Do Photons Have an Optical Phase? (original) (raw)
Posted on 2025-08-18 as part of the Photonics Spotlight (available as e-mail newsletter!)
Permanent link: https://www.rp-photonics.com/spotlight_2025_08_18.html
Author: Dr. Rüdiger Paschotta, RP Photonics AG
Abstract: It is interesting to ask whether photons have an optical phase – but a more precise rephrasing is required to get sensible answers. We also discuss curious two-photon quantum interference.
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Considering the following physics conundrum is rather instructive. Let us start with a simple question: Does a photon have an optical phase?
I can offer two contradicting answers to that question:
- Consider a quantum state with a single photon only. Such quantum states are called Fock states; they have a well-defined photon number, but a completely undefined phase. So the answer to our question would be clearly: No, the phase of a single photon is not defined.
- Consider a Mach–Zehnder interferometer into which we send single photons. It is well known that interference is possible: The probability to get the photons out at a certain port is determined by the relative phase delay in the interferometer. Therefore: Yes, the photons have a phase.
So we have two answers both looking quite convincing, but only one of the those can be correct! Well, we will see that more accurately phrasing the question is an important step towards a convincing resolution.
Various Considerations
Phases of Quantum States
Let us first again consider Fock states. Each of those has a well-defined photon number, and indeed a completely undefined phase; photon number and phase cannot be fully defined simultaneously.
That is definitely so. But does it mean that the 6-photon state ($\left| 6 \right\rangle$), for example, consists of 6 photons, each one having no defined phase? No: The phase is a property of the quantum state as a whole, not of individual photons.
Figure 1: Phasor diagram, showing four coherent states with different values of ($\alpha$), the parameter determining the expectation value of the field phasor. The state at the origin of the coordinate system is the vacuum state.
Let us also consider coherent states (Glauber states), as can be approximated with single-frequency lasers, for example. In a quadrature phase space diagram for the phasors of complex electric fields, such a state is represented by an area. Every such state ($\left| \alpha \right\rangle$) has an expectation value for the phasor, which is the complex number ($\alpha$), and some fluctuations (with Gaussian probability distributions) around that. Except for the vacuum state, every coherent state also has an expectation value for the phase. Again, that phase is a property of the state as a whole, not of individual photons.
During propagation, in the simplest case in vacuum, that phase of a coherent state evolves continuously: ($\left| \alpha(t) \right\rangle = \left| \alpha \: e^{-i \omega t} \right\rangle$). Similarly, the phase of a light field described in classical physics evolves over time. And that phase evolution affects superpositions and interference conditions.
Further, a coherent state is a coherent superposition of Fock states, where each Fock state has a certain complex amplitude and thus a phase. But these are again phases of multi-photon states, not of individual photons.
So how do we now judge the initially given answer: “No, the phase of a single photon is not defined”? We should formulate that more clearly:
- Individual photons in multi-photon states, including the single-photon state, have no individual phases (not even just undefined ones). In that sense, the initial statement would be correct, although not fully clear.
- However, any quantum state (even the vacuum state without any photon!) has a phase which evolves over time — although the absolute value of that phase is unobservable; only changes of that phase are physically relevant, as they affect state superpositions. All those phases together influence the optical phase of a coherent state, for example, but not every quantum state has a defined phase.
Single-photon Interference
When a single photon enters a Mach–Zehnder interferometer, it kind of “probes” both possible paths. After the initial beam splitter, we do not have one photon in each arm with 50% probability, for example, but rather a coherent superposition of two quantum states, each with the photon in one specific arm. At the second (output) beam splitter, the complex amplitudes of the two quantum states interfere with each other, and here the relative phase of the two quantum states matters for the result — thus for the probability of finding the photon in one of the two outputs. Further, that relative phase is just the relative optical phase of a classical wave. If that phase relationship is set properly, it can result in the photon emerging at one specific output port every time.
This is clearly not an interference between two photons having individual phases, but rather an interference between two quantum states, involving a single photon.
Conclusions
We can conclude from all this:
- It does not make sense to assign an optical phase to an individual photon (or in fact to any other Fock state). Phase is a property of the quantum field state, not of a single photon naively considered as a particle.
- Quantum states can carry relative phase information. The absolute/global phase of a quantum state (which, by the way, can have a well-defined value without fluctuations) is unobservable. What matters in experiments are relative phases: between different Fock components in a superposition, or between amplitudes for different spatial modes. Those relative phases directly control interference outcomes.
- Photons themselves don’t “have a phase”, but the state of the light field does, and that is what governs interference.
A Further Curiosity
Let us also consider an even more curious situation, where two indistinguishable photons simultaneously enter a 50:50 beam splitter from different input ports. These photons may be prepared by two single-photon sources, or subsequently by one such source, and sent to the beam splitter with appropriate timing and mode matching.
What will happen here? One might expect that every photon has a 50% chance of being reflected rather than transmitted, and does not care about the other one; after all, we have no optical nonlinearity here. So we would have 25% chance to find both photons at one port, 25% for the other port, and 50% to have both in different outputs.
But that is not what really happens: In the ideal case, one would always observe both photons in the same output port — never the case with two leaving on different ports. That is called the Hong–Ou–Mandel effect and has been proven experimentally, albeit with non-perfect correlations due to non-perfect conditions concerning photon indistinguishability, photon losses, timing, mode matching etc. That is a genuine quantum interference process which cannot be explained by classical wave theory. (Considering two classical waves at the inputs, we do not get that behavior.) A core element is indistinguishability of the photons:
- For the result that the photons exit at different ports, there are two indistinguishable two-photon probability amplitudes: both photons transmitted, and both photons reflected. These amplitudes cancel destructively, making that outcome impossible (in the ideal case).
- For the result with two photons exiting at one specific port, there are again two indistinguishable two-photon probability amplitudes, but these add constructively.
Incomplete cancellation occurs when the photons differ in arrival time, optical spectrum or other properties. In fact, the interference contrast can be used as a measure of the degree of indistinguishability, e.g. for subsequently emitted photons from single-photon sources, or for two different sources.
What we cannot have is both photons always going into one specific port — while with classical waves, that would be possible if their relative phases are adjusted appropriately. So in the quantum world, we have a genuinely probabilistic behavior.
This article is a posting of the Photonics Spotlight, authored by Dr. Rüdiger Paschotta. You may link to this page and cite it, because its location is permanent. See also the RP Photonics Encyclopedia.
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