Superfluid Vortex Front at $T\to 0$: Decoupling from the Reference Frame

Steady-state turbulent motion is created in superfluid ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$ at low temperatures in the form of a turbulent vortex front, which moves axially along a rotating cylindrical container of ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$ and replaces vortex-free flow with vortex lines at constant density. We present the first measurements on the thermal signal from dissipation as a function of time, recorded at $0.2T_c$ during the front motion, which is monitored using NMR techniques. Both the measurements and the numerical calculations of the vortex dynamics show that at low temperatures the density of the propagating vortices falls well below the equilibrium value, i.e., the superfluid rotates at a smaller angular velocity than the container. This is the first evidence for the decoupling of the superfluid from the container reference frame in the zero-temperature limit.

Figure 1

Measuring setup. Two independent cw NMR spectrometers are used to monitor vortex-front motion in the sample container above the $\varnothing$ 0.75 mm orifice. The middle section with the quartz tuning fork oscillators is employed for thermometry and quasiparticle bolometry. The bottom section below the $\varnothing$ 0.3 mm orifice provides the thermal contact to the refrigerator. The superconducting solenoid in the center of the NMR sample produces the $H_b$ field for stabilizing the ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{A}$ phase. This A-phase barrier layer divides the NMR sample in two ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$ sections of equal length.

Figure 2

Two $\Omega$-scaled bolometer signals. Time $t=0$ is assigned to the moment when the trigger signal (cooling due to the latent heat in B to A transition) is observed. This trigger signal and the background heat leak have been removed by subtracting the equivalent signal measured in the equilibrium vortex state. The green curve represents a fit to the experimental record at $\Omega =1.2\mathrm{rad}/\mathrm{s}$. The thermal time constant of the bolometer is about 25 s, which is much shorter than the time scale of front propagation.

Figure 3

Precessing motion in the vortex front and the vortex bundle behind it. Top: Oscillations start in the NMR signal (total transverse magnetization $M_{\perp }$ [17]) when the front arrives at the pick-up coil. Their source is the precessing motion of vortices. Bottom: Fourier transforms of parts of the NMR signal marked in the top panel: The front (red line) and the cluster behind it (blue line). Since the NMR coil is fixed to the rotating frame, the precession frequencies in the laboratory frame are counted from the angular velocity $\Omega$, as shown by the arrows.

Figure 4

Vortex front in numerical calculations. Snapshots of vortex configurations at different temperatures are analyzed when the front is located at the same axial position $z=34\mathrm{mm}$. Top: Azimuthal velocity $⟨v_s{⟩}_{\varphi }={\Omega }_{s}r$ as a function of $z$, when averaged over the vortices in the cross section $0\le r\le 0.8R$. The vortex-free state is before the front ($z>35\mathrm{mm}$) and the quasiequilibrium vortex state with ${\Omega }_s(T)<\Omega$ is behind the front. Inset: The precession frequency of the vortex ends in the front region ($z\ge 29\mathrm{mm}$), value of ${\Omega }_s$ averaged over $z$ behind the front, and the interpolation formula Eq. (6) plotted as a function of the mutual friction parameter $\alpha (T)$. The uncertainty bars denote the $z$-averaged 95% confidence limit. Bottom: Turbulence in the front is supported by vortex reconnections, which have been counted here within bins of width $\Delta z=2\mathrm{mm}$ over a time interval in which the front propagates 0.5 mm. Parameters: cylinder length 40 mm, $R=1.5\mathrm{mm}$, $\Omega =1.0\mathrm{rad}/\mathrm{s}$, $\alpha (T)=33.2\exp (-1.97T_c/T)$.