Separable Schmidt modes of a nonseparable state

Two-photon states entangled in continuous variables such as wave vector or frequency represent a powerful resource for quantum-information protocols in higher-dimensional Hilbert spaces. At the same time, there is a problem of addressing separately the corresponding Schmidt modes. We propose a method of engineering two-photon spectral amplitude in such a way that it contains several nonoverlapping Schmidt modes, each of which can be filtered losslessly. The method is based on spontaneous parametric down-conversion (SPDC) pumped by radiation with a comblike spectrum. There are many ways of producing such a spectrum; here we consider the simplest one, namely, passing the pump beam through a Fabry-Perot interferometer. For the two-photon spectral amplitude (TPSA) to consist of nonoverlapping Schmidt modes, the crystal dispersion dependence, the length of the crystal, the Fabry-Perot free spectral range, and its finesse should satisfy certain conditions. We experimentally demonstrate the control of TPSA through these parameters. We also discuss a possibility to realize a similar situation using cavity-based SPDC.

Figure 1

(a),(b) A simplified picture of the TPSA formed by (a) a short pump pulse and (b) the same pulse after passing through an FP cavity. (c) Squared modulus of the TPSA calculated for a 20 mm KDP crystal pumped by 25 fs pulses at 532 nm transmitted through a 50 $\mu$m FP cavity (left) and the zoom into a single maximum (right).

Figure 2

Measured (red line) and calculated (filled contour) pump spectrum with the central wavelength 430 nm and the width $2.8$ nm, transmitted through a FP interferometer normal to the beam (left), and tilted by ${20}^{\circ }$ (center) and ${45}^{\circ }$ (right).

Figure 3

Calculated squared modulus of TPSA for type-II SPDC in a KDP crystal pumped by 160 fs pulses passed through a 100 $\mu$m FP interferometer. Left: 15 mm crystal; the pump central wavelength is (a) 370, (b) 415, and (c) 450 nm and the FP is at normal incidence. Right: the pump central wavelength is 450 nm; the FP is tilted by 45 degrees and the crystal thickness is (d) 13, (e) 15, and (f) 17 mm.

Figure 4

The experimental setup: frequency-doubled pulses from mode-locked Ti-sapphire laser are fed into an FP interferometer shaping the spectrum as shown in Fig. 2. The beam pumps a type-II nonlinear crystal (BBO or KDP). A dichroic mirror (DM) removes the residual pump beam. Photons of the same pair are separated by a polarizing beam splitter and fed in two identical fibers of 1 km length. Finally, photons are detected by two SPADs and coincidence electronics is used to measure their arrival times with respect to the trigger from the laser. Inset: the measured distribution, recalculated into wavelengths, for a 5 mm BBO crystal.

Figure 5

Left: Measured squared modulus of TPSA (in counts) for type-II KDP crystal of total length 10 mm pumped at (a) 395, (b) 415, (c) 430, and (d) 440 nm. The FP interferometer is tilted at ${45}^{\circ }$. Right: The corresponding theoretical distributions with $1.5$ nm resolution taken into account.

Figure 6

The shape of the Fabry-Perot transmission $F_p\left(\omega \right)$ around a single peak. Red solid line: real part; blue dots: imaginary part (negative values not shown); brown dashed line: squared modulus. Left: tilted FP, $d=100\mu$m, $\varphi ={45}^{\circ }$, $R=0.4$. Right: untilted FP with the spacing changed to have the same width of a single maximum, $d=20\mu$m, $R=0.8$.

Figure 7

The shape of the TPSA modulus around a single peak for the pump spectra shown in Fig. 6, pump central wavelength 440 nm, and the crystal length 1.5 cm.

Figure 8

The shape of the TPSA modulus around a single peak corresponding to the pump wavelength 532 nm, FP spacing $d=50\mu$m, FP transmission $R=0.64$, and crystal length 20 mm.