Precise equation of state measurements of $^4$He near the $\lambda$-point, using dual-mode Superconducting Cavity Stabilized Oscillators

We report on progress towards precise equation of state measurements of $^4$He saturated vapor near the $\lambda$-point using a Superconducting Cavity Stabilized Oscillator (SCSO) system. By operating the SCSO in a dual-mode phase-locked loop configuration we will be able to measure the dielectric constant of $^4$He to parts in $10^{15}$ precision and comparable accuracy. The dielectric constant in turn implies a value of the density to parts in $10^{10}$. Other measured parameters include the temperature to sub-nK precision using paramagnetic salt high-resolution thermometry (HRT) and pressure to parts in $10^9$ using a Straty-Adams type diaphragm gauge. These substantially improved resolutions relative to existing data are expected to provide new insights into the interactions of helium atoms near Bose-Einstein condensation. Numerous error reduction techniques will be discussed, along with other applications of SCSO to precision metrology.