The accepted version of “Interpreting weak value amplification with a toy realist model” is now available online. The work was an interesting exercise involving two topics in the foundations of physics: Weak values and stochastic electrodynamics. Basically we showed that a recent quantum optics experiment from our lab could be re-interpreted with a slightly modified version of classical electrodynamics.
The aim of the work was to develop some intuition on what weak values could mean in a realist model (i.e. one where the mathematical objects represent real “stuff”, like an electromagnetic field). To do this we added a fluctuating vacuum to classical electrodynamics. This framework (at least at the level we developed it) does not capture all quantum phenomena, and so it is not to be taken seriously beyond the regime of the paper, but it did allow us to examine some neat features of weak values and the experiment.
One interesting result was the regime where the model succeeds in reproducing the theoretical results (the experimental results all fall within this regime), which provided insight into weak values. Specifically, the model works only when the fluctuations are relatively small. Going back to the real world, this provide new intuition on the relation between weak values and the weak measurement process.
The experiment we looked at was done in Aephraim Steinberg‘s light matter interaction lab by Matin Hallaji et al. a few months before I joined the group. It was a ‘typical’ weak value amplification experiment with a twist. In an idealized version, a single photon should be sent into an almost balanced Mach Zehnder interferometer with a weak photon counting (or intensity) measurement apparatus on one arm (see figure below). The measurement is weak in the sense that it is almost non-disturbing, but also very imprecise, so the experiment must be repeated many times to get a good result. So far nothing surprising can happen, and we expect to get a result of 1/2 photon (on average) going through the top arm of the interferometer. To get the weak value amplification effect, we post-select only on (those rare) events where a photon is detected at the dark port of the interferometer. In such a case, the mean count on the detector can correspond to an arbitrarily large (or small, or even negative) number of photons.
The twist with the experiment was to use a second light beam for the photon counting measurement. This was the first time this type of weak measurement was done in this way, and is of particular importance since many of the previous experiments could be explained using standard electrodynamics. In this case, the photon-photon interaction is a purely quantum effect.
In reality the experiment described above was not feasible due to various imperfections that could not avoided, so a compromise was made. Instead of sending a single photon, they used a coherent beam with around 10-100 photons. To get the amplification effect on a single photon, they used a trick that would increase the number of photons by 1 and only looked for the change in the signal due to the extra photon. The results showed that the same amplification as the ideal case. However, there remained (and still remains) a question of what that amplification means. Can we really talk about additional photons appearing in the experiment?
As we showed, the results of the experiment can be explained (or at least reproduced) in a model which is more intuitive than quantum theory. Within that model it is clear that the amplification is a real, i.e the post-selected events corresponded to cases where more light traveled through the top arm of the interferometer.
The model was based on classical electrodynamics with a slight modification, we assumed that fields fluctuated like the quantum vacuuum. This turned out to be sufficient to get the same predictions as quantum theory in the regime of the experiment. However, we showed that this model would not work if the intensity of the incoming light was sufficiently small, and in particular it would not work for something like a single photon.
Our model has a clear ‘reality’, i.e the real field is a fluctuating classical EM field, and so it provides a nice start for a more general theory that has weak values as its underlying real quantities. One important feature of the model is the regime where it makes accurate predictions. It turns out that there are two bounds on the regime of validity. The first is a requirement that the light is coherent and is not too weak, this roughly corresponds to being in a semi-classical regime. The second is that the probability of detecting photons at the dark port is not significantly effected by the fluctuation which is similar to the standard weak measurement requirement, i.e. that the measurement back-action is small.
My main takeaway from the work was that the weak values ended up being the most accurate experimentally accessible quantity for measuring the underlying field. Making a leap into quantum theory, we might say that weak measurements give a more accurate description of reality than the usual strong measurement.
The result also set a new challenge for us: Can we repeat the experiment in the regime where the model brakes down (e.g. with single photons)?