# Fundamental Limits for Photon Detection

In quantum electrodynamics "photons don't have positions". The physical relevance and consequences of this fact has been discussed on this site 1. (Further relevant questions about the concept of photon position: 2, 3, 4, 5). The answer to 1 says that this is a consequence of the Reeh-Schlieder theorem (see e.g. arXiv:1803.04993). It implies the impossibility of having a particle detector that is both reliable and of finite size (and measures for a finite length of time). In the non-relativistic limit, the theorem becomes irrelevant for particles with non-zero rest mass, thus allowing wavefunctions and position operators for things like electrons. The issue with photons is that they have zero rest mass and do not allow such a non-relativistic limit.

The difficulties of localizing particles in relativistic quantum field theory (QFT) have led some to argue against the concept of "particles of light" altogether (see e.g. W. E. Lamb, Jr., “Anti-photon.” Applied Phys B60(1995). Also e.g. arXiv:quant-ph/0103041). On the other hand, in fields such as Quantum Optics and Quantum Information it is common to talk about (distributions of) arrival times and even positions of photons and the concept of a photon wave function has also been proposed (see 2, 3, 4).

My question is: What quantitative limits does QFT imply for a finite size single photon detector measuring for a finite time interval? Such limits might concern things like timing jitter, dark counts and detection efficiency and depend on the spectrum and perhaps other properties of the light being measured, as well as a hypothetical rest mass of the photon, for which we can only achieve an upper bound (see e.g. doi:10.1016/j.physletb.2008.07.018).

I'm assuming these limits are nowhere near the achievable precision of current technology and will perhaps never be achieved for other reasons (practical limitations). Nonetheless, I believe knowing these limits may help understand some fundamental aspects of light and of quantum electrodynamics.

Other relevant questions concerning photon detection:

• Possibly relevant: arXiv:math-ph/0607044 Sep 2, 2020 at 13:54

In quantum electrodynamics "photons don't have positions"... It implies the impossibility of having a particle detector that is both reliable and of finite size (and measures for a finite length of time).

The QED was developed for the description of processes between (charged) subatomic particles through photon exchange. For the necessity of the calculation a generally existing electromagnetic field was introduced - without sources having to be present. But does this mean that the sources can be neglected in real processes? I think not. If we want to detect very low frequency photons, we have to generate them first (at least theoretically).

How do we obtain - and this time in practice and not just theoretically - photons below the infrared and terahertz frequencies? We use wave generators and a conducting wire. The electrons in this wire are now accelerated forward and backward, emitting periodically and synchronously polarized photons.

First, we have to be clear: The photons emitted do not have the frequency of the wave generator. The generated wavelength of the radio wave has nothing to do with the wavelength of the emitted photons. Take a powerful wave generator and a thin wire, then you can see how the wire glows through infrared photons. Furthermore, it is dangerous near the antenna rod of a strong radio transmitter because of the X-rays (which oscillate with the frequency of the generator). The same applies to a fighter plane radar.

The consequence is that there is a lower limit for the photon energy. The most obvious (real and not Gedankenexperiment) experiment I can imagine is the jump of an electron in a Rydberg atom.

Secondly, we need a structure for the emission of photons with low energy. The proposal was in the last paragraph. And we should then also use the Rydberg atom as a detector.

The difficulties of localizing particles in relativistic quantum field theory (QFT) have led some to argue against the concept of "particles of light" altogether (see e.g. W. E. Lamb, Jr., “Anti-photon.” Applied Phys B60(1995). Also e.g. arXiv:quant-ph/0103041).

I am not sure, or the paper was misinterpreted, or the author was trapped with his thoughts in the source-free EM field.

My question is: What quantitative limits does QFT imply for a finite size single photon detector measuring for a finite time interval?

To develop an answer, some theoretical assumptions have to be applied:

1. In reality there are electric fields and magnetic fields between charged particles. The EM field is a mathematical construct to describe both fields.
2. The electric and magnetic fields and the photon are considered to have no internal structure. The field is described by the exchange of virtual photons between the influencing particles. Instead, a possible model could be the assumption of real field lines with an internal structure.
3. The constituents of the inner structure determine the lower limit for the minimum energy size of a photon.

I'm assuming these limits are nowhere near the achievable precision of current technology and will perhaps never be achieved for other reasons (practical limitations).

This I agree. How about the Rydberg setup?