity is theoretically predicted to show simultaneous absorption of one photon by two atoms.

However, the study in this paper is explicitly in a non-relativistic context. The Hamiltonian used, a modification of the Jaynes-Cummings Hamiltonian, does not include any propagation of electromagnetic waves at all. It is valid in the limit in which the relevant dynamics are much slower than the time it takes for light to propagate through the cavity.

To be precise, the Hamiltonian for the EM field by itself in this study, as given in Eq. 1 of the paper and the surrounding discussion, is simply:

$$\hat{H}_{\mathrm{c}}=\omega_{\mathrm{c}} \hat{a}^{\dagger} \hat{a} $$

where the $\hat{a}$ correspond to the cavity mode to which the atoms are coupled. In such a model, emission or absorption of a photon changes the EM field everywhere in space within the cavity simultaneously, neglecting the finite speed of light. The essential approximation is the neglect of higher modes, Wolpertinger points out a nice paper where the importance of these modes to avoiding causality violations at short times is studied in detail.

This non-relativistic approximation is widely used in this field. For example, in other cavity studies (example), one uses the cavity mode to generate an "infinite" range of interactions between atoms- which, again, clearly violates special relativity but is a good approximation for the length and time scales relevant to the problem.

If you instead have two atoms that are outside each others' light cones at their times of absorption, this will no longer work because it would lead to superluminal signaling. One person could change their atom so that it is either at the resonant transition or hidden in another atomic state, and by doing so send a message that someone with the other atom could read out by seeing whether their atom (or local EM field) changes values or doesn't.

Edit: I will be more specific about the limits that causality imposes, and doesn't impose, on this process. Let's imagine a cavity with a mode length of one light-year. It is initialized with one photon in the fundamental mode, with energy $E$, and an atom in the ground state with an excited state that is at energy $E/2$. Then, at time zero (in the rest frame defined by the cavity and atom), another identical atom is placed in the cavity one light-year away. The claim is then that any joint transition that occurs faster than one light year would allow you to detect the presence or absence of the atom at the other end of the cavity fast enough to allow superluminal signalling, and is therefore forbidden.

That said, after a longer time there is no fundamental prohibition on this absorption process. The atoms and the EM field will in general be in an entangled state:

$$|g_1g_2,1\rangle +|e_1e_2,0\rangle $$

where $g,e$ are the atomic states and $0,1$ the number of cavity photons. This entangled state has a fundamentally similar nature to two entangled spins that are separated (as in a loophole-free Bell test). Measurement of the state at any point will collapse the entire spatially extended state. However, this cannot be used for superluminal signalling, so there is no problem anymore with relativity.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – SuperCiocia
    Mar 9, 2022 at 22:54
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This answer really just fleshes out @knzhou's comment.

As per the paper Garziano et al. "One Photon Can Simultaneously Excite Two or More Atoms", Phys. Rev. Lett. 117, 043601 (2016), the answer is: yes, such processes can indeed happen. How does this work and why is it somewhat unintuitive?

Let's start with the latter. The reason this process was long deemed impossible or a very small effect is that many light-matter interaction processes obey the so-called rotating-wave approximation. Mathematically, it neglects certain terms in the atom-light interaction by approximating $(\hat{a}+\hat{a}^\dagger)(\hat{\sigma}^++\hat{\sigma}^-)\approx\hat{a}\hat{\sigma}^+ + \hat{a}^\dagger\hat{\sigma}^-$, where $\hat{\sigma}^\pm$ are the atomic transition's ladder operators and $\hat{a}$ is a bosonic mode. The approximated theory is then excitation number conserving. Within this picture, the process described in the question is not possible.

When the coupling between the light mode and the atom becomes very strong, however, the neglected counterrotating terms start to matter. This regime is called the ultra-strong coupling regime (see also this question) and has been realized experimentally in the last decade.

The process that is studied in this paper is pretty much exactly what is described in the question, where the cavity ensures that the coupling between the light field and the atoms is strong enough. The counterrotating terms in the Hamiltonian then allow to couple the $|1\rangle_{\mathrm{ph},2\omega_0}|0\rangle_{\mathrm{atom}_1,\omega_0}|0\rangle_{\mathrm{atom}_2,\omega_0}$ to the $|0\rangle_{\mathrm{ph},2\omega_0}|1\rangle_{\mathrm{atom}_1,\omega_0}|1\rangle_{\mathrm{atom}_2,\omega_0}$ state.

While the paper is a theoretical work, the relevant parameters have been realized experimentally, and such processes lead to interesting phenomena related to non-classical states of light (see e.g. here). Note that one should add some disclaimers. If the question is understood in the sense of "I shoot in one photon and out come two fully excited atoms", this would require intricate design of the light-matter interaction to eliminate competing processes. I am not aware of an experimental realization thereof.

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    $\begingroup$ & knzhou That's certainly the sort of answer I was after, though I'm not well versed enough in quantum mechanics to appreciate the details. I think you're saying that under normal circumstances this is a unlikely event but it will happen if there's enough coupling and no competing events. $\endgroup$
    – Roger Wood
    Mar 4, 2022 at 21:04
  • $\begingroup$ Does this paper address the space like separated component of the question? $\endgroup$
    – Jagerber48
    Mar 4, 2022 at 21:33
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    $\begingroup$ That there is a Hamiltonian that realizes this process is not surprising or interesting IMO. What is more vexing about the question for me is, how can a photon (something with a finite spatial extent, but also purportedly a “single” object) cause two events simultaneously that are space like separated. How does one “part” of the photon “know” the other “part” was absorbed? $\endgroup$
    – Jagerber48
    Mar 4, 2022 at 21:40
  • $\begingroup$ @Jagerber48 yes, the described process is indeed not an initial single atom excitation with subsequent reemission and excitation of the second atom, as required by the question. $\endgroup$ Mar 4, 2022 at 22:05
  • $\begingroup$ How is this compatible with conservation of angular momentum? Is one of the atoms being excited to a state with same Jz quantum number as the ground state? $\endgroup$ Mar 5, 2022 at 10:02
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I'm not exactly an expert on this subject, but I'll give it a go nevertheless:

I would say it is not possible for a single photon to make two different simultaneous interactions in different locations (i.e. being space-like separated).

Why? I think there must be more elegant ways to prove it, but here are my thoughts:

Let's say, for simplicity, that we have a source that emits one photon in either direction A or B, and before the photon interacts, the whole system (photon + its source) is in a superposition of the photon going into these two directions.

The photon now carries a momentum in either direction, and the photon source has also a corresponding negative change in its momentum, since momentum has to be conserved. So the source is also in a superposition of two different momenta. Now, if the photon could interact with half of its energy in both directions simultaneously, the source momentum could not be in either one of its states. This would only be possible, if two photons are created, but that violates the very essential assumption of having a single photon in this problem.

Thus I conclude, that a single photon cannot have two space-like separated interactions in a single state (i.e. a superposition of these two events is, of course, possible). Having these two interactions would mean that it is actually two photons.

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  • $\begingroup$ That's a good point you're making. I hadn't given any thought to the question of momentum. If the photon is presented as a standing wave in a cavity, I assume in quantum-mechanics that's represented as a 50% superposition of two photons each traveling in opposite directions and with zero total momentum. So can I not absorb all the energy (half in each of two atoms) again with no net momentum? $\endgroup$
    – Roger Wood
    Mar 4, 2022 at 19:20
  • $\begingroup$ I think this problem seems harder if you think about a standing wave in a cavity. Still, the superposition of the two directions $1/\sqrt{2}(|\leftarrow\rangle + |\rightarrow\rangle)$ includes also the surrounding system, i.e. the cavity. Si I think the cavity has to have a slightly different momentum between these two states, so that when the photon reflects and its momentum changes, the momentum of the cavity changes accordingly. Note here, that there is no "middle position" for the cavity again, it is either the momentum of $|\leftarrow\rangle$ or $|\rightarrow\rangle$. $\endgroup$
    – JustSaying
    Mar 4, 2022 at 19:38
  • $\begingroup$ Ending up in the middle of the two states would require some kind of a merging of the different states, which I'm pretty sure has not been observed, although I might of course be wrong. As the end momentum would be the same in both cases, I guess I cannot straight away deny its possibility, in this special case that the momenta of the two states are opposite. $\endgroup$
    – JustSaying
    Mar 4, 2022 at 19:43
  • $\begingroup$ I've accepted Wolpertinger's answer. It does seem like such events can happen but they're outside of the 'normal' regime of quantum mechanical effects. You made an interesting point about the recoil. That seems like another case of entanglement. Can we say that the recoil momentum of the source atom and the momentum of the emitted photon are entangled? $\endgroup$
    – Roger Wood
    Mar 4, 2022 at 21:10
  • $\begingroup$ Yes that is a good answer, I would have accepted it myself as well! And yes, you can very well say the source and the emitted photon are entangled, and they will be up until something else disturbs the system. $\endgroup$
    – JustSaying
    Mar 7, 2022 at 9:10
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No, a single photon can't excite two atoms at the same time, space-like separated or not.

But it can excite a superposition of having excited atom A (and not B) AND having excited atom B and not A.

Within each "branch" the excited atom can then go onto decay and excite the other atom, but in both cases the excitation of the second atom would only happen after a speed of light delay after the decay of the first atom.

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  • $\begingroup$ I'm sure you're correct, but why is that? $\endgroup$
    – Roger Wood
    Jan 31, 2022 at 2:25
  • $\begingroup$ @RogerWood Actually I'm not so sure as I think about it more. I think for this problem you need to consider a "flying photon", as opposed to a photon trapped in a cavity, because we are concerned about speed of light decays and cavity trapped photons are coarse-graining over such small time scales. I find flying photons tricky to reason about. Rather than having an energy/photon, flying photons have energy/second/photon. It is actually striking me (as it may have struck you) that the non-local electric field of the "photon" should be able to create 2 non-local excitations. $\endgroup$
    – Jagerber48
    Jan 31, 2022 at 15:47
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    $\begingroup$ The issue is the term "photon". The only case in which I'm able to make complete sense of the term "photon" is when the electric field is confined in space such that the EM field can be expanded into spatial eigenmodes with finite extent. In that case a photon is related to a quantized amplitude excitation of that spatial normal mode. But in the free-space case it is not clear to me what a single-photon state of the electromagnetic field is. Like I said, I can imagine states of fixed photon flux. $\endgroup$
    – Jagerber48
    Jan 31, 2022 at 15:49
  • $\begingroup$ I'll need to think more, but I think my revised (controversial) answer would be that a pulse of light with a spatial mode extending further than the spacing between the two atoms CAN lead to simultaneous (space-like seperated) excitation of two atoms. Though I do think the probability of excitation will depend in non-trivial ways on the temporal and spatial profile of the quantum pulse. For example, the pulse should be something like a pi-pulse, and that spatial profile will likely need to concentrate energy near the atoms. $\endgroup$
    – Jagerber48
    Jan 31, 2022 at 15:54
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    $\begingroup$ Unfortunately even if the atoms are in a cavity the speed of light delay from one atom to the other will be less than the cavity round trip time. Usual cavity dynamics mathematics involves coarse graining over time scales shorter than the cavity round trip time. So usual cavity mechanics doesn’t apply and you just need to consider the free space situation anyways. $\endgroup$
    – Jagerber48
    Jan 31, 2022 at 23:19

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