Signatures of superconducting Higgs mode in irradiated Josephson junctions

This paper predicts that the unambiguous detection of the superconducting Higgs mode can be achieved in microwave-irradiated asymmetric and transparent Josephson junctions by observing resonant enhancements in the second harmonic of the current-phase relation at zero bias and the AC Josephson current at finite bias.

The Gist

Imagine you have a superconductor. In the world of physics, this is like a magical material where electricity flows with zero resistance. Inside this material, electrons pair up and dance in perfect unison. This synchronized dance is described by something physicists call the "order parameter." Think of this order parameter as a giant, invisible drumhead stretched across the material.

Usually, this drumhead is perfectly still. But if you hit it just right, it can vibrate in two different ways:

1. The Phase Wiggle: The whole drumhead tilts back and forth. This is easy to see because it creates electric currents.
2. The Higgs Bounce: The drumhead doesn't tilt; instead, it expands and contracts, like a balloon being squeezed and released. This is the Higgs mode. It's the "heartbeat" of the superconductor.

The Problem:
The Higgs bounce is very shy. It doesn't carry an electric charge, so it's incredibly hard to spot. Scientists have tried to find it using high-speed lasers (THz spectroscopy), but it's like trying to hear a whisper in a hurricane. It's often ambiguous.

The New Idea:
This paper proposes a clever new way to catch the Higgs mode in the act, using a device called a Josephson Junction.

The Analogy: The Two-Story Bridge

Imagine a bridge connecting two islands (the two superconductors).

• Island A (Left): Has a small, weak population (a small superconducting gap).
• Island B (Right): Has a huge, strong population (a large superconducting gap).
• The Bridge: It's a very open, transparent bridge where people (electrons) can cross back and forth easily.

Normally, people cross the bridge in a steady rhythm. But the authors propose two ways to shake things up to reveal the Higgs mode.

Method 1: The Microwave Shaker (Phase Bias)

Imagine you start shaking the bridge with a microwave oven's frequency.

• Without the Higgs: The bridge sways in a simple, predictable pattern. If you measure the "current" of people crossing, the second rhythm (the second harmonic) is weak and points in one direction.
• With the Higgs: When the shaking frequency matches the natural "heartbeat" (mass) of the Higgs mode on the weak island, something magical happens. The bridge starts to bounce up and down in sync with the shake.
• The Clue: This bounce changes the direction of the second rhythm! It flips the sign. Instead of a weak sway in one direction, you get a strong sway in the opposite direction. It's like if a pendulum suddenly decided to swing backward when you pushed it forward. This "sign flip" is a smoking gun that the Higgs mode is there.

Method 2: The Voltage Ladder (Shapiro Steps)

Now, imagine you push a steady DC voltage (a constant slope) through the bridge while still shaking it with microwaves.

• The Shapiro Steps: Usually, when you combine a steady slope with a shake, the current gets stuck at specific "steps" (like rungs on a ladder). These are called Shapiro steps.
• The Higgs Effect: If the Higgs mode is excited, it acts like a super-charged booster. It amplifies a specific "second harmonic" current (a faster rhythm).
• The Clue: This booster makes a specific, usually tiny, step on the ladder suddenly grow huge—almost as big as the main step. If you see this tiny step suddenly become a giant, you know the Higgs mode is resonating and pumping energy into the system.

Why This Matters

The authors used complex math (like a microscopic simulation of every electron) to prove that this isn't just a theory. They showed that if you build a junction with:

1. High Asymmetry: One side weak, one side strong.
2. High Transparency: A very open bridge.
3. Microwave Radiation: A steady shake.

...you can unambiguously detect the Higgs mode.

The Takeaway

Think of the Higgs mode as a ghost that refuses to be seen directly. But if you shake the house (the junction) at just the right frequency, the ghost causes the furniture (the electrical current) to move in a weird, specific way that no other ghost could cause.

This paper gives us a new "ghost detector" for superconductors. Instead of needing expensive, high-tech lasers, we might be able to find the Higgs mode using standard electrical equipment and a bit of microwave radiation. It's a simpler, clearer way to prove that the "heartbeat" of the superconductor exists.

Technical Summary

1. Problem Statement

The Higgs mode in superconductors corresponds to oscillations of the amplitude of the superconducting order parameter (OP), $\Delta(t) = |\Delta(t)|e^{i\phi(t)}$. While the phase mode (Nambu-Goldstone mode) is easily detected, the Higgs mode is electrically neutral and difficult to observe directly. Previous attempts using THz spectroscopy have found signatures, but unambiguous detection remains challenging.

Theoretical proposals have suggested detecting the Higgs mode via transport in Josephson junctions (JJs), specifically by looking for an enhancement of the second harmonic of the AC Josephson current ($2\omega_J$) when the bias voltage $V$ satisfies $2eV \approx \omega_H$ (the Higgs mass). However, a direct detection of this high-frequency current (e.g., ~45 GHz for Aluminum) is experimentally impractical.

The Core Challenge: How to unambiguously detect the Higgs mode in a Josephson junction using standard, lower-frequency transport measurements without requiring direct high-frequency current detection.

2. Methodology

The authors propose a theoretical framework combining phenomenological models with a rigorous microscopic Floquet-Keldysh formalism.

• System Model: They consider a highly asymmetric Josephson junction (two superconducting leads with unequal equilibrium gaps, $\Delta_{0,L} \ll \Delta_{0,R}$) with high transparency. The junction is subjected to microwave irradiation at frequency $\omega_r$.
• Excitation Mechanism:
  • Unlike previous works where the bulk leads are irradiated by a vector potential, this work assumes the radiation induces an oscillating voltage across the barrier ($V_{AC} \cos(\omega_r t)$).
  • This voltage creates a time-dependent Josephson phase $\phi(t)$, which, via gauge invariance, couples to the order parameter amplitude, exciting the Higgs mode in the lead with the smaller gap.
• Theoretical Tools:
  • Phenomenological Model: Uses a coupled order parameter approach to derive the Higgs susceptibility $\chi(\omega)$ and predict the resonant enhancement of the second harmonic current.
  • Microscopic Theory: Employs a self-consistent Keldysh-Gorkov formalism on a lattice model (single-channel JJ). They solve the Dyson equations for retarded/advanced Green's functions and the non-equilibrium gap equation self-consistently.
  • Floquet Analysis: Since the system is driven by both DC voltage ($\omega_J = 2eV$) and AC radiation ($\omega_r$), the Green's functions and self-energies are expanded in a pseudo-Floquet basis to handle the incommensurate frequencies.

3. Key Contributions

The paper proposes two distinct, unambiguous experimental signatures of the Higgs mode in microwave-irradiated JJs:

1. Microwave-Induced Enhancement of the Second Harmonic in the Current-Phase Relation (CPR):

   • In the absence of DC bias, the equilibrium CPR is typically dominated by the fundamental harmonic ($\sin \phi$).
   • The Higgs mode induces a significant enhancement of the second harmonic ($\sin 2\phi$) component of the DC supercurrent.
   • Crucial Distinction: The Higgs-induced second harmonic has a positive sign at zero radiation strength ($\alpha=0$), whereas the conventional (Higgs-free) second harmonic is negative. This sign change skews the CPR peak, providing a clear fingerprint.
   • The amplitude exhibits resonant behavior when the microwave frequency $\omega_r$ is tuned across the Higgs mass ($\omega_r \approx 2\Delta_{0,L}$).

2. Enhancement of Shapiro Steps at Finite DC Bias:

   • When a DC voltage bias is applied alongside microwaves, Shapiro steps (SS) appear in the $I-V$ characteristics.
   • The authors identify a specific step, $SS^2_1$, which occurs when $2\omega_J = \omega_r$.
   • In conventional JJs, this step is negligible because the $2\omega_J$ current is weak. However, near the Higgs resonance ($\omega_J \approx \Delta_{0,L}$), the Higgs mode drastically enhances the $2\omega_J$ current.
   • Signature: The $SS^2_1$ step height becomes comparable to the fundamental step $SS^1_1$ (where $\omega_J = \omega_r$), and its dependence on radiation strength deviates from the standard Bessel function profile ($J_{-1}(\alpha)$) due to the mixing of harmonics.

4. Key Results

The authors performed numerical simulations based on their microscopic theory to validate the phenomenological predictions:

• CPR Analysis (Phase Bias):

  • Sign Reversal: As predicted, the second harmonic amplitude $\bar{I}^{(2)}$ flips from negative (conventional) to positive (Higgs-renormalized) at $\alpha=0$.
  • Resonance: The magnitude of the second harmonic peaks sharply when $\omega_r = 2\Delta_{0,L}$.
  • CPR Skewing: The full CPR curve $I(\phi_0)$ changes shape, developing a "dip" or skewing backward/forward depending on the sign of the second harmonic, which is a direct visual indicator of the Higgs mode.

• Shapiro Steps Analysis (Voltage Bias):

  • Amplitude Enhancement: In the absence of the Higgs mode, the $2\omega_J$ current is suppressed by the gap asymmetry. With Higgs renormalization, the $2\omega_J$ current ($I_{2\omega_J}$) becomes comparable to the fundamental current ($I_{\omega_J}$) at resonance.
  • Step Height Ratios: The height of the $SS^2_1$ step (governed by $I_{2\omega_J}$) increases dramatically near resonance, reaching magnitudes similar to $SS^1_1$.
  • Deviation from Adiabatic Approximation: While the adiabatic approximation (AA) captures the qualitative dependence on radiation strength ($\alpha$), the full microscopic theory reveals deviations at higher frequencies due to electronic retardation effects, confirming the necessity of the non-equilibrium formalism.

• Comparison with Previous Work: The authors clarify that their results differ from recent studies (e.g., Vallet & Cayssol) where the bulk leads are irradiated. In the bulk-irradiated case, the spectrum lacks the $\omega_J$ component and specific harmonics found in their barrier-voltage model, leading to different resonance conditions and step structures.

5. Significance and Practical Implications

• Experimental Feasibility: The proposal offers a practical route to detect the Higgs mode using standard transport measurements (DC $I-V$ curves and CPR) rather than complex THz spectroscopy.
• Material Requirements: The authors suggest using Aluminum-based junctions due to their small superconducting gap (~45 GHz), which falls within the accessible microwave range. The required asymmetry can be achieved by varying the thickness of the Al leads or tuning the temperature.
• Robustness: The signatures are robust even at moderate transparencies (though high transparency enhances the effect) and rely on the fundamental symmetry breaking of the superconducting state, making them applicable to conventional s-wave superconductors.
• Fundamental Physics: This work bridges the gap between non-equilibrium superconductivity and collective mode physics, demonstrating how the Higgs mode can be "read out" via the Josephson effect, effectively turning the JJ into a detector for amplitude fluctuations.

In conclusion, the paper provides a comprehensive theoretical blueprint for identifying the superconducting Higgs mode through specific, measurable anomalies in the current-phase relation and Shapiro steps of microwave-irradiated, asymmetric Josephson junctions.