Figure 1

(a) Cross section of the experimental package. The sapphire resonator is shown in its cavity, which is mounted in a vacuum chamber at the end of an insert in a liquid helium dewar. Herotek DT8016 power detectors are used at the output to generate a voltage proportional to incident microwave power, measured with an oscilloscope. (b) (Not to scale) Schematic of the system described by our theoretical model. ${\Omega }_0$ and ${\Omega }_{-}$ represent fixed microwave WG mode resonances in the sapphire resonator with bandwidths of the order of 10 Hz. ${\Omega }_{+}$ models a lossy resonance at ${\Omega }_{+}=2{\Omega }_0-{\Omega }_{-}$ as no WG mode exists at 12.045 GHz. The applied pump frequency ${\omega }_0$ can be selected in a range of over 4 kHz around ${\Omega }_0$ to successfully result in the generation of ${\omega }_{-}$, whose frequency changes only over a narrow range $<40\mathrm{Hz}$, and ${\omega }_{+}$ with a frequency range of the order of 8 kHz.Reuse & Permissions

Figure 2

Power detected in transmission through the resonator when (a) the 12.037 GHz pump signal is offset 2.671 kHz above resonance and (b) the 12.037 GHz pump is offset 2.659 kHz above resonance. In both cases, the excitation signal was switched on at 0 s.Reuse & Permissions

Figure 3

Output power of the signal and idler as the input pump is swept in frequency over the ${\mathrm{WGH}}_{17,0,0}$ resonance. The power of the signal is upshifted by the ratio of the amplitudes $a_{-}/a_{+}$. This upshifted curve nearly directly overlaps the signal curve. The transmission curve of the ${\mathrm{WGH}}_{17,0,0}$ resonance is shown for reference.Reuse & Permissions

Figure 4

Output frequency of the idler (top) and signal (bottom) as the input pump frequency is swept over the ${\mathrm{WGH}}_{17,0,0}$ resonance at 12.037 GHz. The gradient of the “signal” slope is 2.00, meaning ${\Delta }_{+}\approx 2{\Delta }_0$. The frequency of the idler (${\omega }_{-}$) is strongly locked to the WG mode frequency ${\Omega }_{-}$ as the frequency shift is more than 2 orders of magnitude smaller.Reuse & Permissions

Figure 5

Effective nonlinearity and threshold power as a function of the normalized detuning from resonance. The subscripts $u$ and $l$ refer, respectively, to the upper doublet of the ${\mathrm{WGH}}_{17,0,0}$ mode (for positive detuning) and lower doublet (for negative detuning). Here, $g^{\prime }$ ranges between ${10}^{-18}$ and ${10}^{-17}$ and decreases as the pump tunes towards resonance, with FWM turning off between 20 and 40 unloaded half bandwidths (HBWs) from resonance. A maximum in $g^{\prime }$ occurs around 500 unloaded HBWs, with an associated inflection in the required $P_{\mathrm{in}}^{\mathrm{thresh}}$ to enable FWM.Reuse & Permissions