Derivation of the density matrix of a single photon produced in parametric down-conversion

We discuss an effective numerical method of density matrix determination of fiber coupled single photon generated in process of spontaneous parametric down conversion in type I noncollinear configuration. The presented theory has been successfully applied in case of source utilized to demonstrate the experimental characterization of spectral state of single photon, what was reported in Wasilewski, Kolenderski, and Frankowski [Phys. Rev. Lett. 99, 123601 (2007)].

Figure 1

(Color online) The setup is comprised of ${\chi }^{\left(2\right)}$ nonlinear crystal of length $L$, pumped by a Gaussian beam of $w_p$ comprised of pulses described by a spectral envelope function $A_{sp}$. The idler photon is collected to a single mode fiber SMF, provided it is emitted into a spatial mode of characteristic width $w_f$ at the crystal.

Figure 2

(Color online) The spectral density matrix $\rho \left(\omega ,{\omega }^{\prime }\right)$ of photons transmitted through 20 nm spectral filter. The plots were generated for $L=1\text{mm}$ long BBO crystal cut for ${\theta }_c=30°$, pumped using Gaussian beam of $w_p=100\mu \text{m}$ waist. The pulse duration was set to ${\tau }_p=100\text{fs}$. The collecting optics and the single mode fiber was set to couple the Gaussian mode of $w_f=100\mu \text{m}$. The observation angle was set to (a) $\alpha =2.2°$ (direction of perfect phase matching) and (b) $\alpha =3.2°$.