1.2.3. Measurement of Subpicosecond Time Intervals between Two Photons by Interference [2]

This experiment is widely used to ensure the indistinguishability of two bosonic particles in the experiment.

It should be emphasized that the signal and idler photons have no definite phase, and are therefore mutually incoherent, in the sense that they exhibit no second-order interference when brought together at detector D1 or D2. However, fourth-order interference effects occur, as demonstrated by the coincidence counting rate between D1 and D2.

Although the sum frequency $\omega_1+\omega_2$ is very well defined in the experiment, the individual down-shifted frequencies $\omega_1,\omega_2$ have large uncertainties, that, in practice, are largely determined by the pass bands of the in- terference filters IF inserted in the down-shifted beams, as shown in Fig. 1.

The joint probability is $$P_{12}(\tau)=K|G(0)|^2\{T^2|g(\tau)|^2+R^2|g(2\delta\tau-\tau)|^2-RT[g^*(\tau)g(2\delta\tau-\tau)+c.c]\}$$ where $G(t)​$ is the Fourier transform of the weight function $\phi(\omega_0/2+\omega,\omega_0/2-\omega)​$ with respect to $\omega ​$ $$G(\tau)=\int\phi(\omega_0/2+\omega,\omega_0/2-\omega)e^{-\mathrm{i}\omega\tau}\mathop{}\!\mathrm{d}\omega$$ and $g(\tau)\equiv G(\tau)/G(0)$

The expected number $N_c$ of observed photon coincidences is then given by $$N_c=C\left[R^2+T^2-2RT\frac{\int^\infty_{-\infty}g(\tau)g(\tau-2\delta\tau)\mathop{}\!\mathrm{d}\tau}{\int^\infty_{-\infty}g^2(\tau)\mathop{}\!\mathrm{d}\tau} \right]$$ In the special case when $g(\omega_0/2+\omega,\omega_0/2-\omega)$is Gaussian in w with bandwidth $\Delta\omega$, then $g(\tau)$ has the Gaussian form $$g(\tau)=e^{-(\Delta\omega\tau)^2/2}$$

$$N_c=C(T^2+R^2)\left[1-\frac{2RT}{R^2+T^2}e^{-(\Delta\omega\delta\tau)^2} \right]$$