Ernesto F. Galvão - genuine

Here I would like to describe, in accessible terms, the motivation for the problem we study in our paper "Witnessing genuine multiphoton indistinguishability", recently published in Physical Review Letters (February 2019). A slightly different, early version of the paper is freely available here.

The paper describes a way to test whether a set of single-photon sources is producing a state of n genuinely indistinguishable photons, as opposed to some convex combination of states in which effectively less than n photons interfere. The question of state indistinguishability is a foundational one, but it is also important for applications. For example, photonic quantum computers depend on high photon indistinguishability to function well. As it turns out, the question we study can be motivated by a simple logic problem, as I explain below. Then I briefly describe how our theoretical ideas were tested in the quantum optics lab headed by our collaborator Fabio Sciarrino in Rome, which resulted in our PRL paper.

The problem of three bags of sweets

Let’s start by thinking of the following logic problem. Alice, Bob and Charlie each fills a bag with 100 sweets, each choosing some distribution over the many different flavours available. Now assume we know the following about the three bags:

1- Alice and Bob have exactly nAB sweets with matching flavours;

2- Alice and Charlie have exactly nAC sweets with matching flavours.

To clarify the accounting of sweets, consider the following contents for the bags of Alice and Bob:

Alice’s bag has 20 caramel sweets, 20 vanilla sweets, and 60 chocolate sweets.
Bob’s bag has 10 caramel sweets, 30 vanilla sweets, and 60 lime sweets.

Then , as the bags have 10 caramel sweets and 20 vanilla sweets in common.

Now, the question we’d like to ask is: what’s the condition on and that guarantees that the three bags have at least one sweet of the same flavour?

Let’s use a Venn diagram to visualize the problem:

In terms of sets: we're given the sizes of sets A, B, C, and the sizes of the intersections of A and B, and of A and C. We'd like to find a condition on these two two-set intersections that guarantees a non-empty triple intersection of A, B and C.

As with many problems in statistics and probability, it's easy to let our intuition fool us. It is, however, easy to show that if the sum of the two intersections is larger than 100 then there's some triple intersection - the two pairwise intersections "crowd out" the whole of set A, forcing the existence of a triple intersection. This is curious: it's possible to guarantee a non-empty three-set intersection without ever directly measuring it.

Back to photons

So what does this have to do with photons? Well, quantum theory describes each photon by a wavefunction, which gives us a recipe to calculate the probability distributions of any property one may care to measure: colour, time-of-arrival, momentum, etc. The probability that photon A passes in a test for photon B can be directly calculated as the overlap , where and are the two photons’ wavefunctions. Interestingly, this overlap can be measured experimentally even if we don’t know and . The trick is to use the famous Hong-Ou-Mandel test: send each photons in one input mode of a 50/50 beam-splitter. It can be shown that the probability that the two photons will come out in a single mode (that is, that they will bunch) is . In particular, perfectly indistinguishable photons always bunch, and perfectly distinguishable photons, described by orthogonal wavefunctions, bunch 50% of the time.

We can represent the properties of each photon by a set. The size of the set intersection can be interpreted as the probability that photon A passes for photon B (or vice-versa); naturally the size of each set is 1, as a photon passes its own test with probability 1.

So we’re back to using sets to describe properties of objects, but now instead of 3 bags of multi-flavoured sweets we have 3 photons (each with its “multi-flavoured” wavefunction).

Genuine 3-photon indistinguishability

If we have three photons, how can we guarantee they’re genuinely indistinguishable? Quantum theory allows us to measure two-photon overlaps, using the HOM test, but there’s no such thing as a “three-photon overlap”.

Our solution uses the idea on two-set intersections above. We designed a 4-mode interferometer that allows us to do two HOM tests in parallel, by first putting photon A in a superposition over two spatial modes, then using those modes for two independent HOM tests with photons B and C. The design is such that the probability of bunching at the outputs of the interferometer allows us to estimate the sum of the two overlaps . By experimentally generating photons with high indistinguishability and using our interferometer, we managed to estimate that , guaranteeing (i.e. witnessing) the presence of genuine 3-photon indistinguishability.

On the theory side, we obtain sufficient conditions to witness n-photon indistinguishability (for any n), and design interferometers that do the job. For any n, we prove a theorem on set intersections that extends the result above, for 3 sets, to any number of sets. I and Daniel have already found other witnesses which were not reported on this paper, and we’re working on a new theoretical paper exploring these ideas.