Frequency-dependent hydrodynamic inductance and the determination of the thermal and quantum noise of a superfluid gyroscope

We reexamine mass flow in a superfluid gyroscope containing a superfluid Josephson weak link. We introduce a frequency-dependent hydrodynamic inductance to account for an oscillatory flow of the normal fluid component in the sensing loop. With this hydrodynamic inductance, we derive the thermal phase noise, and hence the thermal rotational noise of the gyroscope. We examine the thermodynamic stability of the system based on an analysis of the free energy. We derive a quantum phase noise, which is analogous to the zero-point motion of a simple harmonic oscillator. The configuration of the studied gyroscope is analogous to a conventional superconducting RF SQUID. We show that the gyroscope has very low intrinsic noise $(1.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}\phantom{\rule{0.3em}{0ex}}\mathrm{rad}\phantom{\rule{0.2em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}∕\sqrt{\mathrm{Hz}})$, and it can potentially be applied to study general relativity, Earth science, and to improve global positioning systems (GPS).