Angular Schmidt modes in spontaneous parametric down-conversion

We report a proof-of-principle experiment demonstrating that appropriately chosen set of Hermite-Gaussian modes constitutes a Schmidt decomposition for transverse momentum states of biphotons generated in the process of spontaneous parametric down-conversion. We experimentally realize projective measurements in the Schmidt basis and observe correlations between appropriate pairs of modes. We perform tomographical state reconstruction in the Schmidt basis, by direct measurement of single-photon density matrix eigenvalues.

Figure 1

Experimental setup realizing projective measurements in Schmidt basis (see text for details). Phase mask (PM) is shown as transparent for simplicity; actual realization uses double reflection from SLM as shown in the inset.

Figure 2

Coincidence counting rates for ${\text{HG}}_{\mathit{mn}}$ modes selected in the transmitted arm of the setup; ${\text{HG}}_{00}$ mode (a) and ${\text{HG}}_{10}$ (b) modes are selected in the reflected arm.

Figure 3

(a) Single counts rates for ${\text{HG}}_{\mathit{mn}}$ modes selected in the transmitted arm of the setup. Not normalized, only background noise was subtracted. (b) Single-photon density matrix eigenvalues ${\lambda }_{\mathit{mn}}$ in a HG-modes basis, which are the weights in the Schmidt decomposition. (c) One-dimensional projection on the plane $n=0$. Red (dark) bars are normalized experimental results, gray (light) bars are eigenvalues for HG modes predicted by the double-Gaussian model.

Figure 4

Ghost images of the first three Schmidt modes: ${\text{HG}}_{00}$ (a), ${\text{HG}}_{10}$ (b), and ${\text{HG}}_{20}$ (c). Solid curves are HG fits.