Shaping the joint spectrum of down-converted photons through optimized custom poling

We present a scheme for engineering the joint spectrum of photon pairs created via spontaneous parametric down-conversion. Our method relies on customizing the poling configuration of a quasi-phase-matched crystal. We use simulated annealing to find an optimized poling configuration which allows almost arbitrary shaping of the crystal's phase-matching function. This has direct application in the creation of pure single photons—currently one of the most important goals of single-photon quantum optics. We describe the general algorithm and provide code, written in C++, that outputs an optimized poling configuration given specific experimental parameters.

Figure 1

Periodic poling produces a sinc-shaped phase-matching function. Alternative phase-matching functions, e.g., Gaussian, can be generated by customizing the poling configuration. The phase mismatch, $\mathrm{\Delta }k$, has units of rad/m.

Figure 2

Comparison of phase-matching functions (top), joint spectral amplitudes (middle), and Schmidt coefficients (bottom), which are all dimensionless, for different apodization schemes. The Gaussian target function is also shown in red. (a) Standard first-order periodically poled crystal (no apodization). We define the peak nonlinearity of this crystal to be unity and scale all other peak nonlinearities accordingly. (b) Customized duty cycle method proposed by Dixon et al. [26]. In this scheme, the peak center is shifted, but this does not affect the photon purity. The dashed, red line shows the Gaussian target function shifted for comparison. (c) Custom-poled crystal generated using simulated annealing. We used a randomized initial trial configuration, then ran the algorithm with $2\times {10}^5$ iterations, using $q=0.999$ and $A=1000$. We used $M=2001$ samples to compute the objective function $d_{\mathbf{s}}$, over the range $[a,b]=\left[2\pi \right(1-0.025)/\mathrm{\Lambda },2\pi (1+0.025)/\mathrm{\Lambda }]$. The height of the target function was $H=0.8N{l}_c/\pi$. All plots were generated using the Sellmeier equations given in [34, 35]. We define $\mathrm{\Delta }{\omega }_j={\omega }_j-{\overline{\omega }}_j$ where ${\overline{\omega }}_j={\omega }_p/2$. The number of domains is (a) $N=740$, (b) $N=860$, and (c) $N=1300$. We note that $N=1300$ is roughly the optimal number of domains needed for a custom-poled crystal, and increasing $N$ does not improve the performance of the algorithm, but rather degrades it.

Figure 3

Phase-matching functions (dimensionless) for (a) triangular and (b) rectangular phase-matching functions. Target functions are in red; optimized custom-poled functions are in black. We define the peak nonlinearity of a first-order-poled crystal of the same length as unity and scale all other peak nonlinearities accordingly. We define $\mathrm{\Delta }{k}^{\prime }=\mathrm{\Delta }k-2\pi /\mathrm{\Lambda }$. We note that sharp edges make the top-hat function very difficult to approximate with a decomposition of smooth functions. For comparison with other methods, see [12, 25].