Bistable Fourth Sound Resonance in Superfluid $^3$He-B due to Gap Suppression

Abstract

Superfluidity in $^3$He exhibits many unique properties that are of interest to modern condensed matter research, including multiple superfluid phase transitions, topological defects, and exotic classes of excitations like Majorana and Weyl fermions. Many of the most interesting theoretical proposals, which remain underexplored, are realized in highly confined geometries, where surface effects play a dominant role in the thermodynamic and hydrodynamic properties. We have developed nanofluidic resonators capable of exciting a fourth-sound acoustic mode in thin channels with a highly confined dimension ($750-1800$ nm) that is only $1-2$ orders of magnitude larger than the superfluid coherence length. When a sufficiently large drive force is applied, we observe a non-linear softening of the resonance that we interpret as due to the flow suppression of the superfluid gap. We have developed a model of the device that allows the resonance amplitude to be calibrated into a superfluid velocity, which exhibits critical behavior at particular velocities. We identify one of the observed critical velocities as being the velocity at which the gap component parallel to the flow is suppressed to zero. We compare the calibrated velocity to the prediction of a Ginzburg-Landau model, and find reasonable agreement. This measurement represents an ongoing effort to link the hydrodynamic measurements of these nanofluidic devices to theoretical predictions regarding surface gap suppression and surface-bound states.

Figures (4)

• Figure 1: a) Example of a bistable frequency response curve measured by sweeping the drive frequency of the Helmholtz resonator. Due to the hysteretic nature of the bistability, the measured branch depends on the initial frequency when sweeping through the resonance. Here, red data points correspond to the lower branch measured by sweeping the frequency from low to high (dubbed the "forward" frequency sweep), and blue data points show a frequency sweep from high to low frequencies ("reverse" sweep). b) Simplified diagram of the bridge circuit used to measure the time-varying capacitance signal from the Helmholtz resonator. c) Simplified cross-section of a Helmholtz resonator device where the basin has been compressed by $\delta z$. This has the effect of increasing the fluid pressure to $P_1$, which differs from the external pressure by $\delta P = P_2-P_1$, which drives motion in the fluid with velocity $v_s$. d) Diagram of the confined volume of the Helmholtz resonator with dimensions. e) Phase diagram of superfluid $^3$He. The solid black line denotes the superfluid critical temperature, and the dashed black line the bulk A to B transition curve. The red and blue dashed lines denote the A-B transition line for a parallel plate geometry with confined dimensions of 750 and 1800 nm respectively. f) Plot of the reduction in superfluid density due to velocity-dependent gap suppression. Here, a simple parabolic dependence is assumed, which does not account for the multiple order parameter components of the planar distorted B-phase. The velocity at which $v_s=v_c$, such that $\rho_s = 0$, is the maximal pair-breaking critical velocity.
• Figure 2: a) Plot of the non-linear drive scaling of the 1800 nm device at 25 bar and $T = 0.72 T_c$ for drive voltages of 10-50 mV. The maximum values are marked with squares in a color corresponding to the drive voltage in the legend. For each drive voltage, two parameters are extracted: the maximum amplitude, and the frequency at which the maximum amplitude occurs $f_c$. These coordinates trace the backbone of the non-linear resonance. b) The maximum amplitude of the voltage resonance is converted into a mass flux using Equation \ref{['eq:vcal']}. This is then converted to normalized by the critical velocity expected by the GL theory $v_{sb}/v_c = (2m\xi/\hbar)v_{sb}$. The square of this normalized velocity is then plotted against $1-(\omega_{b}/\omega_{00})^2$, to test the relationship predicted by the Duffing like backbone equation seen in equation \ref{['eq:backbone']}.
• Figure 3: a) Plot of the calibrated superfluid velocity as a function of drive voltage at 25 bar and $T=0.72 T_c$ for the 1800 nm device. Two data sets are superimposed (one plotted as squares, the other circles), which were taken on different dates but are otherwise identical. A discontinuity in the slope can be identified at a critical velocity $v_{c1}^{\textrm{cal}} = 31$ mm/s, which in normalized units natural to the GL theory is $(2m^* \xi/\hbar) v_{c1}^{\textrm{cal}} = 0.345$. b) Plot of the superfluid fraction calibrated from the resonant frequency. A change in slope is observed at the same critical velocity. c) A plot from a Ginzburg-Landau simulation performed for an 1800 nm parallel plate geometry at 25 bar and $T=0.72 T_c$ of the order parameter components $a_{11}(z=d/2)$ (blue), and $a_{22}(z=d/2) \approx a_{33}(z=d/2)$ (orange) at the center of the slab (here only $a_{22}$ is plotted as the difference is too small to resolve). The component $a_{11}(z=d/2)$ can be seen to go to zero at the critical velocity $v_{c1}^{\textrm{GL}}/v_c = 1/\sqrt{5}$ (marked by the dashed black line), which represents a phase transition from a B-like phase to a planar-like phase. d) Plot of the normalized mass current as a function of normalized velocity calculated using the same GL model.
• Figure 4: a) The Helmholtz resonance for the 750 nm device is plotted for the reverse sweep direction at a fixed drive voltage (50 mV) for a range of temperatures at 15.7 bar. The resonance shifts up in frequency due to the temperature dependence of the thermodynamic (i.e., velocity-independent) superfluid density $\rho_{s,0}(T)$. The shape of the curve can also be seen to change, becoming more strongly non-linear at lower temperatures. This is due to the temperature dependence of the critical velocity changing the relative scale of the Duffing non-linearity. b) The same resonances are shown, but shifted by subtracting the center frequency $f_0$. The colored boxes mark the point at which the frequency drops discontinuously due to the switching between two branches. For a Duffing oscillator, this point coincides with the maximum of the curve. Here, some distortion of the Duffing-like line shape can be observed for the lower temperatures, due to the neglected term in Equation \ref{['eq:u_full']}. Despite this, the frequency discontinuity points can be fit to a quadratic function, shown as a dashed black curve.