What the paper says!?
In this paper, with Sana Khalid, I show that if a light source has a compact support of exactly zero auto-correlations—thereby implementing a perfect single-photon source resilient to detectors time uncertainty—then it develops oscillations as a result of the high ordering of the photons in a stream of successive heralders.
We provide a closed-form expression for the Glauber g^{(2)} that ensues (Eq. (4)): g^{(2)}(\tau) = (1+\gamma t_\mathrm{G})e^{-\gamma(|\tau|-t_\mathrm{G})}\sum^{\infty}_{n = 0} \frac{\big[\gamma\big(|\tau| - (n+1)t_\mathrm{G}\big)e^{\gamma t_\mathrm{G}}\big] ^{n}}{n!}\mathbb{1}_{[(n+1)t_\mathrm{G}, \infty [} (|\tau|)\,.
Copy-paste the following in a Mathematica notebook to plot it:
Ind[t1_, t2_, t_] := If[t1 <= t <= t2, 1, 0] (* Indicator function *)
imax = 20; (* number of peaks - increase until converged *)
max\[Tau] = 75; (* max autocorrelation time *)
\[Gamma] =
1/5; tG = 10; (* source radiative lifetime and temporal gap *)
Plot[(\[Gamma] tG + 1) Exp[-\[Gamma] (Abs[t] - tG)] \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 0\), \(imax\)]\(
\*FractionBox[\(
\*SuperscriptBox[\((\[Gamma]\ \((Abs[t] -
n\ tG)\) Exp[\[Gamma]\ tG])\), \(n -
1\)]\(\ \)\), \(\((n - 1)\)!\)] Ind[n*tG, \[Infinity],
Abs[t]]\)\), {t, 0, max\[Tau]},
PlotStyle -> {{Red, Thick, Opacity[.75]}}, PlotPoints -> 1000,
PlotRange -> {-.1, 2}, Filling -> Axis]
This paper was the first in a series studying such types of quantum light.[1][2] It is discussed in this Springer Nature blog post as well as in several Blog posts on this web, namely on the pair correlation function and on a mechanism for implementing such a perfect single-photon source. It is also discussed in various threads on 𝕏:
- The Monty Hall Problem in Quantum Optics:
- Correlations in circular cascades.