AC-flux-driven SQUID diode spectroscopy as a probe of current-phase relations

This paper proposes and validates a method to unambiguously extract individual current-phase relation harmonics in asymmetric SQUIDs by analyzing the distinct Bessel-function-modulated signatures of the ac-flux-driven diode effect, offering a robust spectroscopic tool for investigating unconventional superconductors.

The Gist

Imagine you have a special kind of electrical switch called a Josephson junction. In a perfect, normal world, electricity flows through this switch the same way no matter which direction you push it. But in the strange world of superconductors (materials that conduct electricity with zero resistance), things get weird. Sometimes, these switches act like a diode: they let electricity flow easily in one direction but block it in the other.

The scientists in this paper wanted to figure out why a specific switch acts like a diode. They suspected the answer lies in the "secret recipe" of the material inside the switch, known as the Current-Phase Relation (CPR). Think of the CPR as the unique musical signature or DNA of the superconductor. Some materials have a simple, smooth signature (like a pure sine wave), while others have complex, jagged signatures with extra beats (harmonics) or even half-beats (subharmonics).

The Problem:
Usually, trying to read this "musical signature" is like trying to identify a song by listening to a muffled, distorted recording. Different songs can sound almost identical when played through a standard filter, making it hard to tell them apart. Plus, the equipment itself (the device) can add its own noise, hiding the true sound.

The Solution: The "Flux-Driven Diode" Trick
The authors propose a new way to listen to the music. Instead of just pushing a steady current, they wiggle the magnetic environment around the switch using an AC (alternating) magnetic flux. Imagine shaking a box of marbles back and forth. How the marbles (the electrons) react to the shaking depends entirely on the shape of the box (the CPR).

They built a device called an asymmetric SQUID (basically a loop with two different switches). One switch is "normal," and the other is the "mystery" switch they want to test.

The Magic of the "Bessel Dressing"
Here is the clever part: When they shake the magnetic field, the different parts of the "musical signature" get wrapped in a special mathematical coat called a Bessel function.

• If the mystery switch has a simple signature, the shake makes it wiggle a little bit in a predictable way.
• If it has a half-beat signature (like a 4π-periodic wave, often found in topological materials), the shake creates a pattern of "arcs" (like rainbow bands) that are wide apart.
• If it has a double-beat signature (like a π-periodic wave, found in multi-band superconductors), the shake creates a pattern of arcs that are very crowded and dense.
• If it has both, the patterns interfere with each other, creating a complex, beat-like mess.

The Result: A Fingerprint Map
By measuring how well the device acts as a diode while changing two things—the strength of the shake (amplitude) and the speed of the shake (frequency)—they can draw a 2D map.

• Simple Switch: The map is mostly empty or has very faint, weak lines.
• Half-Beat Switch: The map shows wide, spaced-out rainbow arcs.
• Double-Beat Switch: The map shows a dense forest of rainbow arcs.
• Mixed Switch: The map shows a unique, interwoven pattern.

Why It Matters
This method is like having a high-resolution scanner that can see the "DNA" of the superconductor without needing to build complex microwave circuits or guess based on static measurements. It works even if the device is "dampened" (sluggish) or "underdamped" (bouncy).

Who Can Use This?
The paper suggests this technique is perfect for testing:

1. Topological materials (which might host exotic particles called Majorana fermions).
2. Multi-band superconductors (like iron-based superconductors).
3. Exotic states of matter where electrons pair up in groups of four instead of two.

In short, the authors have invented a new "spectroscopy" tool. Instead of just asking "Does it conduct?", they can now ask, "What is the exact shape of the superconducting state inside?" by simply shaking the magnetic field and watching the resulting rainbow patterns.

Technical Summary

Technical Summary: AC-Flux-Driven SQUID Diode Spectroscopy as a Probe of Current-Phase Relations

Problem Statement
The current–phase relation (CPR) of a Josephson junction encodes microscopic details of superconducting states, particularly through higher-order and fractional harmonics (e.g., $\sin(\phi/2)$ in topological systems or $\sin(2\phi)$ in high-transparency and multiband junctions). However, unambiguously extracting these components is challenging. Different CPR compositions can produce nearly identical static interference patterns, which are further obscured by device asymmetries, damping, and dynamical effects. Existing methods, such as flux-dependent DC transport and Shapiro-step analysis, face limitations: the former suffers from inverse-problem ambiguities where distinct CPRs yield similar flux dependencies, while the latter is susceptible to microwave transmission-line distortions. Furthermore, previous studies utilizing microwave power to tune diode polarity have largely explored only fixed-frequency parameter spaces, leaving the two-dimensional dependence on both amplitude and frequency unexplored.

Methodology
The authors propose a spectroscopic probe based on the AC magnetic-flux-driven diode effect in asymmetric DC SQUIDs. The system consists of two Josephson junctions with unequal critical currents: one conventional junction with a purely sinusoidal CPR ($\sin\phi$) and a second junction potentially hosting subharmonic ($\sin(\phi/2)$) and second-harmonic ($\sin(2\phi)$) components. The SQUID is biased by a DC current and threaded by an external magnetic flux comprising a static component ($\Phi_{dc}$) and an AC component ($\Phi_{ac}\cos\omega t$).

The study employs two complementary theoretical approaches to analyze the fast-driven dynamics:

1. Kapitza-type perturbation theory: Applied to the conventional junction in the limit of small loop inductance ($\beta_L \ll 1$) and small fast oscillation amplitude ($\xi_0 \ll 1$). This yields analytical expressions for the diode efficiency $\eta$ as a function of AC flux amplitude and frequency.
2. Jacobi–Anger averaging: Applied to the general CPR to handle the non-perturbative regime. This method reveals that the AC flux modulation "dresses" each CPR harmonic with a distinct Bessel function factor dependent on the effective phase-modulation amplitude $\xi_0$.

The analytical predictions are verified and extended through numerical solutions of the coupled resistively and capacitively shunted junction (RCSJ) equations for various CPR scenarios, including pure amplitude asymmetry, pure subharmonic, pure second-harmonic, and mixed cases. The analysis covers both overdamped and underdamped regimes.

Key Contributions and Results
The central finding is that the two-dimensional phase diagram of the diode efficiency, $\eta(\phi_{ac}, \omega)$, encodes distinct, identifiable fingerprints of the underlying CPR components. The AC flux modulation dresses each harmonic with a specific Bessel function:

• Conventional $\sin\phi$: Dressed by $J_0(\xi_0)$.
• Subharmonic $\sin(\phi/2)$ (4$\pi$-periodic): Dressed by $J_0(\xi_0/2)$.
• Second-harmonic $\sin(2\phi)$ ($\pi$-periodic): Dressed by $J_0(2\xi_0)$.

These dressings manifest as characteristic arc-like nodal lines (where $\eta=0$) in the $(\phi_{ac}, \omega)$ plane:

• Pure Asymmetry (Conventional): Exhibits a weak response with faint, widely spaced arcs governed by $J_0(\xi_0)$.
• Subharmonic Only: Displays widely spaced concentric arcs. The spacing is determined by the zeros of $J_0(\xi_0/2)$, which are separated by larger intervals of $\xi_0$ (and thus $\phi_{ac}$) compared to the conventional case, reflecting the 4$\pi$ periodicity.
• Second-Harmonic Only: Shows a much denser pattern of arcs. The spacing corresponds to the zeros of $J_0(2\xi_0)$, which are roughly four times denser than the subharmonic case, reflecting the $\pi$ periodicity.
• Mixed Case: Exhibits intermodulation features where the two contributions interfere. This results in fewer visible arcs than the pure second-harmonic case due to partial cancellation of polarity-switching boundaries, creating a unique nonlinear signature.

The study demonstrates that these distinguishing features are robust across different damping regimes (from overdamped $\beta_c=0.1$ to underdamped $\beta_c=3$). While the Stewart-McCumber parameter $\beta_c$ affects the curvature of the arcs via the inductive resonance frequency $\omega_L$, the relative density and amplitude of the arc patterns remain consistent indicators of the CPR composition.

Significance
The paper establishes AC-flux-driven SQUID diode spectroscopy as a practical transport-based diagnostic for nontrivial CPRs. By sweeping both the amplitude and frequency of the applied AC flux, researchers can access a two-dimensional parameter space where different microscopic CPR components produce distinct geometric signatures. This approach offers a solution to the inverse problem of CPR reconstruction, providing a method to distinguish between conventional asymmetric junctions, topological junctions (via fractional Josephson effects), and multiband or high-transparency systems without the ambiguities of static measurements or the circuit complexities of Shapiro-step analysis. The protocol is applicable to systems including topological insulators, multiband superconductors (e.g., iron-based superconductors, MgB$_2$), and potentially charge-4e condensates.