Second-Sound Spectroscopy of a Nonequilibrium Superfluid-Normal Interface

An experiment is proposed to test a previously developed theory of the hydrodynamics of a nonequilibrium heat current-induced superfluid-normal interface. It is shown that the interfacial “trapped” second-sound mode predicted by the theory leads to a sharp resonant dip in the reflected signal from an external second-sound pulse propagated toward the interface when its horizontal phase speed matches that of the interface mode. The influence of the interface on thermal fluctuations in the bulk superfluid is shown to lead to slow power dependence of the order parameter, and other quantities, on distance from it.

Figure 1

Scaled mean field steady state temperature, $M$, and order parameter magnitude, $|\Psi |$, profiles through the interface as derived from (3), using parameters $a_0=b_0=1$, $d_0=2$, and $c_0=0$, which produce $M_{\infty }=-1.936$, $|{\Psi }_{\infty }|=1.216$, and $K_{\infty }=0.676$.

Figure 2

Scattering geometry: An incoming pulse with amplitude $A_{\mathrm{in}}$ and spectrum narrowly centered on horizontal wave vector $\mathbf{q}$ and frequency $\omega$ approaches the interface at velocity ${\mathbf{v}}_g^{-}$. The reflected pulse, with relative amplitude ${\alpha }_{\mathrm{out}}$, moves away at velocity ${\mathbf{v}}_g^{+}$, accompanied by an excitation of the interface itself with relative amplitude ${\alpha }_N$ moving at speed $C_S^0$ along the direction of $\mathbf{q}$. When the resonance condition $\omega /q=C_S^0$ holds, ${\alpha }_{\mathrm{out}}\simeq 0$ (see Fig. 3) and the pulse is strongly absorbed by the interface.

Figure 3

Magnitude and phase (inset) of the reflection coefficient ${\alpha }_{\mathrm{out}}={\alpha }_0^{\mathrm{out}}+{\alpha }_1^{\mathrm{out}}\sqrt{q}$ plotted as functions of the horizontal phase speed $C_S$ for $q=0,0.1,0.2,0.3,0.4,0.5$, using scaled Model F parameters $a_0=b_0=1$, $d_0=2$, $c_0=0$. In both figures, sharper curves correspond to smaller $q$. The minimum allowed phase speed $C_S^{\min }$ occurs for $q_z^{\pm }=q_{z,c}$, hence a purely horizontal group velocity. The large $C_S$ asymptote of ${\alpha }_0^{\mathrm{out}}$ corresponds to a high frequency pulse propagated nearly straight at the interface. For small $q$ the minimum occurs at $C_S=C_{S,0}+D_{S,0}\sqrt{q}$ with value $B_0{D}_{S,0}\sqrt{q}$, $B_0=2a_0^2{M}_{\infty }^2(|{\Psi }_{\infty }{|}^2-2K_{\infty }^2)/9K_{\infty }^2|{\Psi }_{\infty }{|}^2(C_S^0{)}^3$, scaling with the interfacial dissipation constant. The phase switches rapidly (instantaneously for $q\to 0$) between near-$\pi$ to near-zero as $C_S$ increases through $C_S^0$.