What does $τ$ mean in second-order correlation function?

Question

I have this question to which I cannot find a simple explanation to. In most articles about single photon sources people use second order correlation function $g$$^{(2)}$$(τ)$, to show that their source actually produce antibunched photons. If I understand correctly, if the value of $g$$^{(2)}$$(τ=0)$ is <1, then we truly have a single photon source. This single value then should be enough to prove that we're working with a single photon source, and it can be evaluated experimentally the same way as in this example https://www.qutools.com/files/quED/worksheets/qutools_HBT.pdf.

However, in many articles I see a full graph of $g$$^{(2)}$$(τ)$ drawn with a dip around $τ=0$. What extra information do we get from this graph compared to the single value calculation of $g$$^{(2)}$$(τ=0)$ ? Also what is the meaning of $τ$ and how does it manifest in real life experiment? Is it just a delay between the clicks of the two detectors?

Answer 1

Welcome to Physics.SE!

The $\tau$ in $g^{(2)}(\tau)$ is the time delay between detected photons. I often found it useful to think of $g^{(2)}(\tau)$ in terms of conditional probabilities^[1] $$g^{(2)}(\tau) = \frac{P(t+\tau | t)}{P(t)}.$$ $P(t)$ is the probability density to detect a photon at time $t$ and $P(t+\tau | t)$ is the probability density to detect a photon at time $t+\tau$, conditioned on a photon detection at time $t$. As you already pointed out in your question, the light from a single-photon source shows $g^{(2)}(0) < 1$, because two photons at the same time are unlikely (or impossible for a perfect single-photon source).

The value of $g^{(2)}(0)$ alone is already a good measure for the purity of single photons. However, there are various reasons, why one often shows the intensity autocorrelation function $g^{(2)}(\tau)$ for a range of $\tau$s:

1. One might want to highlight the dynamics after the detection of the first photon. The detection of a photon from a two-level system projects the system into the ground state. From there, it takes some time to get excited again. This depends on the Rabi frequency $\Omega$ of the driving laser. At high laser intensity, the system might even undergo Rabi oscillations.
2. The range of $\tau \gg 1/\gamma$ can be used to normalize $g^{(2)}(\tau)$. In the experiment, one measures coincidences within histogram bins (see the plots of points 3. and 4.). However, the normalization is not always obvious. In principle, the mean countrate can be used for normalization, but only if the experiment delivers a constant countrate. One example where this is not the case, is a blinking molecule. The plot below shows $g^{(2)}(\tau)$ on different timescales. Every $\sim 10^4 \, \text{ns}$ the molecule enters a dark state from which it doesn't emit photons. The emission of the single-photon streams is therefore "bunched" compared to the overall mean countrate. But one might be interested in only the behavior of the molecule during its active phases. Looking at only short timescales, comparable to the lifetime of the molecule $1/\gamma$, it looks like $g^{(2)}(\tau \gg 1/\gamma) > 1$. It is quite common to normalize $g^{(2)}(\tau)$ to its value at $\tau \gg 1/\gamma$.
3. Other people looking at your data might want to know something about the statistical certainty of your data set. You might have exactly $0$ coincidences in the histogram bin corresponding to $\tau = 0$, but if there are $0$ photons in most other histogram bins, you can't make any claims about the photon source. The plot below shows a histogram of $2774$ coincidences distributed over $45$ bins. If we only use every 10^th photon, the shot noise in the histogram is much higher. In that case, not only the $\tau = 0$ bin, but also 3 neighboring ones are completely empty.
4. Usually, it doesn't take additional effort to retrieve $g^{(2)}(\tau)$ from the data, instead of only $g^{(2)}(0)$. These histograms are computed from a list of photon detection times, so the data for any $\tau$ is there anyway. Practically, the timetags recorded with different single-photon detectors are likely to be slightly delayed relative to each other. This causes features, like the antibunching dip in $g^{(2)}(\tau)$ to appear at $\tau$ other than $0$, as in the plot below. If one has $g^{(2)}(\tau)$ for a range of $\tau$ available, one can simply shift the time axis in a plot, to make the data dip at $\tau = 0$.