Surface State Dissipation in Confined ${}^3\mathrm{He}\text{−}A$

We have studied the power dependence of superfluid Helmholtz resonators in flat (750 and 1800 nm) rectangular channels. In the $A$ phase of superfluid ${}^3\mathrm{He}$, we observe a nonlinear response for velocities larger than a critical value. The small size of the channels stabilizes a static uniform texture that eliminates dissipative processes produced by changes in the texture. For such a static texture, the lowest velocity dissipative process is due to the pumping of surface bound states into the bulk liquid. We show that the temperature dependence of the critical velocity observed in our devices is consistent with this surface-state dissipation. Characterization of the force-velocity curves of our devices may provide a platform for studying the physics of exotic surface bound states in superfluid ${}^3\mathrm{He}$.

Figure 1

$A$-phase flow. (a) Simplified phase diagram showing the range of temperatures where the $A$ phase exists for 750 and 1800 nm channels. The red and blue points are the transition temperatures measured for both devices and the dashed lines are fits. The $A$ phase is stabilized to lower temperatures and pressures by the tight confinement [27]. (b) Drawing of the confined Helmholtz resonator volume. The dimensions of the channel are $D\times 1.6\mathrm{mm}\times 1.38\mathrm{mm}$, where $D=750$ or 1800 nm. (c) Momentum space plot of the Fermi surface (red) and the gap which goes to zero at the poles aligned with the anisotropy vector $\widehat{\ell }$. The plot shows the spatial dependence of ${\mathrm{\Delta }}_A(z)$, in the highly confined dimension, where the distance from the wall is in units of the coherence length $\xi$. The spatial dependence has been computed using Ginzburg Landau theory [2].

Figure 2

Characterizing the nonlinear regime. (a) The mass current, calibrated from the detector peak voltage, for the 750 nm device at 22.45 bar and 2.08 mK is plotted as a function of the pressure gradient across the channels $|\nabla P|$. The bias voltage $V_{\mathrm{dc}}$ is held constant throughout the experiment such that $⟨j_s⟩a\propto V_{\mathrm{DET}}$ and $|\nabla P|\propto V_{\mathrm{ac}}$. The inset highlights the linear drive regime and a critical value at which the slope abruptly changes. The dashed line is a fit to the linear regime data used to highlight the change in slope. The dotted lines indicate the point at which the slope changes. (b) Log-plot of the Helmholtz resonance with drive voltages ranging from 1 to 50 mV. The resonances are normalized by the drive voltage. Near resonance, the line shape distorts above the critical drive.

Figure 3

Temperature scaling. (a) Plot of the peak voltage as a function of temperature for the 750 nm device at 22.45 bar (green squares), and 27.94 bar (blue squares), as well as 1800 nm device at 22.45 bar (red circles), and 27.94 bar (orange circles). The dashed lines show fits to functions of the form $(1-BT/T_{c,1800}{)}^{n/2}$. The parameter $B$ is used to rescale the critical temperature for the 750 nm device. The 22.45 bar and 27.94 bar datasets are fit together for the 1800 nm device but separately for the 750 nm device. (b) Measured frequency dependence of the Helmholtz modes. (c) Superfluid density of each device as calculated from the resonant frequency. The gray curve is the bulk superfluid fraction.

Figure 4

Critical velocity. (a) The critical current of the Helmholtz resonator is computed from the resonance amplitude at which the linear regime ends, and the superfluid density from the center frequency of the resonance as $v_c=⟨j_s⟩/{\rho }_{s,c}$. The ratio of these values is the critical velocity. The dashed line is a fit to all datasets of the form $v_0\sqrt{1-T/T_c}$. (b) Plot of the quasiparticle dispersion relation when the fluid is at rest. The solid lines represent the bulk dispersion relation, and the dashed lines the bound state dispersion. (c) Plot of the quasiparticle dispersion relation at a finite critical velocity when the maximum energy value of the lower-band bound states is equal to the minimum energy value of the upper-band bulk states (highlighted by the dotted line).

Figure 5

Plot of the resonance peak height as a function of drive voltage, measured for the 1800 nm device at 28 bar for a variety of temperatures near the $A\text{−}B$ critical temperature. The $A$-phase data are plotted in circles, and the $B$ phase in squares. Lower temperature measurements have higher amplitudes due to the increased superfluid fraction. For low drives, the drive-amplitude relationship is linear for both $A$ and $B$ phases. Above the $A$-phase critical velocity, the trend deviates for the $A$ phase as the line shape becomes distorted with a flat top, whereas in the $B$ phase this flattening does not occur.