Frequency Conversion in a High Q-factor Sapphire Whispering Gallery Mode Resonator due to Paramagnetic Nonlinearity

Nonlinear frequency conversion is a well known and widely exploited family of effects in optics, often arising from a Kerr nonlinearity in a crystal medium. Here, we report high stability frequency conversion in the microwave regime due to a $\chi^{(3)}$ nonlinearity in sapphire introduced by a dilute concentration of paramagnetic spins. First, we produce a high stability comb from two microwave fields at 12.029 and 12.037 GHz corresponding to two high $Q$-factor Whispering Gallery (WG) modes within the Electron Spin Resonance (ESR) bandwidth of the Fe$^{3+}$ ion. The resulting comb is generated by a cascaded four-wave mixing effect with a 7.7 MHz repetition rate. Then, by suppressing four-wave mixing by increasing the threshold power, third harmonic generation is achieved in a variety of WG modes coupled to various species of paramagnetic ion within the sapphire

High Resolution Flicker-Noise-Free Frequency Measurements of Weak Microwave Signals

Amplification is usually necessary when measuring the frequency instability of microwave signals. In this work, we develop a flicker noise free frequency measurement system based on a common or shared amplifier. First, we show that correlated flicker phase noise can be cancelled in such a system. Then we compare the new system with the conventional by simultaneously measuring the beat frequency from two cryogenic sapphire oscillators with parts in 10^15 fractional frequency instability. We determine for low power, below -80 dBm, the measurements were not limited by correlated noise processes but by thermal noise of the readout amplifier. In this regime, we show that the new readout system performs as expected and at the same level as the standard system but with only half the number of amplifiers. We also show that, using a standard readout system, the next generation of cryogenic sapphire oscillators could be flicker phase noise limited when instability reaches parts in 10^16 or betterComment: Accepted for publication in IEEE Transactions on Microwave Theory & Technique

High Q-factor Sapphire Whispering Gallery Mode Microwave Resonator at Single Photon Energies and milli-Kelvin Temperatures

The microwave properties of a crystalline sapphire dielectric whispering gallery mode resonator have been measured at very low excitation strength (E/hf=1) and low temperatures (T = 30 mK). The measurements were sensitive enough to observe saturation due to a highly detuned electron spin resonance, which limited the loss tangent of the material to about 2e-8 measured at 13.868 and 13.259 GHz. Small power dependent frequency shifts were also measured which correspond to an added magnetic susceptibility of order 1e-9. This work shows that quantum limited microwave resonators with Q-factors > 1e8 are possible with the implementation of a sapphire whispering gallery mode system

Single Crystal Sapphire at milli-Kelvin Temperatures: Observation of Electromagnetically Induced Thermal Bistability in High Q-factor Whispering Gallery Modes

Resonance modes in single crystal sapphire ($\alpha$-Al$_2$O$_3$) exhibit extremely high electrical and mechanical Q-factors ($\approx 10^9$ at 4K), which are important characteristics for electromechanical experiments at the quantum limit. We report the first cooldown of a bulk sapphire sample below superfluid liquid helium temperature (1.6K) to as low as 25mK. The electromagnetic properties were characterised at microwave frequencies, and we report the first observation of electromagnetically induced thermal bistability in whispering gallery modes due to the material $T^3$ dependence on thermal conductivity and the ultra-low dielectric loss tangent. We identify "magic temperatures" between 80 to 2100 mK, the lowest ever measured, at which the onset of bistability is suppressed and the frequency-temperature dependence is annulled. These phenomena at low temperatures make sapphire suitable for quantum metrology and ultra-stable clock applications, including the possible realization of the first quantum limited sapphire clock.Comment: 5 pages, 4 figure

Extremely Low-Loss Acoustic Phonons in a Quartz Bulk Acoustic Wave Resonator at Millikelvin Temperature

Low-loss, high frequency acoustic resonators cooled to millikelvin temperatures are a topic of great interest for application to hybrid quantum systems. When cooled to 20 mK, we show that resonant acoustic phonon modes in a Bulk Acoustic Wave (BAW) quartz resonator demonstrate exceptionally low loss (with $Q$-factors of order billions) at frequencies of 15.6 and 65.4 MHz, with a maximum $f.Q$ product of 7.8$\times10^{16}$ Hz. Given this result, we show that the $Q$-factor in such devices near the quantum ground state can be four orders of magnitude better than previously attained. Such resonators possess the low losses crucial for electromagnetic cooling to the phonon ground state, and the possibility of long coherence and interaction times of a few seconds, allowing multiple quantum gate operations