Probing superfluid ${}^4\mathrm{He}$ with high-frequency nanomechanical resonators down to millikelvin temperatures

Superfluids, such as superfluid ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$, exhibit a broad range of quantum phenomena and excitations which are unique to these systems. Nanoscale mechanical resonators are sensitive and versatile force detectors with the ability to operate over many orders of magnitude in damping. Using nanomechanical-doubly clamped beams of extremely high quality factors ($Q>{10}^6$), we probe superfluid ${}^4\mathrm{He}$ from the superfluid transition temperature down to millikelvin temperatures at frequencies up to 11.6 MHz. Our studies show that nanobeam damping is dominated by hydrodynamic viscosity of the normal component of ${}^4\mathrm{He}$ above 1 K. In the temperature range of 0.3–0.8 K, the ballistic quasiparticles (phonons and rotons) determine the beams' behavior. At lower temperatures, damping saturates and is determined either by magnetomotive losses or by acoustic emission into helium. It is remarkable that all these distinct regimes can be extracted with just a single device, despite damping changing over six orders of magnitude.

Figure 1

False color scanning electron microscope image (SEM) of a $150-\mu \mathrm{m}$-long composite aluminum on a silicon nitride nanobeam. The device is driven by a network analyzer through 80 dB of attenuation distributed over several temperature stages of the cryostat. At the output, two 40-dB low-noise amplifiers are used at room temperature to improve the signal-to-noise ratio. The inset shows an example of a frequency response taken in vacuum at 7 mK with a magnetic field of 10 mT demonstrating a quality factor of $5\times {10}^6$.

Figure 2

(a) Magnetic-field dependence of damping for 150- and $30-\mu \mathrm{m}$-long beams operated in vacuum and superfluid ${}^4\mathrm{He}$ at 10 mK. Damping at high magnetic fields follows $B^2$ dependence (shown with dashed lines) as a result of magnetomotive loading in nanobeams [21]. Towards low magnetic fields, $Q_{\mathrm{tot}}^{-1}$ becomes constant as we approach the internal damping of the beams in vacuum along with acoustic emission in the liquid. The inset: Frequency dependence for acoustic damping in superfluid ${}^4\mathrm{He}$. The dashed line shows the $f_0^2$ dependence calculated by Eq. (4) for dipole acoustic emission with no fitting parameters [22]. (b) Temperature dependence of damping for a $150-\mu \mathrm{m}$-long sample immersed in liquid ${}^4\mathrm{He}$ measured in a range of magnetic fields. Notations on the top indicate dominant damping mechanisms for each temperature range with $T_{\lambda }=2.17\mathrm{K}$ indicating the superfluid transition temperature. The damping of the nanobeam below 1 K is explained by ballistic scattering of the thermal excitations in superfluid ${}^4\mathrm{He}$ on the nanobeam surface. The combined damping due to phonons Eq. (2) and rotons Eq. (3) is shown by the blue dashed line. The red solid line shows total damping calculated as a sum of phonon, roton, and acoustic emission term Eq. (4). For magnetic fields exceeding 0.1 T, the magnetomotive damping becomes dominant at low temperatures.