Seeded and unseeded high-order parametric down-conversion

Spontaneous parametric down-conversion (SPDC) has been one of the foremost tools in quantum optics for over five decades. Over that time, it has been used to demonstrate some of the curious features that arise from quantum mechanics. Despite the success of SPDC, its higher-order analogs have never been observed, even though it has been suggested that they generate far more unique and exotic states than SPDC. An example of this is the emergence of non-Gaussian states without the need for postselection. Here we calculate the expected rate of emission for $n\mathrm{th}$-order SPDC with and without external stimulation (seeding). Focusing primarily on third-order parametric down-conversion, we estimate the photon detection rates in a rutile crystal for both the unseeded and seeded regimes.

Figure 1

The distributions of (a) the interaction volume in which the nonlinear process takes place and (b) the phase-matching function in the reciprocal ($k$) space.

Figure 2

Frequency-angle contour for a ${\mathrm{TiO}}_2$ crystal with type-I ($\mathrm{o}\to \mathrm{eee}$) phase matching at different orientation angles. The pump wavelength is ${\lambda }_p=532\mathrm{nm}$, and the parameters for mode 3 are fixed at ${\lambda }_3=1596\mathrm{nm}$ and ${\theta }_3=0^{\circ }$.

Figure 3

The differential single-photon count rate for TOPDC in a 5-mm ${\mathrm{TiO}}_2$ crystal pumped by 100 mW as a function of the crystal orientation angle. The pump wavelength is ${\lambda }_p=532\mathrm{nm}$, and the parameters for mode 3 are ${\lambda }_3=1596\mathrm{nm}$ and ${\theta }_3=0^{\circ }$.

Figure 4

The frequency-angle spectrum of the single-photon differential generation rate in mode 3 for unseeded TOPDC in a ${\mathrm{TiO}}_2$ crystal oriented at $68.{24}^{\circ }$. The crystal length is $L=5\mathrm{mm}$, and the pump power is $P_p=100\mathrm{mW}$.

Figure 5

The frequency-angle spectrum of mode 2 for seeded TOPDC in ${\mathrm{TiO}}_2$ of length $L=5\mathrm{mm}$ for the pump power $P_p=100\mathrm{mW}$ and the crystal orientation $68.5^{\circ }$.

Figure 6

The expected triple-, double-, and single-photon count rates of seeded and unseeded TOPDC in 5-nm ${\mathrm{TiO}}_2$ for (a) the broadband regime and (b) the narrow-band regime. The seed beam is centered at 1620 nm, the pump power is $P_p=100\mathrm{mW}$, and the overlapped seed peak intensity $I_s=1.6\times {10}^{13}\mathrm{W}{\mathrm{m}}^{-2}$. The count rates calculated for the broadband regime assume collection of wavelengths from 1200 to 1600 nm and angles from $-{10}^{\circ }$ to ${10}^{\circ }$ with a detection efficiency of 15%. The count rates for the narrow-band regime assume collection of wavelengths from 1579 to 1589 nm and angles from $-0.{25}^{\circ }$ to $0.{25}^{\circ }$ with a detection efficiency of 80%.