# New paper & talk @UFF: Witnessing genuine multi-photon indistinguishability

Daniel Brod and I (Ernesto Galvão) have recently posted a new preprint on the arxiv, the latest result of our long-standing collaboration with the experimental group of Fabio Sciarrino at Sapienza University of Rome. I’ll be giving a talk about it next Friday, April 20th, at 11.00 in room A5-01, come if you can!

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. As it turns out, this question can be motivated by a simple logic problem, as I explain below. Then I briefly describe how the Rome lab experimentally tested our idea.

### 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 $n_{AB}$ sweets with matching flavours;

2- Alice and Charlie have exactly $n_{AC}$ 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 $n_{AB}=30$, 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 $n_{AB}$ and $n_{AC}$ 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 $|A \cap B|$ and $|A \cap C|$. We’d like to find a condition on $|A \cap B|$ and $|A \cap C|$ that guarantees a non-empty triple intersection $A \cap B \cap C$.

As with many problems in statistics and probability, it’s easy to let our intuition fool us. It is, however, easy to show that $|A \cap B|+|A \cap C|>|A|=100$ guarantees 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 $||^2$, where $|A>$ and $|B>$ are the two photons’ wavefunctions. Interestingly, this overlap can be measured experimentally even if we don’t know $|A>$ and $|B>$. 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 $p_b=\frac{1+||^2}{2}$. 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 $|A \cap B|$ 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 $||^2 +||^2$. By experimentally generating photons with high indistinguishability and using our interferometer, we managed to estimate that $||^2 +||^2 >1$, 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.

So if you’re interested, show up at UFF next Friday, and let’s discuss!

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