Spatially entangled four-photon states from a periodically poled potassium-titanyl-phosphate crystal

Spatial quantum correlations are an important resource for quantum imaging and future quantum technologies based on high-dimensional entanglement. We experimentally explore spatial correlations in four-photon states generated by a highly efficient parametric down-conversion process in a 2 mm long periodically poled KTP crystal. The four-photon states produced in the crystal contain a contribution due to independent double pairs produced via spontaneous pair emission and a highly correlated four-photon state produced by stimulated emission of a second photon pair. We separate this contribution and introduce a joint spatial density for stimulated pair emission. We observe a maximum visibility $\chi =0.25$ of the four-photon state for a 1.5 nm bandpass filter.

Figure 1

Setup to measure spatially entangled four-photon states via parametric down-conversion in a 2 mm PPKTP crystal. UV pump pulses from a frequency doubled Ti:sapphire laser are focused by lens $L1$ ($f_1=250$ mm). Photons are collected by the lenses $L2$ ($f_2=270$ mm) and detected by the detector assemblies $D1\text{--}D3$ that use photon counting APDs to detect single photons in a particular direction defined by the apertures $A$ placed in the combined focal planes of $L2$ and $L3$ ($f_3=4.5$ mm). The beamsplitter ($BS$) allows us to record if two photons are in the same spatial mode. To this end detector assemblies $D1$ and $D2$ are placed on computer controlled translation stages. Correlations in the coincidence events are due to four-photon states that are created by the process of stimulated emission of photon pairs.

Figure 2

Measured single count rates $R_2$ (open symbols) and twofold coincidence rates $R_{23}$ (filled symbols) as a function of position $x_2$, demonstrating spatial correlations in the two-photon field. The pulsed laser beam was focused in the center of the crystal using a $f=250$ mm lens. Data are collected using a 1.5 nm FWHM bandpass filter and a 1.5 mm aperture in the far field. The solid and dashed lines indicate Lorentzian and Gaussian fits to the single and coincidence rates, respectively. The inset shows a far field image of the SPDC light emitted by the PPKTP crystal at a temperature of 20${}^{\circ }$C. The approximate positions of detectors $D2$ and $D3$ are shown by circles on the ring.

Figure 3

Measured joint spatial distribution of stimulated pair emission. (a) False-color plot of the measured difference in coincidence rate $(R_{21}^{0\mathrm{ns}}-R_{21}^{12\mathrm{ns}})$ as a function of the positions of $x_1$ and $x_2$ of detectors $D_1$ and $D_2$. Data are collected with a 4.6 nm FWHM bandpass filter and a 1.5 mm aperture size. The observed maximum coincidence rate $R_{21}^{0\mathrm{ns}}$ is typically 25 000 s${}^{-1}$. The data clearly demonstrate that stimulated pair emission occurs when the photons are emitted in the same spatial mode, that is, when $x_1=x_2$. (b) Visibility $\chi$ of the four-photon state as a function of position $x_2$. The different symbols (triangles, circles, squares) correspond to different positions $x_1$ (44.4, 50.0, 55.6 mrad) of detector $D_1$. Data are collected using a 1.5 nm FWHM bandpass filter and a 1.5 mm aperture. The solid lines through the data are Gaussian fits.

Figure 4

Visibility $\chi$ of the four-photon state as a function of aperture size using a 1.5 nm FWHM bandpass filter (triangles) and a 4.6 nm FWHM bandpass filter (circles). These data are obtained by subtracting the measured coincidence rate at a delay of 12 ns from the measured coincidence rate at zero delay. Typical single count rates on the detector are $1.4\times {10}^6$ and $3\times {10}^5$ s${}^{-1}$ for apertures $>$2 mm and drop to $5\times {10}^5$ and $1\times {10}^5$ s${}^{-1}$ for a 1 mm aperture size. The solid lines represent a calculation with no free parameters (see text). For apertures larger than 2 mm, the aperture is limited by lens $L3$ and the numerical aperture of the fiber and the visibility saturates as indicated by the horizontal dashed lines. The inset shows the measured coincidence rate $R_{12}$ as a function of electronic delay for a 1.5 nm FWHM bandpass filter and a 1 mm aperture size ($\chi =0.25$), indicated by the arrow.