Roger Bourquin

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 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 × 1016 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.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 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 × 1016 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.

Extremely Low Loss Phonon-Trapping Cryogenic Acoustic Cavities for Future Physical Experiments

Low loss Bulk Acoustic Wave devices are considered from the point of view of the solid state approach as phonon-confining cavities. We demonstrate effective design of such acoustic cavities with phonon-trapping techniques exhibiting extremely high quality factors for trapped longitudinally-polarized phonons of various wavelengths. Quality factors of observed modes exceed 1 billion, with a maximum Q-factor of 8 billion and Q × f product of 1.6 · 1018 at liquid helium temperatures. Such high sensitivities allow analysis of intrinsic material losses in resonant phonon systems. Various mechanisms of phonon losses are discussed and estimated.

Losses in high quality quartz crystal resonators at cryogenic temperatures

Measurement of the mechanical losses of quartz crystal is a topic of interest for communities dealing with the gravitational wave detectors and also the time and frequency domain. About the latter, the authors describe Q-factor measurements of quartz crystal resonators at cryogenic temperatures under 10 K, thanks to a cryocooler-based experimental set-up. A Q-factor of 325 millions at 4 K, on the fifth overtone of the quasilongitudinal mode at 15.9 MHz, has been recorded. As shown, the acoustic wave trapping is suspected to limit the Landau–Rumer regime below 6 K [Landau and Rumer, Phys. Z. Sowjetunion 11, 18 (1937)].

Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. I. Theory

After some necessary recalls on the nonlinear theory of thermoelectroelasticity in piezoelectric crystals, asserting the need of constitutive equations which derive from a rotationally invariant energy function, this paper presents the governing equations for a small vibration superimposed on a bias originated by a slow and homogeneous temperature variation from a well-defined reference state. Thereafter, the authors define the effective coefficients appearing in the linearized incremental dynamic balance equations for linear momentum and electrical charge in Lagrange configuration, not omitting associated boundary conditions. The main features of these coefficients are discussed and explicit relations with more conventionally defined coefficients are given. Determination of numerical values of the proposed effective coefficients and examples of their use in the higher order modeling of static frequency-temperature characteristics of either bulk acoustic wave or surface acoustic wave devices are given in a companion paper.

Recent investigations on BAW resonators at cryogenic temperatures

This work presents the results of investigations on different aspects of cryogenic operation of BAW resonators. For the quartz crystal resonators the losses mechanisms, the temperature sensitivity, the amplitude-frequency effect are described for liquid helium temperatures. The corresponding problems are discussed. To overcome some of these problems, the possible solution of operation at higher modes is considered. Some of these higher modes exhibit outstanding quality factors. The highest (for BAW resonators) quality factor value and quality factor-frequency product are measured for the 11th and 13th overtones respectively. In addition, two LGT resonators have been characterized in a wide temperature range. Finally, some preliminary results on utilization of cryogenic quartz resonators as a part of a frequency stabilization closed loop system are given.

Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. II. Applications and numerical values for quartz

This paper presents a set of numerical values of temperature derivatives of Lagrangian effective elastic coefficients suitable for the modeling of small vibrations in quartz devices submitted to slow and homogeneous temperature variations. After a short description of the proper writing of vibration problems in practical applications with the help of this kind of coefficient, we determine the proposed set of numerical values from the frequency-temperature characteristics of an heterogeneous set of bulk and surface acoustic wave devices.

Investigations on LGS and LGT crystals to realize BAW resonators

The LGS family are promising materials for the design of high quality bulk acoustic wave resonators. We have manufactured many plano-convex 10 MHz 5th overtone Y-cut resonators using langasite (LGS, La3Ga5SiO14) and langatate (LGT, La3Ga5.5Ta0.5O14) crystals. We observed that the quality factor strongly depends on the polishing method, the supplier of the material, and on the energy trapping. For quartz crystals, we have found that resulting IR spectra exhibit absorption peaks more or less deep, linked to defects. These predominant criteria are not surprising, but they have to be defined in manner similar to that used for quartz crystal. A satisfying machining and polishing method has been first applied to elaborate high Q resonators, and a comparison between samples of LGS and LGT materials from different suppliers is established. In addition, LGT resonators are characterized by their motional parameters and frequency-temperature curves. Nevertheless, one of the main results is that the measured Q times f product is not the expected one. We present results of Q-factor versus radius of curvature: it appears that an optimization should be performed and that this last one cannot be directly transposed from that of quartz crystal resonator. Currently, the best resonator that we have made has a Q times f product of 1.4 times 1013 on its 5th overtone (1.7 times 1013 on its 9th overtone). This result is slightly higher than the similar parameter obtained on a state-of-the-art SC-cut quartz crystal resonator working at the same frequency.

Experimental study of temperature effects in vibrating beam and thickness-shear resonators of GaPO/sub 4/ machined by ultrasonic milling

We report progress in modeling and measuring temperature effects in vibrating beam gallium orthophosphate (GaPO/sub 4/) resonators. In addition to the well known thickness-shear AT-cut, temperature compensated cuts exist in GaPO/sub 4/ for length extensional, flexural and torsional modes. Analytical models of temperature effects are compared with experimental measurements on devices fabricated on AT (Y-16.1/spl deg/) and Z-cut plates by ultrasonic machining. The agreement between theory and experiment is fairly good taking into account that temperature coefficients of GaPO/sub 4/ have been experimentally tested for only a few thickness-shear resonators.

Isochronism defect for various doubly rotated cut quartz resonators

It has been shown in earlier work, that the amplitude-frequency effect (also called isochronism defect or anisochronism) could be a limitation factor on ultra-stable oscillators. Theoretical studies based on the non-linear theory of the piezoelectricity have already been developed to explain the amplitude-frequency effect. So it is possible to estimate the dependence of the isochronism defect versus design parameters of resonators (radius of curvature, electrodes diameter, overtone rank, ...). However, due to the unknown of the fourth order elastic coefficients, it is not possible to predict isochronism defect of any resonant frequency of a given energy trapped resonator. To tentatively find the orientation of a plate which does not exhibit an isochronism defect, we have realised electroded resonators with different orientations and curvatures. We present results which verify, particularly, R/sup -1/2/ dependence of amplitude-frequency effect versus radius of curvature. Moreover, we show that the isochronism defect can be positive or negative and can vary from one orientation to other one of about two orders of magnitude.

Fluctuation-dissipation theorem and 1/f noise of bulk acoustic wave cavities

The short time stability in quartz crystal ultra-stable oscillators is limited by some intrinsic frequency fluctuations with a 1/f power spectral density. The physical origin of this 1/f noise is still unknown. In this paper, the fluctuation dissipation theorem is used to numerically evaluate the importance of the contribution of internal damping of thickness fluctuations on the level of noise for bulk acoustic wave cavities. A 1/f contribution to the power spectral density of frequency noise for the shear mode usually used is obtained. Its study provides a possible explanation for the fact that, for the best resonators, the 1/f noise amplitude is usually disconnected from the quality factor measured at resonance.