Fractional AC Josephson Effect

The Josephson effect stands as a cornerstone of macroscopic quantum physics, demonstrating how pairs of superconductors can communicate through a coherent, resistance-free current. This phenomenon has not only led to revolutionary technologies but also serves as a precise probe into the quantum world. However, the frontiers of physics have pushed beyond conventional materials, predicting exotic "topological superconductors" that are theorized to host elusive quasiparticles known as Majorana fermions. The central challenge, which this article addresses, is how to experimentally verify the existence of these strange "half-electron" states. This article will guide you through this fascinating discovery process. The first section, "Principles and Mechanisms," will lay the groundwork by explaining the conventional Josephson effect before revealing how the presence of Majorana fermions fundamentally alters the rules, leading to the fractional AC Josephson effect. Subsequently, the "Applications and Interdisciplinary Connections" section will explore how these principles are applied, from defining our electrical standards to acting as a definitive "smoking gun" in the hunt for Majoranas, and reveal its universal nature across different quantum systems.

To truly appreciate the wonder of the fractional Josephson effect, we must first take a step back and admire the original masterpiece: the conventional Josephson effect. It is a story of quantum mechanics writ large, a story of how two individual droplets of a quantum fluid can communicate and dance in perfect synchrony.

Imagine two superconductors, separated by a whisper-thin insulating barrier. Each superconductor is not just a cold piece of metal; it is a single, colossal quantum entity, a macroscopic wave described by a phase, much like the phase of a light wave. Let's call the phases of our two superconductors $\theta_{\mathrm{L}}$ and $\theta_{\mathrm{R}}$. When these two quantum giants are brought close, they can "shake hands" across the barrier. This handshake is no ordinary interaction; it is a flow of charge without any resistance, a ​​supercurrent​​.

The brilliant insight of Brian Josephson was that the magnitude and direction of this supercurrent, $I_s$, depend solely on the difference in their phases, $\phi = \theta_{\mathrm{L}} - \theta_{\mathrm{R}}$. The relationship is one of elegant simplicity. The energy of this coupling, dictated by fundamental symmetries like time-reversal, takes the form $E_{\mathrm{J}}(\phi) = -E_c \cos(\phi)$. The current, which you can think of as the system's attempt to flow towards lower energy, is proportional to the derivative of this energy with respect to the phase. This gives us the first Josephson relation:

Notice the $\sin(\phi)$ term. It tells us that the physics is ​​$2\pi$-periodic​​. If you advance the phase difference by $2\pi$, you get back to the exact same physical state. This is the natural rhythm of a conventional superconductor. Why $2\pi$? Because the charge carriers in this quantum fluid are not single electrons, but bound pairs of them called ​​Cooper pairs​​, each carrying a charge of $2e$. The entire condensate is made of these pairs, and its quantum wavefunction must be single-valued, leading to this fundamental $2\pi$ periodicity.

Now for the second act. What happens if we apply a DC voltage, $V$, across our junction? A voltage is an energy difference. In the quantum world, an energy difference is a frequency difference. Applying a voltage makes the phase difference, $\phi$, evolve in time. This is the second Josephson relation:

Again, the charge $2e$ of the Cooper pair appears, dictating the tempo of the phase evolution. If we put these two relations together, we discover something marvelous. A constant voltage $V$ causes the phase to advance linearly, $\phi(t) = \phi_0 + (2eV/\hbar)t$. Plugging this into the current relation gives an oscillating supercurrent: $I_s(t) = I_c \sin(\phi_0 + (2eV/\hbar)t)$. The junction has become a perfect quantum converter, turning a DC voltage into an AC current! The frequency of this current, $f = \omega/2\pi = 2eV/h$, is set only by fundamental constants of nature and the applied voltage. This means our tiny junction is now a microwave emitter, radiating electromagnetic waves with a precisely defined frequency—a direct consequence of the choreographed dance of Cooper pairs.

For decades, this was the whole story. The charge was $2e$, the rhythm was $2\pi$. But physics, in its relentless search for the new and strange, was preparing a surprise. It came from the abstract realm of topology, which studies properties of shapes that are unchanged by continuous deformations. Physicists predicted that certain exotic materials could exist in a "topological superconducting" phase.

What makes these materials so special? If you take a wire of such a material, its interior is a superconductor like any other, but its ends are cursed with a peculiar kind of existence. They are forced to host special, localized quasiparticles called ​​Majorana zero modes​​. A Majorana fermion is its own antiparticle; in a sense, you can think of it as "half" of a regular electron. The two halves, a pair of Majorana modes, can be separated at the two ends of a wire, yet they remain inextricably linked, defining a single fermionic state between them. This isn't just a theorist's daydream; experimental recipes involving semiconductor nanowires, spin-orbit interaction, and magnetic fields provide a plausible path to realizing these strange states of matter.

Now, let's perform a new experiment. We build our Josephson junction not from conventional superconductors, but from two topological ones, bringing their Majorana-hosting ends close together. The handshake across this junction is now a conversation between two Majorana modes. How do they communicate? Theory tells us they do so by allowing a single electron to tunnel across the junction.

This changes everything.

Remember how the physics of the junction depends on the phase acquired by the tunneling charge? For a Cooper pair of charge $2e$, the relevant phase coupling is to $\phi$. But for a single electron of charge $e$, the coupling is to $\phi/2$. The energy of the Majorana-mediated coupling thus takes a new form, $E_M(\phi) \propto \cos(\phi/2)$. And the current, its derivative, must now be:

Look closely at that $\sin(\phi/2)$. This function is not $2\pi$-periodic. To get back to where you started, you must advance the phase difference $\phi$ by a full ​​$4\pi$​​! It's as if the fundamental clock of the superconductor has had its period doubled. This profound change in periodicity is a direct consequence of the charge-$e$ nature of the tunneling process between Majoranas.

What does this mean for our AC effect? The phase itself still evolves according to the universal law, $\frac{d\phi}{dt} = 2eV/\hbar$, because the macroscopic superconducting banks are still made of Cooper pairs. But the current flowing across the junction is now marching to a different drummer. It completes one cycle only when $\phi$ advances by $4\pi$. The resulting AC frequency is therefore:

The frequency has been cut precisely in half! This is the ​​fractional AC Josephson effect​​. The frequency of the emitted microwaves now acts as a perfect spectrometer for the charge of the tunneling particle. A frequency of $2eV/h$ signals Cooper pairs. A frequency of $eV/h$ signals the effective tunneling of single electrons, the smoking-gun signature of Majorana zero modes.

This beautiful halved frequency, however, is an extraordinarily delicate phenomenon. The $4\pi$-periodic energy-phase relation $E_M(\phi) \propto \cos(\phi/2)$ actually describes two distinct energy branches, $E_\pm(\phi) = \pm E_0 \cos(\phi/2)$, which cross at $\phi=\pi, 3\pi, \ldots$. These two branches correspond to the two possible states of the fermion formed by the two Majoranas—an "unoccupied" state and an "occupied" state. This two-level property is called ​​fermion parity​​. The fractional Josephson effect is observable only if the system gets locked into one of these parity states and stays there as the phase evolves—a condition known as ​​parity conservation​​.

But the universe is a noisy place. Several effects can conspire to break this fragile coherence.

The primary villain is ​​quasiparticle poisoning​​. Any stray, thermally excited electron (a quasiparticle) in the surrounding superconductor can be accidentally trapped by the junction. Such an event flips the fermion parity, causing the system to jump from one energy branch to the other. If these poisoning events happen very frequently—at a rate $\Gamma$ much faster than the rate of phase evolution $\Omega = 2eV/\hbar$—the system doesn't have time to follow the $4\pi$ path. Instead, it constantly relaxes to the lowest available energy state. The ground state energy, $-E_0|\cos(\phi/2)|$, is a $2\pi$-periodic function! Consequently, fast poisoning washes out the fractional effect and restores the conventional $2\pi$ periodicity. The observation of the fractional effect is thus a race: the quantum evolution must outrun the environmental decoherence, a condition expressed as $\Omega \gg \Gamma$.

This competition has a stunning experimental consequence. In a conventional junction, applying an additional microwave field creates plateaus in the voltage-current curve at all integer multiples of a fundamental voltage—the famous ​​Shapiro steps​​. In a pristine topological junction with a pure $4\pi$ periodicity, a subtle symmetry of the $\sin(\phi/2)$ current ensures that all the odd-numbered steps vanish! The re-emergence of these missing odd steps as one increases temperature or external noise is a telltale sign that quasiparticle poisoning is taking over and restoring the $2\pi$ rhythm.

Even in a perfect, poison-free world, subtle quantum mechanics can complicate the picture. If the Majoranas from opposite sides of the junction overlap slightly, their energy levels no longer cross but form an "avoided crossing" with a tiny energy gap. If the phase evolves slowly (low voltage), the system can adiabatically follow the gapped ground state, which is $2\pi$-periodic. If the phase evolves quickly (high voltage), the system can perform a ​​Landau-Zener transition​​, "jumping" the gap and effectively staying on its original parity branch, thus preserving the $4\pi$ dynamics.

The fractional Josephson effect is more than a mere curiosity. It is a powerful lens into a hidden quantum world. The frequency of radiation from a tiny circuit reveals the nature of the exotic quasiparticles tunneling within it, and the very fragility of the effect teaches us profound lessons about quantum coherence and the ceaseless struggle between the elegant laws of quantum dynamics and the unavoidable noise of the macroscopic world.

In our previous discussion, we delved into the strange and wonderful mechanics of the fractional AC Josephson effect, a dance of quantum phase and exotic particles. You might be left with a feeling of awe, but also a question: What is this all for? Is it merely a curiosity for the physicists, a beautiful but esoteric piece of music played in the quiet of a low-temperature laboratory? The answer is a resounding no. The principles we have uncovered are not only the bedrock of some of our most precise technologies but also a golden thread connecting seemingly disparate realms of the quantum world. They serve as both a master craftsman's tool and a detective's magnifying glass.

Let us begin with the conventional, "vanilla" AC Josephson effect. When we shine microwaves of a precise frequency, $f$, onto a superconducting junction, it doesn't just respond randomly. As we saw, the junction's own oscillation can lock onto the external drive. This phase-locking forces the average voltage, $\overline{V}$, across the junction into a series of exquisitely sharp and stable plateaus, known as Shapiro steps.

The truly breathtaking feature of these steps is the voltage at which they appear. For a conventional junction, these voltages are given by a remarkably simple and profound formula:

where $n$ is an integer (1, 2, 3, ...), $h$ is Planck’s constant, and $e$ is the charge of an electron. Look closely at this equation. The voltage does not depend on the material of the superconductor, the size or shape of the junction, the temperature (as long as it's cold enough), or even the power of the microwaves you’re using. It depends only on an integer you choose, a frequency you can measure with atomic-clock precision, and two of nature's most fundamental constants, $h$ and $e$.

This isn't just an elegant piece of physics; it's the foundation of modern electrical metrology. If you want to create a voltage source of unparalleled accuracy, you don't build a better battery. You take thousands of Josephson junctions, link them in a series, and irradiate them with a stable microwave source. By locking all of them to the first step ($n=1$), the total voltage becomes $N \times (h/2e)f$, where $N$ is the number of junctions. This relation is so robust that the international community has used it to define the volt. The Josephson Voltage Standard is not just an application of the effect; it is its ultimate practical expression, a ruler for electricity forged directly from the laws of quantum mechanics. The very stability that makes this possible—the fact that the voltage steps are independent of messy material details—is a direct consequence of the coherent, collective quantum state of the superconductor.

Now, let us turn our attention from engineering perfection to fundamental discovery. If the conventional effect is a story of impeccable order, the fractional effect is a detective story. The prime suspect is the Majorana fermion, a particle that is its own antiparticle, hiding in the quantum shadows of topological superconductors. How do we prove it's there? We can't just look at it. We need to find its unique fingerprint. The fractional AC Josephson effect provides just that.

Imagine you are an experimentalist presented with three mysterious junctions, and your only tool is a microwave source and a voltmeter. This is precisely the challenge scientists face.

• ​​Device X​​ shows a perfect staircase of voltage steps at multiples of $(h/2e)f$. This is our baseline, our "control" subject. It behaves exactly as a conventional junction should.

• ​​Device Y​​ is more puzzling. It shows the expected integer steps, but also a new set of stable plateaus exactly halfway in between them—so-called "half-integer" steps. Is this the Majorana? The excitement is premature. This is a classic red herring. Further investigation reveals that such steps can arise if the junction's current-phase relation (CPR) is not a pure sine wave, but contains higher harmonics (e.g., a $\sin(2\phi)$ term). The underlying physics is still $2\pi$-periodic. We know such skewed CPRs exist because we have clever, independent methods to measure them, for instance by embedding the junction in a tiny superconducting loop, a SQUID, and mapping its magnetic response.

• ​​Device Z​​ presents the true smoking gun. Here, the steps at $V_1, V_3, V_5$, and so on are all missing. Only the even-numbered steps ($V_2, V_4$, ...) appear robustly. This isn't just the appearance of new steps; it's the systematic disappearance of expected ones.

Why is this the crucial signature? As we learned, the presence of Majorana modes can make the junction's energy $4\pi$-periodic in phase. A stable phase-locked state—a Shapiro step—requires the junction's internal state to return to an equivalent configuration after each cycle of the microwave drive. For a $4\pi$-periodic system, advancing the phase by an odd multiple of $2\pi$ does not return the system to its original state; it inverts the supercurrent's sign. Therefore, stable locking cannot occur, and the odd steps are suppressed. The absence of something can be the most powerful evidence of all.

But a good detective always seeks corroborating evidence. We can go beyond just measuring voltage steps and listen to the junction's quantum "song." A DC-biased conventional junction emits radiation at the Josephson frequency, $\omega_J = 2eV/\hbar$. However, an ideal topological junction should sing at exactly half that frequency, $\omega_M = eV/\hbar$. Modern experiments can build tiny on-chip "radios" (resonators) tuned to listen for this telltale half-frequency emission. We can even add a weak radio-frequency drive and look for a whole comb of "photo-assisted" sidebands centered on this fractional frequency, providing a rich spectral fingerprint.

Another, even more subtle clue lies in the non-linear response. A system with a $4\pi$-periodic potential behaves like a pendulum that must swing around twice to be back where it started. If we "kick" such a system with a drive at frequency $\omega_{rf}$, it can resonate and emit power at the subharmonic frequency $\omega_{rf}/2$. Detecting such a phase-locked subharmonic emission is a powerful, dynamic confirmation that we are indeed dealing with the strange world of $4\pi$ periodicity.

Perhaps the deepest beauty revealed by this physics is its universality. The Josephson effect is not just about superconductors. It is a theme that nature plays again and again, a universal symphony performed by different quantum orchestras.

Consider a Bose-Einstein Condensate (BEC), a cloud of ultracold neutral atoms that have all collapsed into a single macroscopic quantum state. If we trap these atoms in a double-well potential, we create a system analogous to a Josephson junction. There are no charges, no currents, and no magnetic fields. Yet, the atoms can tunnel between the two wells. The relative population of atoms in the wells, $z(t)$, and their relative quantum phase, $\phi(t)$, become the key variables. And what happens when we slightly perturb this system? The atom populations slosh back and forth between the wells in a perfect oscillation—the internal Josephson effect. The same equations we use for superconductors describe the sloshing of these neutral atoms, revealing the deep unity of macroscopic quantum phenomena.

Let's take one final, mind-bending step. The story of the fractional Josephson effect is about a system whose periodicity is doubled, from $2\pi$ to $4\pi$. But what if the fundamental charge itself were fractional? This is not a fantasy. In the bizarre two-dimensional world of the Fractional Quantum Hall Effect, the elementary charge carriers are not electrons, but quasiparticles with charges like $e/3$. If we form a weak junction between the edges of such a system and apply a DC voltage, it too sings a song. The frequency of the resulting AC current is given by the same fundamental relation, $f = \Delta E/h$, but now the energy per tunneling event is $\Delta E = e^*V = (e/3)V$. The frequency becomes $f = eV/(3h)$. Here we have another kind of "fractional AC effect," one born not from a doubled phase period, but from a fractionated charge.

From the precision ruler that defines our volt to the detective's subtle probe for new fundamental particles, and finally to a universal principle echoing through cold atoms and fractional-charge liquids, the Josephson effect is a profound testament to the interconnectedness of physics. It teaches us that by listening carefully to the subtle harmonies of quantum phase, we can measure our world with astonishing precision and, at the same time, catch glimpses of new worlds yet to be fully discovered.