Fourth Sound

Superfluid helium presents a remarkable paradox of nature: a single liquid behaving as two intertwined fluids. One component is normal and viscous, while the other, the superfluid, flows with zero friction—a manifestation of macroscopic quantum mechanics. This raises a fundamental challenge for physicists: how can one study the pure, frictionless superfluid component, free from the dissipative influence of its normal counterpart? This article addresses this question by exploring the unique phenomenon of "fourth sound," a wave mode that exists only under specific conditions designed to isolate the superfluid's motion.

In the following chapters, we will first delve into the "Principles and Mechanisms" of fourth sound, explaining how a porous medium can act as a mechanical filter to "freeze" the normal fluid and allow a pure superfluid pressure wave to propagate. We will uncover the elegant physics that dictates its speed and connects it to other fundamental wave modes. Subsequently, the "Applications and Interdisciplinary Connections" chapter will reveal how fourth sound transforms from a theoretical curiosity into a powerful experimental tool, used to build quantum gyroscopes, probe critical phase transitions, and map the hidden textures of the most exotic forms of matter.

Imagine you are faced with a peculiar substance, a liquid that behaves as if it were two distinct fluids intertwined, occupying the very same space. One of these fluids is perfectly ordinary—it has viscosity, it feels sticky, it behaves just as you'd expect a normal liquid to. But the other, the "superfluid" component, is an entity of pure quantum magic. It flows without any friction whatsoever and possesses a host of other bizarre properties. This is not science fiction; this is superfluid helium, a real-world quantum marvel.

Our challenge, as curious physicists, is to isolate and study the properties of this ghostly superfluid component, free from the influence of its normal, sticky counterpart. How can we possibly separate two fluids that are perfectly mixed at the atomic level? The answer is not to use a chemical filter, but a mechanical one. This clever trick is the gateway to understanding a whole new type of wave: the fourth sound.

Let’s try a thought experiment. Suppose we soak a very fine sponge in our two-component liquid. The sponge is riddled with incredibly narrow, twisting channels. What happens when we try to push the liquid through it? The normal fluid, being viscous and "sticky," will get caught on the walls of these tiny pores. If the channels are narrow enough, the normal fluid will be completely immobilized, or "clamped," stuck fast to the sponge's matrix.

But the superfluid component, having zero viscosity, feels no such drag. It glides effortlessly through the labyrinthine passages, completely unhindered. This special type of porous material, which allows the superfluid to pass while blocking the normal fluid, is known as a ​​superleak​​.

This isn't just a convenient assumption; it's a very real physical effect. We can even predict when it will happen. The "stickiness" of a wall in an oscillating fluid extends a certain distance into the fluid, a distance called the ​​viscous penetration depth​​, $\delta$. This depth depends on the fluid's properties and the frequency of the oscillation. If we build a superleak whose channels have a radius $R$ that is smaller than this penetration depth, the entire volume of the channel is under the influence of the wall's drag. As a result, the whole normal component inside is locked in place. By confining the superfluid, we have ingeniously created a scenario where only the superfluid component is free to move. We have effectively "frozen" the normal fluid.

Now that we have isolated the motion of the superfluid, what happens if we try to create a sound wave? Ordinary sound, which we call ​​first sound​​ in this context, is a pressure wave where both fluid components oscillate together, in phase. But in our superleak, the normal fluid is clamped. It cannot oscillate.

This unique situation gives birth to a new mode of propagation: a pressure and density wave carried exclusively by the mobile superfluid component. This is ​​fourth sound​​.

Let’s first consider the simplest case, at temperatures approaching absolute zero. Here, the normal fluid component vanishes ($\rho_n \to 0$), and the liquid is almost entirely superfluid ($\rho_s \approx \rho$). What is the speed of fourth sound, $c_4$, now? One might naively think it's just the speed of first sound, $c_1$. The actual result is far more subtle and beautiful. The speed is given by:

This is a wonderful piece of physics! It tells us that the restoring force for the wave is indeed related to the liquid's compressibility (which sets $c_1$), but the wave's inertia involves the total density $\rho$ of the fluid. However, only the superfluid fraction, $\rho_s$, is "active" and participating in the wave motion. The speed of fourth sound is thus the speed of first sound, scaled by the square root of the fraction of the fluid that is superfluid. This formula provides a powerful tool: by measuring the speeds $c_1$ and $c_4$, we can directly determine the superfluid density $\rho_s$, a fundamental quantity in the theory of quantum fluids.

What happens when we are at a higher temperature, where there is a significant amount of normal fluid ($\rho_n > 0$)? It's clamped, yes, but it hasn't disappeared. The normal fluid is the carrier of all the heat and entropy in the system.

In an unconfined superfluid, there exists another fascinating wave mode called ​​second sound​​. It is not a pressure wave, but a temperature wave, where the superfluid and normal fluid oscillate out of phase, sloshing against each other in such a way that the total density remains constant but the temperature fluctuates.

In our superleak, the normal fluid cannot slosh. So, can second sound exist? No. But its ghost lingers and profoundly affects our fourth sound wave. When the superfluid component oscillates in a fourth sound wave, it creates regions of higher and lower density. Due to fundamental thermodynamic relationships, these density fluctuations are inextricably linked to temperature fluctuations. The superfluid wave creates a temperature wave.

The clamped normal fluid feels this oscillating temperature but cannot move in response. Nonetheless, its presence, and the temperature wave it would normally carry, couples into the dynamics. The result is one of the most elegant formulas in the field:

Look at this equation. It's like a Pythagorean theorem for wave speeds! It reveals that fourth sound is a hybrid wave. Its nature is a mixture, a weighted average of a pressure wave (first sound) and a temperature wave (second sound). The weighting factors are simply the relative densities of the superfluid and normal fluid components. When $\rho_n \to 0$, we recover our low-temperature result. When $\rho_s \to 0$ near the transition temperature, the first sound term vanishes. This remarkable formula shows how fourth sound beautifully unifies the pressure and thermal dynamics of the superfluid state.

So far, we have a wonderfully clean picture. But real-world superleaks are not perfect, idealized channels. They are often complex, random mazes of interconnected pores. To make our theory match reality, we must account for the geometry of this maze.

Two key parameters emerge. The first is ​​porosity​​ ($\phi$), which is simply the fraction of the material's volume that is empty space for the fluid to fill. The second, more subtle, is ​​tortuosity​​ ($\alpha$). This parameter measures how twisted and convoluted the paths are. If the pores were straight cylinders, $\alpha$ would be 1. In a random packing of spheres, the superfluid has to take a winding, tortuous path to get from A to B, making its effective path length longer. This increased path length adds to the fluid's inertia, slowing the wave down.

Furthermore, our waves are not perfectly immortal. In the real world, they fade away, or ​​attenuate​​. While the superfluid has zero shear viscosity, other, more subtle dissipative processes exist. One such process, described by "second viscosity," involves the continuous microscopic conversion between the superfluid and normal components as pressure and temperature fluctuate. This process is not perfectly efficient and leads to a slight loss of energy from the wave, causing its amplitude to decrease as it propagates.

The story does not end with pores completely filled with superfluid. What if we only have a tiny amount of helium in our porous medium, just enough to form a thin film coating the inner surfaces of the pores? The physics changes again. Now, a wave can travel along this film, a wave of varying thickness. This is known as ​​third sound​​.

It may seem that third and fourth sound are entirely different beasts. One is a bulk wave of pressure in filled pores, the other a surface wave of thickness in a thin film. But physics often reveals that seemingly disparate phenomena are just two faces of the same coin. An advanced model demonstrates that as we gradually add more and more helium to our porous medium, there is a smooth transition from third sound to fourth sound. When the film is thin, the wave is dominated by surface forces. As the pores fill up, bulk compressibility takes over. Fourth sound and third sound are not distinct; they are the two limiting behaviors of a single, more general type of wave, beautifully illustrating the unifying power of physical principles.

As a final touch, we can even imagine a superleak that is itself anisotropic, perhaps made of aligned fibers. In such a material, the superfluid might find it easier to move in one direction than another. This anisotropy is imprinted on the wave, making the speed of fourth sound dependent on its direction of travel. The speed is no longer a simple number, but a more complex mathematical object called a tensor, which precisely captures this directional character.

From the simple idea of "freezing" a sticky fluid in a sponge, we have uncovered a rich tapestry of physics. Fourth sound is more than just another wave; it is a powerful probe, a stethoscope that allows us to listen to the inner workings of the quantum world, measure its fundamental properties, and witness the beautiful symphony of pressure, temperature, and geometry that governs the strange and wonderful state of a superfluid.

If ordinary sound is like listening to a symphony orchestra in its entirety—the booming percussion, the soaring brass, the sweeping strings—then fourth sound is like possessing a magical microphone that filters out everything but the violins. In the quantum orchestra of a superfluid, the "normal fluid" component is the noisy, viscous percussion section, while the "superfluid" component is the ethereal, frictionless string section. Ordinary sound waves, being pressure waves in the total density, jostle every instrument. But fourth sound, by its very nature, can only exist when the normal fluid is clamped, locked in place by a porous medium. This allows us to listen exclusively to the pure music of the superfluid condensate. This specialized listening proves to be an astonishingly powerful and versatile tool, allowing us to dissect some of the most profound and beautiful phenomena at the frontiers of physics.

One of the most elegant applications of fourth sound is as a detector of rotation, a demonstration of a deep principle known as the Sagnac effect. Imagine a superfluid confined to a narrow ring, like a tiny, circular moat. If we excite two fourth sound waves traveling in opposite directions around this ring, they should, in a stationary ring, circle back to their starting point in exactly the same amount of time.

But what if the entire ring is rotating? From the perspective of an observer in the laboratory, the wave traveling in the direction of rotation has a slightly longer path to cover to complete a lap, while the wave traveling against the rotation has a slightly shorter path. This is not just a classical race; in quantum mechanics, the path a wave travels affects its phase. The difference in path lengths for the two counter-propagating waves leads to a shift in their arrival times, which manifests as a measurable difference in their frequencies. The magnitude of this frequency splitting turns out to be directly and simply proportional to the angular velocity of the ring. In essence, the ring of superfluid has become a highly sensitive gyroscope, capable of detecting rotation by listening to the beat frequency between two quantum waves. This transforms a simple condensed matter system into a probe of the fundamental connection between rotation and quantum phase, a principle that also underlies sophisticated navigation devices like ring laser gyroscopes.

What happens when we add impurities to a pure superfluid? For instance, what if we dissolve a small amount of Helium-3 (a fermion) into superfluid Helium-4 (a boson)? The Helium-3 atoms, along with thermal excitations, behave as the normal fluid component. In a porous medium where fourth sound can propagate, these ³He atoms are held stationary. The superfluid ⁴He must then flow through a static, microscopic maze of ³He impurities.

This journey is no longer a straight line. The superfluid must take a winding, convoluted path, a property described by a geometric factor called "tortuosity." This more complex path, along with the fundamental interactions between the ⁴He and ³He atoms, alters the speed of the fourth sound wave. By carefully measuring the speed of fourth sound in such a mixture, we can deduce these microscopic properties. It allows us to probe how the superfluid "feels" the presence of the impurities, providing information about the effective mass of the ³He quasiparticles and their coupling to the superfluid background. Fourth sound thus becomes a tool for a kind of "quantum chemistry," allowing us to characterize the properties of quantum solutions and alloys at the most fundamental level.

Phase transitions are the most dramatic moments in the life of a material, where its properties change suddenly and profoundly. Fourth sound provides an unparalleled view of the action right at the critical point where superfluidity is born.

As we cool liquid helium toward its lambda transition temperature, $T_\lambda$, the system enters a state of turmoil. It can't quite decide whether to be a normal fluid or a superfluid. The "order parameter," which measures the degree of superfluidity, begins to fluctuate wildly, flickering in and out of existence. This is the realm of critical phenomena. A fourth sound wave propagating through this tumultuous state perturbs these fluctuations. The system tries to respond and settle back to equilibrium, but it is incredibly sluggish near the transition—a phenomenon called "critical slowing down." This slow, dissipative response drains energy from the sound wave, causing it to be absorbed, or attenuated. By measuring this attenuation as a function of how close we are to the transition temperature, we are directly measuring the characteristic relaxation time of these fundamental critical fluctuations. This provides a direct, macroscopic window into the microscopic dynamics governing the phase transition, offering a powerful test for the predictions of dynamic scaling theory.

The story gets even more remarkable in two dimensions. Here, a unique topological transition known as the Berezinskii-Kosterlitz-Thouless (BKT) transition can occur. BKT theory makes a startling prediction: at the transition temperature, the superfluid density does not fade smoothly to zero but jumps discontinuously from a finite value to zero. This universal jump is a fundamental signature of the transition. But how can one measure it? The speed of fourth sound, $c_4$, is directly related to the superfluid density. Therefore, a measurement of $c_4$ as the system is cooled should reveal a sudden, sharp increase precisely at the BKT temperature. The magnitude of this jump in the sound speed provides a direct confirmation of the universal jump in superfluid stiffness, a beautiful fingerprint of this exotic topological transition. This very principle has been used in experiments with thin films of helium and ultracold atomic gases to provide stunning verification of BKT theory.

Beyond the "simple" case of superfluid Helium-4, fourth sound serves as an indispensable cartographer for mapping even stranger quantum landscapes.

Consider the A-phase of superfluid Helium-3. Unlike Helium-4, this is an anisotropic superfluid. The Cooper pairs of ³He atoms possess an internal orbital angular momentum, giving the fluid a directional "texture" or "grain," much like a piece of wood. This preferred direction is described by a vector $\hat{l}$. While this texture is invisible to the naked eye, it is not invisible to fourth sound. The attenuation of a fourth sound wave in ³He-A is primarily caused by heat flow, which is carried by quasiparticle excitations. Because of the anisotropic energy gap, heat flows much more easily along certain directions relative to $\hat{l}$ than others. Consequently, the attenuation of fourth sound becomes strongly dependent on its direction of propagation relative to this internal quantum compass. By measuring the damping of the sound as it travels parallel versus perpendicular to the grain, we are literally mapping the orientation of the hidden quantum order parameter, using sound as a form of echolocation for quantum textures.

Perhaps the most paradoxical state of matter is the supersolid, a material that is simultaneously a rigid, ordered crystal and a flowing, frictionless superfluid. In this bizarre state, mass can flow without resistance through the solid lattice. This superfluid component can support a fourth sound wave. Measuring the properties of this unique sound mode gives us direct insight into the nature of the supersolid state. The damping, or attenuation, of the wave reveals the internal friction (viscosity) between the normal and superfluid components, telling us about the dissipative processes within this strange marriage of solid and fluid. Furthermore, imperfections in the crystal lattice, such as dislocations, create strain fields that can scatter the fourth sound wave. Studying this scattering process allows physicists to understand how the superfluid component is coupled to the elastic, mechanical properties of the solid crystal it inhabits. Fourth sound becomes a bridge connecting the worlds of quantum hydrodynamics and solid-state mechanics.

From fundamental tests of quantum mechanics to the detailed characterization of complex materials, fourth sound is a testament to the power of isolation in experimental science. By silencing the clamor of the normal fluid, we amplify the pure signal from the quantum condensate, transforming a simple pressure wave into a sophisticated instrument for exploring the frontiers of our physical world.