1 INTRODUCTION

Experiments with an aerogel oscillating in superfluid 3He are widely used to study the effect of impurities (magnetic or nonmagnetic) on the superfluid properties of 3He. As is known, in the case of p-wave pairing, even nonmagnetic impurities can lead to the suppression of superfluidity in the system [1]. In addition to a trivial suppression of the order parameter, the symmetry of the aerogel gives rise to superfluid phases, the existence of which is energetically unfavorable in pure 3He [2–4]. One of these phases is the polar phase of superfluid 3He, for which an analogue of the Anderson theorem [5] is valid under certain conditions; i.e., impurities do not affect its thermal properties. The first experiments of the type mentioned above, in which silica aerogels were used, i.e., those formed by SiO2 strands, were reported in [6–8]. One of the aims of these works was to demonstrate that superfluid 3He in the aerogel is in the Larkin–Imry–Ma state characteristic of the A phase [9]. This technique is currently used to study the properties of superfluid 3He in the so-called nematic aerogels, which are formed by co-directed strands [10]. In the course of experiments with this type of aerogels, the β phase of superfluid 3He was discovered [4], and the effect of magnetic impurities on the phase diagram of superfluid 3He was studied [11].

In all experiments of this type, the aerogel was attached to a thin U-shaped superconducting wire, placed in a cell filled with 3He, and a dc magnetic field was applied to the system. Next, an alternating electric current was passed through the wire, which, due to the Lorentz force acting on the wire, led to the excitation of mechanical vibrations of the system. In the simplest consideration of these oscillations, the frequency squared should be inversely proportional to the total mass of the system \({{\omega }^{2}} \sim \frac{1}{{{{M}_{\Sigma }}}}\), which in turn includes the masses of the wire, aerogel, normal 3He component in the aerogel, and the added mass of 3He outside aerogel, which is involved in the system motion. The emergence of a superfluid phase in the aerogel changes the streamlines of the superfluid component inside and outside the aerogel, which, in turn, changes the total added mass of the system and, as a consequence, the frequency of vibrations. To calculate the currents induced by vibrations, it is necessary to know the boundary conditions at the aerogel surface. Two types of boundary conditions were considered in [7, 12, 13]. In particular, it was assumed in [12, 13] that the superfluid current and the phase of the Cooper pair condensate are continuous at the surface of the aerogel. The latter condition is not always valid and can only be justified under the condition of continuity of the order parameter at the aerogel boundary [13]. It is important for further discussion that the aerogel in [13] was treated as an absolutely rigid body. Despite significant differences in the boundary conditions considered in [7, 12], the asymptotic temperature dependence of the vibration frequency far from the superfluid transition point in the aerogel is almost the same in both cases. This occurs because the imposed boundary conditions lead to the situation, where the superfluid component both inside and outside the aerogel at sufficiently low temperatures is generally not carried away by the aerogel motion, and the entire temperature dependence is due only to a decrease in the mass of the normal component involved in the motion. It is surprising that the superfluid density in the aerogel calculated within the aforementioned models is significantly higher than that of bulk 3He at the same temperatures, which clearly contradicts the assumption that the order parameter is suppressed by impurities. In [12], it was assumed that this discrepancy is due to the interaction of the mechanical vibration mode with another one, which was also excited in the reported experiments and is described as an analogue of the second sound for this complex system [14]. However, since the second branch of vibrations was not observed in [8], and the calculated superfluid density was also high, this interaction is not significant for the interpretation of the observed effect.

In this work, we demonstrate that a faster increase in the frequency of mechanical vibrations of the system on cooling can be explained by imposing the hydrodynamic boundary conditions at the surface of the aerogel implying the continuity of hydrodynamic flows. Thus, in contrast to [13], the condition of continuity for the phase of the order parameter should be replaced by the continuity of the components of the momentum flux tensor in the direction normal to the aerogel surface. As we show below, the results of solving hydrodynamic equations indicate that, a phase difference arises between two superfluid states in a narrow layer near the aerogel boundary, and the superfluid current flowing across the boundary is proportional to this phase difference. This type of the relation between the current and the phase jump corresponds to the case of a Josephson junction between superfluid states inside and outside the aerogel, i.e., to the weak coupling case.

2 EQUATIONS OF MOTION OF AEROGEL IN A SUPERFLUID

In this section, we consider the effective equation of motion of the spherical aerogel that is located in superfluid 3He and oscillates due to the elasticity of the wire, to which it is attached. The exact solution of the problem requires the accurate calculation of the stresses in the aerogel and the wire, which, in particular, depend on the method of attaching the aerogel. We consider a composite system in the form of a simple oscillator with a given effective stiffness. Since the depth of viscous penetration for the vibration frequencies under study far exceeds the distance between the aerogel strands, we assume that the normal component of 3He in the aerogel moves together with the aerogel backbone. First, we consider the motion of the system disregarding the effects of viscosity that arise outside the aerogel. Since the motion of the superfluid liquid is potential, we introduce the corresponding potentials \(\varphi _{s}^{{{\text{in}}}}\) and \(\varphi _{s}^{{{\text{out}}}}\) inside and outside the aerogel. The gradients of these functions determine the vector field of superfluid velocities inside and outside the aerogel. The motion of the normal component outside the aerogel in the first approximation (at small vibration amplitudes and disregarding the viscosity) can also be considered as potential and is specified by the field \(\varphi _{n}^{{{\text{out}}}}\). Let \(u_{i}^{{(0)}}\) be the average displacement vector of the aerogel from the equilibrium position, i.e., \(u_{i}^{{(0)}} = \frac{1}{{{{V}_{0}}}}\int dV{{u}_{i}}({\mathbf{r}})\), where \({{u}_{i}}({\mathbf{r}})\) is the displacement field in the aerogel, the integral is taken over the volume of the aerogel, and V0 is the volume of the aerogel at equilibrium. Then, the hydrodynamic equation integrated over the volume, which corresponds to the conservation law of the total momentum in the aerogel and in 3He as a whole [15], determines the effective equation of motion of the system in the form

$$\begin{gathered} {{V}_{0}}({{{\tilde {\rho }}}_{a}}{{\delta }_{{ij}}} + {{(\rho _{n}^{{{\text{in}}}})}_{{ij}}})\ddot {u}_{j}^{{(0)}} + {{(\rho _{s}^{{{\text{in}}}})}_{{ij}}}\int {{\nabla }_{j}}\frac{{\partial \varphi _{s}^{{{\text{in}}}}}}{{\partial t}}dV \\ + \,{{V}_{0}}{{{\tilde {\rho }}}_{a}}\omega _{0}^{2}u_{i}^{{(0)}} + \oint {{n}_{j}}\delta \sigma _{{ij}}^{{{\text{out}}}}dS = 0, \\ \end{gathered} $$
(1)

where \({{\tilde {\rho }}_{a}}\) is the effective aerogel density (taking into account a nonzero mass of the wire, to which the aerogel is attached), ρl is the density of the liquid, \({{(\rho _{{s,n}}^{{{\text{in}}}})}_{{ij}}}\) are the density tensors for the normal and superfluid components of the liquid in the aerogel, \(\delta \sigma _{{ij}}^{{{\text{out}}}}\) is the change in the momentum flux tensor outside the aerogel related to the liquid flow, \({{n}_{i}}\) is the outer normal to the aerogel surface, and \({{\omega }_{0}}\) is the frequency of vibrations for the system in vacuum; in the last term, the integral is taken over the aerogel surface. The first two terms in Eq. (1) determine the change in the momentum of the chosen volume of the system, and the second two terms determine the force acting on the system from the wire and from the liquid surrounding the aerogel. The representation \({{V}_{0}}{{\tilde {\rho }}_{a}}\omega _{0}^{2}u_{i}^{{(0)}}\) for the force acting on the chosen volume of the system from the wire is a simplification discussed at the beginning of the section.

First, we calculate the force acting on the spherical particle from the normal component of the liquid surrounding the aerogel. From the condition of potential character of the motion outside the aerogel, the momentum flux tensor can be expressed in the form

$$\delta \sigma _{{ij}}^{{{\text{out}}}} = - {{(\rho _{s}^{{{\text{out}}}})}_{{ij}}}\frac{{\partial \varphi _{s}^{{{\text{out}}}}}}{{\partial t}} - {{(\rho _{n}^{{{\text{out}}}})}_{{ij}}}\frac{{\partial \varphi _{n}^{{{\text{out}}}}}}{{\partial t}},$$
(2)

where the second term determines the required contribution of the normal component. Since the normal component of the liquid does not flow through the   aerogel, we have the boundary condition \({{(\rho _{n}^{{{\text{out}}}})}_{{ij}}}({{\nabla }_{j}}\varphi _{n}^{{{\text{out}}}} - \dot {u}_{j}^{{(0)}}){{n}_{i}}\) = 0 at the aerogel surface. For the isotropic B phase or for the A phase with the “hedgehog” texture of the vector orbital l (the vector l is everywhere perpendicular to the surface), this condition reduces to the standard expression \(\dot {u}_{i}^{{(0)}}{{n}_{i}}\) = \(({{\nabla }_{i}}\varphi _{n}^{{{\text{out}}}}){{n}_{i}}\). Since the liquid can be treated as incompressible in the frequency range under study, we use the well-known solution for \(\varphi _{n}^{{{\text{out}}}}({\mathbf{r}},t)\) = \( - \frac{{{{R}^{3}}}}{{2{{r}^{2}}}}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}{{n}_{i}}\)of the equation \(\Delta \varphi _{n}^{{{\text{out}}}} = 0\), satisfying the specified boundary conditions for the spherical sample of the radius R [16]. The force acting on the sphere from the normal component of the liquid is determined by the integration over the surface of the particle:

$$ - \oint {{n}_{j}}{{(\rho _{n}^{{{\text{out}}}})}_{{ij}}}\frac{{\partial \varphi _{n}^{{{\text{out}}}}}}{{\partial t}}dS = \frac{{2\pi }}{3}{{R}^{3}}{{(\rho _{n}^{{{\text{out}}}})}_{{ij}}}\frac{{{{\partial }^{2}}u_{j}^{{(0)}}}}{{\partial {{t}^{2}}}},$$
(3)

where \(\frac{1}{2}{{(\rho _{n}^{{{\text{out}}}})}_{{ij}}}{{V}_{0}}\) specifies the tensor of the added mass arising due to the motion of the normal liquid around the aerogel.

Equation (1) with substituted Eq. (2) includes two unknown functions \(\varphi _{s}^{{{\text{in}}}}\) and \(\varphi _{s}^{{{\text{out}}}}\), which are determined as follows. First, we determine the phase of the superfluid in the aerogel from the condition of the potential character of the superfluid motion:

$$\frac{{\partial \varphi _{s}^{{{\text{in}}}}}}{{\partial t}} = - \delta \mu _{l}^{{{\text{in}}}},$$
(4)

where \(\delta \mu _{l}^{{{\text{in}}}}\) is the change in the chemical potential of the liquid in the aerogel at the vibrational motion of the system. Let \({{u}_{i}}({\mathbf{r}},t) = u_{i}^{{(0)}}(t) + u_{i}^{{(1)}}({\mathbf{r}},t)\) be the field of displacements in the aerogel, \(u_{i}^{{(1)}} \ll u_{i}^{{(0)}}\). Then, as demonstrated for the anisotropic aerogel in [14], we have

$$\delta \mu _{l}^{{{\text{in}}}} = c_{{l1}}^{2}\frac{{\delta {{\rho }_{l}}}}{{\rho _{l}^{{(0)}}}} + c_{{ul}}^{2}u_{{zz}}^{{(1)}} - \tilde {c}_{{ls}}^{2}u_{{ll}}^{{(1)}},$$
(5)

where \({{c}_{{l1}}}\) is the velocity of first sound in the system, \(c_{{ul}}^{2}\) and \(c_{{ls}}^{2}\) are the combinations of the elastic constants of the system, \(\delta {{\rho }_{l}}\) is the change in the velocity of the liquid, and \(u_{{ll}}^{{(1)}}\, = \,{{\partial }_{l}}u_{l}^{{(1)}}\), \(u_{{zz}}^{{(1)}}\, = \,{{\partial }_{z}}u_{z}^{{(1)}}\). Note that under the condition \({{\rho }_{l}} \gg \rho _{s}^{{{\text{in}}}}\), the mass conservation equation gives us the relation between \(\delta {{\rho }_{l}}\) and \(u_{{ll}}^{{(1)}}\): \(\delta {{\rho }_{l}} \approx - {{\rho }_{l}}u_{{ll}}^{{(1)}}\). For the further analysis, it is important that the velocity of first sound far exceeds all other velocities related to the elastic properties of the system (in particular, sound velocities in the aerogel); therefore, we can write the approximate equality

$$\delta \mu _{l}^{{{\text{in}}}} \approx - c_{{l1}}^{2}u_{{ll}}^{{(1)}}.$$
(6)

In the main approximation, the momentum flux tensor of the system is isotropic and is described by one scalar variable, namely, the pressure, which in the approximation under study, is determined in the aerogel by the expression

$$\delta {{p}^{{{\text{in}}}}} \approx - \tilde {c}_{{l1}}^{2}{{\rho }_{l}}u_{{ll}}^{{(1)}},$$
(7)

where \(\tilde {c}_{{l1}}^{2}\) is the sum of several elastic constants, but further on, we can neglect the difference between it and \(c_{{l1}}^{2}\). Excluding \(u_{{ll}}^{{(1)}}\) from Eqs. (6) and (7), we can find the following relation between the change in the pressure in the aerogel and the change in the potential of the superfluid motion:

$$\delta {{p}^{{{\text{in}}}}} \approx {{\rho }_{l}}\delta {{\mu }_{l}} = - {{\rho }_{l}}\frac{{\partial \varphi _{s}^{{{\text{in}}}}}}{{\partial t}}.$$
(8)

The continuity of the momentum flux tensor suggests that the pressure changes outside and inside the aerogel should be the same: \(\delta {{p}^{{{\text{in}}}}} = \delta {{p}^{{{\text{out}}}}}\). Note also that the components of superfluid density tensor in a wide temperature range are small not only inside the aerogel but outside it as well, \({{(\rho _{s}^{{{\text{out}}}})}_{{ij}}} \ll {{\rho }_{l}}\). Therefore, in such a relation determining the pressure balance, we can assume that the outside pressure is mainly determined by the flow of the normal component of the liquid with an isotropic density tensor around the aerogel; i.e., \(\delta {{p}^{{{\text{out}}}}} \approx {{\rho }_{l}}\frac{{\partial \varphi _{n}^{{{\text{out}}}}}}{{\partial t}}\). Summarizing, we can obtain the following approximate relation valid at the aerogel boundary:

$$\frac{{\partial \varphi _{s}^{{{\text{in}}}}}}{{\partial t}} \approx \frac{{\partial \varphi _{n}^{{{\text{out}}}}}}{{\partial t}}.$$
(9)

This boundary condition is satisfied by the function

$$\varphi _{s}^{{{\text{in}}}}({\mathbf{r}},t) = - \frac{1}{2}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}{{r}_{i}},$$
(10)

corresponding to the flow of the superfluid component with the same superfluid velocity throughout the entire volume of the aerogel (\(\Delta {{\varphi }_{s}} = 0\)). Thus, we find that the pressure difference between its ends created by the motion of the normal component outside the aerogel leads to the inflow of the superfluid component into the aerogel in the direction opposite to the motion of the body itself (Fig. 1).

Fig. 1.
figure 1

When a spherical aerogel sample moves with the velocity \(\frac{{\partial {{{\mathbf{u}}}_{0}}}}{{\partial t}}\) in superfluid 3He, pressure and chemical potential gradients arise the liquid in the direction of motion in the volume occupied by the aerogel. According to the law of conservation of superfluid velocity, this leads to the emergence of a superfluid flow in the opposite direction. The conservation law for the total momentum of the system implies that such motion induces an additional force acting on the aerogel along the direction of the motion, which leads to an increase in the resulting frequency of oscillations.

Full size image

Then, the function \(\varphi _{s}^{{{\text{out}}}}\) is determined using the continuity condition for the superfluid flow across the aerogel boundary. We seek the function \(\varphi _{s}^{{{\text{out}}}}\) that decreases at infinity and satisfies the incompressibility condition for the superfluid component of the liquid (\({{\nabla }_{i}}{{({{j}_{s}})}_{i}} = 0\), where js is the current of the superfluid component flowing relative to the normal one [13]:

$$\varphi _{s}^{{{\text{out}}}} = b\frac{{{{R}^{{\gamma + 1}}}}}{{{{r}^{\gamma }}}}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}{{n}_{i}} - \frac{{{{R}^{4}}}}{{2{{r}^{3}}}}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}{{n}_{i}},$$
(11)

where \(\gamma = 3\) for the isotropic B phase and \(\gamma = \frac{3}{2} + \frac{{\sqrt {17} }}{2}\) for the A phase with the hedgehog texture. The continuity condition for the mass flow at the aerogel boundary (in this case, for the superfluid current) can be written as

$${{(\rho _{s}^{{{\text{in}}}})}_{{ij}}}({{\nabla }_{j}}\varphi _{s}^{{{\text{in}}}} - \dot {u}_{j}^{0}){{n}_{i}} = (\rho _{s}^{{{\text{out}}}}{{)}_{{ij}}}({{\nabla }_{j}}\varphi _{s}^{{{\text{out}}}} - \dot {u}_{j}^{0}){{n}_{i}}.$$
(12)

Substituting Eqs. (10) and (11) into Eq. (12), we find

$$b = \frac{3}{2}\frac{1}{{\gamma - 1}}\frac{{\rho _{s}^{{{\text{in}}}}}}{{\rho _{s}^{{{\text{out}}}}}},$$
(13)

where \(\rho _{s}^{{{\text{in}}}}\) is the component of the superfluid density tensor in the aerogel along the direction of vibrations (along or across the aerogel anisotropy axis) and \(\rho _{s}^{{{\text{out}}}}\) is the component of the superfluid density tensor outside the aerogel along the normal to its surface (for two aforementioned phases). Note that the normal and superfluid components of the liquid in the temperature range, where \(\rho _{s}^{{{\text{in}}}} \simeq \rho _{s}^{{{\text{out}}}}\), move in the opposite directions.

After determining all unknown functions, the effective equation of vibrations in the system can be written in the form

$$\left( {{{{\tilde {\rho }}}_{a}} + \frac{3}{2}{{\rho }_{l}} - \frac{3}{2}\frac{\gamma }{{\gamma - 1}}\rho _{s}^{{{\text{in}}}}} \right)\ddot {u}_{i}^{{(0)}} + {{\tilde {\rho }}_{a}}\omega _{0}^{2}u_{i}^{{(0)}} = 0.$$
(14)

The frequency of vibrations in the system can be expressed from this equation as a function of \(\rho _{s}^{{{\text{in}}}}\):

$$\omega (T) = \frac{{{{\omega }_{n}}}}{{\sqrt {1 - \frac{\gamma }{{\gamma - 1}}\frac{{\rho _{s}^{{{\text{in}}}}(T)}}{{{{\rho }_{l}}}}\left( {1 - \frac{{\omega _{n}^{2}}}{{\omega _{0}^{2}}}} \right)} }}{\kern 1pt} ,$$
(15)

where \(\omega _{n}^{2} = \omega _{0}^{2}\frac{{{{{\tilde {\rho }}}_{a}}}}{{{{{\tilde {\rho }}}_{a}} + \frac{3}{2}{{\rho }_{l}}}}\) is the square of the vibration frequency of the system in 3He in the limit of the absence of decay. At temperatures above the superfluid transition temperature in the aerogel, T > Tca, the vibration frequency (disregarding the viscosity of surrounding 3He) is independent of the temperature \(\omega (T) = {{\omega }_{n}} = {\text{const}}\), whereas the frequency at T < Tca, where the Ginzburg–Landau theory is applicable, should increase linearly with decreasing temperature since \(\rho _{s}^{{{\text{in}}}} \sim \left( {1 - \frac{T}{{{{T}_{{{\text{ca}}}}}}}} \right)\). The factor \(\frac{\gamma }{{\gamma - 1}}\) in the radicand in Eq. (15) is a geometric factor; i.e., it depends on the sample shape, and can be written in a more general form as \(1 + \tilde {\alpha }\), where \(\tilde {\alpha }\) = \(\frac{1}{{\gamma - 1}}\) for the spherical sample. Note also that this factor depends on the type of the outer superfluid phase: \(\tilde {\alpha } = 0.5\) and \(\tilde {\alpha } \approx 0.4\) for the spherical sample in the B phase and in the A phase with the hedgehog texture, respectively. This means that the rate of the increase in the frequency with decreasing temperature in the B phase is slightly higher than that in the A phase. Just this numerical factor exceeding unity can explain why the frequency in experiment increases with decreasing temperature faster than it was expected. In the next section, we use the following formula for the fitting of experimental data for the aerogel of an arbitrary shape:

$$\omega (T) = \frac{{{{\omega }_{n}}}}{{\sqrt {1 - (1 + \tilde {\alpha })\frac{{\rho _{s}^{{{\text{in}}}}(T)}}{{{{\rho }_{l}}}}\left( {1 - \frac{{\omega _{n}^{2}}}{{\omega _{0}^{2}}}} \right)} }}{\kern 1pt} ,$$
(16)

with an unknown parameter \(\tilde {\alpha } > 0\).

3 COMPARISON WITH EXPERIMENT

To check the obtained relations, let us consider experimental data on the frequencies of two excited vibrational modes \({{\omega }_{{1,2}}}(T)\) and on the widths \({{\zeta }_{{1,2}}}(T)\) of the resonance lines as functions of the temperature for the case where a polar phase is formed in a nematic aerogel [12]. The damping of the first vibrational (mechanical) mode is mainly due to the viscosity of the surrounding 3He aerogel. In addition, the viscosity (the deviation from the potential character of the liquid motion near the aerogel surface) contributes to the added mass of the system and hence to the vibrational frequency. This contribution was disregarded in the previous section. Furthermore, the contribution of this additional inertial viscous force to \(\varphi _{s}^{{{\text{in}}}}\) was also ignored since this would require solving the problem of finding \(u_{{ll}}^{{(1)}}\) and \(u_{{zz}}^{{(1)}}\) in the aerogel, which was unnecessary in our approximation. The result of a more accurate calculation shows that the contribution of the viscosity to the potential of superfluid motion in the aerogel is small in terms of the ratio \(\delta {\text{/}}R\), where \(\delta \) is the viscous penetration depth for liquid 3He. Below the temperature of the bulk superfluid transition, this ratio decreases rapidly with decreasing temperature, which justifies the approximation made. Thus, for a more accurate comparison with experiment, it is necessary to exclude the contribution of the viscosity to the experimentally observed vibrational frequency of the system, which is related to a change in the added mass of the liquid outside the aerogel due to the viscous contribution. This can be easily done for the spherical sample oscillating in the viscous liquid since if the damping is small, there is a simple relation between the corresponding addition to the vibrational frequency and the width of the resonance line: \({{\omega }_{1}}(T) = \omega _{1}^{'}(T) + \frac{1}{2}{{\zeta }_{1}}(T)\), where \(\omega _{1}^{'}\) is the vibrational frequency of the system including a small contribution from the nonpotentiality of the motion of the normal component near the aerogel surface and \({{\zeta }_{1}}\) \( \ll \) ω1 is the width of the resonance curve. As before, the coefficient 1/2 in the above expression for ω1 is a geometric factor that could depend on the sample shape. Excellent agreement of this dependence with the experimental data at T > Tca was demonstrated in [12].

The second difficulty in processing experimental data is the effective coupling of two observed vibration modes, the nature of which is not quite clear. To make the discussion simpler, let us consider a mode-coupling model, which involves a matrix element specified by the frequency \({{\omega }_{{12}}}\), so that

$${{\omega }_{1}} = \frac{{\omega _{1}^{{(0)}} + \omega _{2}^{{(0)}}}}{2} + \frac{1}{2}\sqrt {{{{(\omega _{1}^{{(0)}} - \omega _{2}^{{(0)}})}}^{2}} + 4\omega _{{12}}^{2}} ,$$
(17)
$${{\omega }_{2}} = \frac{{\omega _{1}^{{(0)}} + \omega _{2}^{{(0)}}}}{2} - \frac{1}{2}\sqrt {{{{(\omega _{1}^{{(0)}} - \omega _{2}^{{(0)}})}}^{2}} + 4\omega _{{12}}^{2}} ,$$
(18)

where \(\omega _{{1,2}}^{{(0)}}\) are the frequencies of two uncoupled modes. The temperature dependence of the frequency \(\omega _{1}^{{(0)}}\) is determined by Eq. (16), whereas the temperature dependence of the frequency of the second mode was found in [14] in the Ginzburg–Landau limit

$$\omega _{2}^{{(0)}}(T) = \frac{{{{\omega }_{{a \bot }}}}}{{\sqrt {\left[ {\left( {1 + 3\frac{{{{\rho }_{l}}}}{{{{\rho }_{a}}}}} \right) + \frac{{\rho _{l}^{2}}}{{\rho _{s}^{ \bot }(T){{\rho }_{a}}}}\frac{{c_{{{\text{ul}}}}^{2}}}{{c_{{l1}}^{2}}}} \right]} }},$$
(19)

where \(\rho _{s}^{ \bot }\) is the component of the superfluid density tensor for the polar phase in the direction perpendicular to the aerogel anisotropy axis, \({{\omega }_{{a \bot }}} \sim 2000\) Hz, the square of the ratio of velocities is \(\frac{{c_{{{\text{ul}}}}^{2}}}{{c_{{l1}}^{2}}} \sim 0.01\), and it is nearly independent of the pressure. By analogy with the second sound, the real part of the frequency of these vibrations arises only at T < Tca. The small factor \(\frac{{c_{{{\text{ul}}}}^{2}}}{{c_{{l1}}^{2}}}\) in the denominator of (19) leads to a fast square root increase in the vibrational frequency in the temperature range 1 – \(\frac{T}{{{{T}_{{{\text{ca}}}}}}}\) ~ 0.01; then, the frequency is saturated at about 1700 Hz. Since the frequency of the mechanical mode is about 550 Hz, the crossing and coupling of the modes are important only in a narrow temperature range near Tca. According to the suggested temperature dependences of these two coupled vibrational modes, the mechanical mode is described by Eq. (17) at \(T \gtrsim {{T}_{{{\text{ca}}}}}\) and by Eq. (18) at \(T \lesssim {{T}_{{{\text{ca}}}}}\).

The results of numerical fitting of the temperature dependence of the vibrational frequency of the mechanical mode for pressures of 7.1, 15.6, and 29.3 bar are presented in Fig. 2. Since the rate of the increase in the frequency with decreasing temperature depends on the type of the order parameter in the superfluid surrounding the aerogel, we consider only the case of the A phase outside the aerogel and a narrow temperature range near Tc where the effects nonlinear in T – Tca can be neglected. The temperature dependence of the superfluid density of the polar phase is taken in the following form valid in the Ginzburg–Landau approximation:

$$\rho _{s}^{ \bot } = \frac{{\left( {1 - \frac{T}{{{{T}_{{{\text{ca}}}}}}}} \right)}}{{{{\beta }_{{12345}}}}}\frac{{{{\rho }_{l}}}}{{\left( {1 + \frac{{F_{1}^{s}}}{3}} \right)}},$$
(20)

where \({{\beta }_{{12345}}} = {{\beta }_{1}} + {{\beta }_{2}} + {{\beta }_{3}} + {{\beta }_{4}} + {{\beta }_{5}}\), \({{\beta }_{i}}\) are the coefficients in the expansion of the free energy of 3He in the Ginzburg–Landau theory, and \(F_{1}^{s}\) is the Landau parameter for the Fermi liquid.

Fig. 2.
figure 2

Temperature dependence of the frequency of the mechanical vibrational mode observed in the experiment [12], for pressures of (○) 7.1, (×) 15.6, and (+) 29.3 bar. Solid lines correspond to the fitting by Eqs. (17) and (18) with the use of Eqs. (16) and (19). The dependence \(\rho _{s}^{ \bot }(T)\) is given by Eq. (20), where the coefficients \({{\beta }_{i}}\) of pure 3He are used. The fitting is performed for the case of the A phase occurring outside the aerogel.

Full size image

All coefficients in Eq. (20) depend on the pressure and are taken equal to those for pure 3He. For all three pressures, the coupling frequency \({{\omega }_{{12}}}\) of two modes turns out to be about 80 Hz. The decrease in the vibrational frequency with increasing pressure at T > Tca is well described only by an increase in the 3He density [12]. The coefficient \(\tilde {\alpha }\), which depends on the shape of the aerogel and causes a faster increase in the frequency with decreasing temperature, turned out to be equal to 0.71, 0.86, and 0.74 for pressures of 7.1, 15.6, and 29.3 bar, respectively. Some spread in values of this coefficient could be due to the use of the \({{\beta }_{i}}\) coefficients for pure 3He, which could have a different pressure dependence in the presence of a nematic aerogel. In addition, the approximation under discussion includes only the terms linear in \(\left( {1 - \frac{T}{{{{T}_{{{\text{ca}}}}}}}} \right)\), which can also somehow limit the possible values of the fitting parameters. Since the aerogel used in the experiment described above had the shape of a cuboid, the difference of the found coefficient \(\tilde {\alpha }{\text{'}}\) from its theoretical value of 0.4 expected for the spherical aerogel is quite natural.

4 CONCLLUSIONS

The physical picture of the considered effect is quite simple: when the aerogel moves in 3He, the normal component of the liquid flows around it. This creates a pressure difference between the sides of the aerogel, which causes the flow of the superfluid component across the aerogel in the direction opposite to the direction of the initial motion (Fig. 1). The resulting reactive force acting on the aerogel is directed in such a way that it gives rise to an additional increase in the frequency of vibrations in the system. Another distinctive feature of the model under study is that a phase difference between two superfluid states inside and outside the aerogel appears at the aerogel boundary

$$\varphi _{s}^{{{\text{out}}}} - \varphi _{s}^{{{\text{in}}}} = \frac{3}{2}\frac{1}{{\gamma - 1}}R\frac{{\rho _{s}^{{{\text{in}}}}}}{{\rho _{s}^{{{\text{rout}}}}}}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}{{n}_{i}},$$
(21)

where the coefficient \(\frac{3}{2}\frac{1}{{\gamma - 1}}\) is related to the spherical shape of the aerogel. Let us analyze the currents inflowing across an element of the aerogel surface at point A and outflowing across another element of the surface located near point B (see Fig. 1). Due to the symmetry of the problem, the currents flowing across these surfaces are the same and equal to \(\left| {\frac{3}{2}\rho _{s}^{{{\text{in}}}}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}n_{i}^{A}} \right|\) and the additional phase incursion along the AB segment due to the crossing both aerogel boundaries turns out to be \(\delta \varphi _{s}^{{AB}}\) = \( - 3\frac{1}{{\gamma - 1}}R\frac{{\rho _{s}^{{{\text{in}}}}}}{{\rho _{s}^{{{\text{out}}}}}}\frac{{\partial u_{i}^{{(0)}}}}{{\partial t}}n_{i}^{A}\), where \(n_{i}^{A}\) is the outer normal to the surface at point A. Since the amplitude of the order parameter outside the aerogel is the same at points A and B, the superfluid current flowing across the aerogel between these points corresponds to the linear regime of the Josephson current, \(j_{s}^{{AB}} \sim {{\Delta }_{A}}{{\Delta }_{B}}\delta \varphi _{s}^{{AB}} \sim \rho _{s}^{{{\text{out}}}}\delta \varphi _{s}^{{AB}}\), where ΔA(ΔB) is the amplitude of the order parameter at point A(B), and ρs ~ Δ2.

The current flow across through the aerogel boundary has a Josephson character only in the low velocity limit, where the phase difference is small. The velocity of motion of the aerogel with characteristic dimensions of the order of 1 mm should be much lower than 0.1 mm/s, which in principle corresponds to the experimental conditions [12]. Note also that according to the results obtained in [13], the phase difference at the aerogel boundary of the macroscopic spherical aerogel is given by the expression

$$\begin{gathered} \Delta {{\varphi }_{s}} = \left[ {\left( {{\mathbf{v}}_{s}^{{{\text{out}}}} - \frac{{\partial {{{\mathbf{u}}}^{{(0)}}}}}{{\partial t}}} \right){\mathbf{n}}} \right]\left( {\int\limits_{ - \infty }^0 \frac{{{{\rho }_{s}}(r) - \rho _{s}^{{{\text{out}}}}}}{{{{\rho }_{s}}(r)}}dr} \right. \\ + \;\left. {\left[ {\left( {{\mathbf{v}}_{s}^{{{\text{in}}}} - \frac{{\partial {{{\mathbf{u}}}^{{(0)}}}}}{{\partial t}}} \right){\mathbf{n}}} \right]\int\limits_0^{ + \infty } \frac{{{{\rho }_{s}}(r) - \rho _{s}^{{{\text{in}}}}}}{{{{\rho }_{s}}(r)}}dr} \right). \\ \end{gathered} $$
(22)

The change in the superfluid density at the aerogel boundary occurs at the coherence length \(\xi (\tau )\) of superfluid 3He, which is much smaller than the sphere radius. Consequently, for the phase difference (22) to be of the order of \(R\frac{{\partial {{{\mathbf{u}}}^{{(0)}}}}}{{\partial t}}\), which follows from Eq. (21), the superfluid density ρs(r) should have a singularity at the aerogel boundary. This fact again indicates a weak coupling between two volumes with different superfluid states. Thus, the qualitative behavior of the order parameter at the aerogel boundary can be indirectly estimated from the reported experimental results. To determine the change in the order parameter at the aerogel boundary more accurately, a microscopic description within the Abrikosov–Gorkov theory is required. In conclusion, it is noteworthy that the polar phase studied in this work belongs to the class of nontrivial topological phases, at the boundary of which topologically stable edge current-carrying states can exist [17, 18]. It is of interest to study the coupling of these currents to vibrations in the system, which can become significant at sufficiently low temperatures, when the normal component of the liquid density in the bulk of the system is low.