**Contact:**** serge.galliou@femto-st.fr **

**Context and objective of the activity**

For decades, acoustic resonators, i.e. acoustic cavities, have been widely used in on-board frequency sources due to a relative simplicity of implementation. As an example, the so-called Oven Controlled Xtal Oscillators (OCXOs) exhibit fractional frequency instabilities which can be less than 1 10-13over integration times of a few seconds for the best ones. Their performance depends closely on the intrinsic noise and quality-factorQ(the inverse of the mechanical losses) of the resonator. The most popular material for acoustic cavities is quartz. Indeed, in addition to attractive elastic constants, very pure synthetic crystals can be achieved leading to low intrinsic loss. Moreover, it is piezoelectric making easier the electronics to mechanics coupling.

Losses in bulk acoustic waves (BAW) resonators are obviously the sum of losses coming from various origins including engineering losses and intrinsic ones. The engineering process developed for quartz crystal devices during tens of years at FEMTO reduces the former in such a way that they can become negligible in comparison with the latter: two-level-system losses due to impurities are avoid by sweeping the crystal, surface scattering is rejected by chemical-mechanical polishing, losses through holders are minimized by using a monolithic structure with an isolated rim and also mainly by trapping the acoustic energy. Among the remaining intrinsic losses, it can be shown that the thermoelastic ones – irrelevant for shear waves but that might exist for longitudinal waves - become negligible at the operating frequencies,f> 5 MHz. As a consequence, the dominant remaining losses come from the interaction of the acoustic wave with thermal phonons. At room temperatures, an acoustic resonator vibrating on a mechanical mode at a frequencyfexhibits a quality-factorQ, i.e. the inverse of losses, limited byQ´f = constant» 1013. This relationship is derived from the Akheiser’s regime. When the temperature goes down below 10 K, theQ-factorincreases andcan reach incredible values of a few billionwith our state-of-the-art acoustic cavities. Indeed, mechanisms of phonon-phonon interactions at cryogenic temperatures are different from those at 300 K because of the increase of the thermal phonon lifetimeat low temperature(2pftTph>> 1wheretTphis the thermal phonon lifetime):theQbehavioris changed and theoreticallydoes not depend onfanymore, according to Landau-Rumer’s theory predicting that the absorption coefficienta(f)is proportional tof, more preciselya(f)is proportional toT4´f,Tbeing the temperature. Actually, typically beyond 200 MHz, scattering due the surface roughness leads to a decrease of the Q-factor according to af-1law. Fig. 1 shows measurements of such Q-values [1]. Also, a new source of loss was discovered, the acoustic equivalent to Rayleigh Scattering [2].

**Images**

An acoustic cavity exhibiting Q-factor greater than a billion is of course very attractive for various applications. Firstly they can be used as the core of an ultra-stable frequency source. Beyond their use in clocks, these low-loss phonon-trapping cavities offer the opportunity to demonstrate the quantum behavior of mechanical systems when the conditionhf > kBTcan be met (h: Planck const.,kB: Boltzmann const.). Although a thermal phonon occupation number close to 6 has already been achieved [3], investigations are still in progress with these quartz crystal resonators potentially leading to extraordinarily large coherence times beyond the capability of any other competing technology. Below are given research works on more fundamental physics referring to some results mentioned above, and collaborations related to this topic:

- The ARC discovery project “Precision Tests of Fundamental Physics at the Electroweak Unification Scale,” of M. Tobar, Centre of Excellence for Engineered Quantum Systems, UWA, Australia [4],

- Works of the Ion Storage group at NIST, Boulder, CO USA [5],

- Works of P. T. Rakich’s group at Yale University, New Haven, CT USA [6],

- Works of A. Heidmann’s group at Lab. Kastler Brossel, Paris, France [7],

- Works of P. Bushev and S. Danilishin groups, Experimentalphysik, Universität des Saarlandes, Saarbrücken / Institut für Theoretische Physik, Leibniz Universität and Max-Planck Institut für Gravitationsphysik (Albert-Einstein-Institut), Hannover, Germany [8], and others [9],...

Furthermore, because of the low mechanical losses, such an acoustic cavity could also be operated as an optomechanical system, where the standing acoustic wave would be activated by radiation pressure [10]. As a consequence, non-piezoelectric materials, but nevertheless low mechanical loss materials, could also be used...

**Images**

** Bibliography:**

[1] S. Galliou, M. Goryachev, R. Bourquin, Ph. Abbé, J. P. Aubry, and M. E. Tobar, “Extremely Low Loss Phonon-Trapping Cryogenic Acoustic Cavities for Future Physical Experiments,” Nature: Scientific Reports, Sci. Rep. 3, 2132; DOI:10.1038/srep02132 (2013).

http://www.nature.com/srep/2013/130704/srep02132/full/srep02132.html

[2] M. Goryachev, D. L. Creedon, S. Galliou, and M. E. Tobar, “Observation of Rayleigh phonon scattering through excitation of extremely high overtones in low-loss cryogenic acoustic cavities for hybrid quantum systems,” Physical Review Letters 111, 085502, 2013.

[3] M. Goryachev, D. L. Creedon, E. N. Ivanov, S. Galliou, R. Bourquin, M. E. Tobar, “Extremely High Q-factors in milligram Scale Bulk Acoustic Wave Quartz Resonators at milli-Kelvin Temperature,” Appl. Phys. Lett. 100, 243504 (2012).

[4]. A. Lo, P. Haslinger, E. Mizrachi, L. Anderegg, H. Müller, M. Hohensee, M Goryachev and M. E. Tobar, “Acoustic Tests of Lorentz Symmetry Using Quartz Oscillators,” Phys. Rev. X 6, 011018, 2016.

[5]. S. Kotler, R. W. Simmonds, D. Leibfried, and D. J. Wineland, “Hybrid quantum systems with trapped charged particles,” Physical Review A 95, 022327, 2017.

[6]. W. H. Renninger, P. Kharel, R. O. Behunin, and P. T. Rakich, “Bulk crystalline optomechanics” Nature Physics, 14, 601–607, 2018.

[7]. S. Galliou, S. Deléglise, M. Goryachev, L. Neuhaus, G. Cagnoli, S. Zerkani, V. Dolique, J. Bon, X. Vacheret, Ph. Abbé, L. Pinard, C. Michel, T. Karassouloff, T. Briant, P.-F. Cohadon, A. Heidmann, M. E. Tobar, R. Bourquin, “A new method of probing mechanical losses of coatings at cryogenic temperatures”, Rev. of Scientific Instruments, Vol. 87, Issue 12, 123906 (2016); http://doi.org/10.1063/1.4972106. https://arxiv.org/abs/1605.07697

[8]. P. A. Bushev, J. Bourhill, M. Goryachev, N. Kukharchyk, E. Ivanov, S. Galliou, M. E. Tobar, and S. Danilishin, “Testing of Quantum Gravity with Sub-Kilogram Acoustic Resonators,” arXiv:1903,03346v1 [quant-ph], 8 Mar 2019.

[9]. M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” Review of Modern Physics, 86, 4, 139, 2014.

[10] J. Bon, L. Neuhaus, S. Deléglise, T. Briant, Ph. Abbé, P.-F. Cohadon, S. Galliou, “Cryogenic optomechanic cavity in low mechanical loss material”, Journal of Applied Physics, 124, 073104, August 2018. https://doi.org/10.1063/1.5042058