Two-dimensional coherent spectroscopy of disordered superconductors in the narrow-band and broad-band limits

This paper theoretically analyzes two-dimensional coherent spectroscopy signals in disordered superconductors across narrow-band and broad-band limits, revealing distinct nonlinear susceptibility relationships and resonance behaviors tied to quasiparticle and Higgs-mode excitations at the superconducting gap frequency.

The Gist

Imagine a superconductor as a bustling dance floor where electrons pair up and move in perfect unison. Sometimes, this dance floor gets a bit messy (disordered), with obstacles scattered around. Physicists want to understand how these pairs react when hit with light, but standard "flash photography" (linear spectroscopy) often misses the subtle, collective moves of the crowd.

This paper introduces a more advanced technique called Two-Dimensional Coherent Spectroscopy (2DCS). Think of this not as a single flash, but as a sophisticated light show using two laser pulses with a specific delay between them. By analyzing how the electrons respond to this two-pulse "duet," researchers can map out hidden behaviors that are invisible to standard methods.

Here is the breakdown of what the paper discovered, using simple analogies:

1. The Two Ways to Shine a Light

The authors studied two extreme ways of shining these laser pulses on the superconductor:

• The Narrow-Band Limit (The Tuning Fork): Imagine hitting the system with a pure, steady tone, like a tuning fork that rings forever. In this scenario, the paper confirms that the signal you get is related to how the material reacts to a specific "echo" of the light (called the ac Kerr effect).

  • The Result: The signal acts like a threshold. It's like a light switch that stays off until the light frequency hits a specific "gap" size (the energy needed to break an electron pair). Once you cross that threshold, the signal turns on and grows. It doesn't "sing" loudly at a specific note; it just starts working once the volume is high enough.

• The Broad-Band Limit (The Drumstick): Now, imagine hitting the system with a super-short, sharp tap, like a drumstick hitting a drum. This is a "delta-function" pulse.

  • The Result: This creates a completely different signal, related to the dc Kerr effect. Instead of just turning on, this signal resonates. It's like hitting a bell; when the frequency of the tap matches the natural "ringing" frequency of the electron pairs, the signal explodes with intensity.

2. The Mystery of the "Higgs Mode"

In the world of superconductors, there is a special collective vibration called the Higgs mode. You can think of this as the "heartbeat" or the "breathing" of the electron pairs.

• The Problem: Usually, this heartbeat is hard to hear because the individual dancers (quasiparticles) are also moving and making noise at similar frequencies.
• The Discovery:
  • In the Narrow-Band (steady tone) case, the heartbeat is actually off-beat. The signal is mostly driven by a "ghost" of the heartbeat that isn't really resonating. It's like trying to hear a drumbeat by listening to the silence between the beats; you get a signal, but it's not the main drum sound.
  • In the Broad-Band (sharp tap) case, the signal does catch the heartbeat. When the tap frequency matches the heartbeat's natural rhythm, the signal peaks sharply. This is the "resonance" the authors found.

3. The Role of "Messiness" (Disorder)

The paper looked at superconductors that are "dirty" (full of impurities) versus "clean."

• In the Dirty Regime: The "heartbeat" (Higgs mode) is very loud and dominates the signal, especially in the broad-band limit. The messiness of the material actually helps the heartbeat stand out against the background noise of individual dancers.
• In the Clean Regime: As the material gets cleaner, the "heartbeat" gets quieter, and the individual dancers (quasiparticles) start to dominate the signal again.

4. Why This Matters for Experiments

The authors compared their theory to real-world experiments done on a material called NbN.

• The Puzzle: Experiments showed a sharp peak (resonance) at a specific frequency.
• The Explanation: Previous theories using the "steady tone" (narrow-band) model couldn't fully explain this peak because that model only shows a threshold, not a sharp peak.
• The Solution: The authors suggest that even though experiments use "narrow" pulses, they aren't perfectly narrow. They have a little bit of "broadness" (like a drumstick that isn't infinitely sharp). This small broadness allows the dc Kerr effect (the resonance) to sneak in, explaining why the experiments see a sharp peak that matches the heartbeat of the superconductor.

Summary

This paper acts as a translator between two different languages of light. It tells us that if you shine a steady light, you see a "switch-on" behavior. If you hit the material with a sharp tap, you see a "ringing" behavior. By understanding this difference, we can finally explain why real-world experiments see a sharp resonance peak in superconductors: it's the material's "heartbeat" (Higgs mode) finally being heard clearly through the right type of light pulse.

Technical Summary

Technical Summary: Two-Dimensional Coherent Spectroscopy of Disordered Superconductors in the Narrow-Band and Broad-Band Limits

Problem Statement
Two-dimensional coherent spectroscopy (2DCS) has emerged as a powerful tool for probing nonlinear optical responses in quantum materials, including superconductors, by mapping signals in multi-dimensional frequency space. While 2DCS has been applied to detect collective modes such as the Higgs mode (the amplitude mode of the superconducting order parameter) in materials like NbN and MgB2, a theoretical understanding of the signal's origin remains challenging. Specifically, distinguishing the contribution of the Higgs mode from quasiparticle excitations is difficult because both contribute to signals at similar frequencies (near the superconducting gap, $2\Delta$). Furthermore, previous theoretical analyses of disordered superconductors have largely focused on the narrow-band limit (sinusoidal pulses) and relied on real-space simulations that struggle to satisfy the necessary length-scale separations for the "dirty" regime. There is a need to systematically analyze 2DCS signals in both narrow-band and broad-band limits to clarify the relationship between pulse characteristics and the underlying nonlinear susceptibilities.

Methodology
The authors theoretically analyze 2DCS signals for disordered superconductors using a lattice model based on BCS mean-field theory and the self-consistent Born approximation (SCBA) for impurities.

1. Formalism: The paper derives a general formula for the 2DCS signal induced by two arbitrary pulses, expressing the nonlinear current as a frequency integral of the third-order nonlinear susceptibility $\chi^{(3)}$ multiplied by the Fourier spectra of the pulse fields.
2. Limits: The study focuses on two specific limits:
   • Narrow-band limit: Pulses are modeled as monochromatic sinusoidal waves.
   • Broad-band limit: Pulses are modeled as delta-function pulses.
3. Numerical Simulation: The authors numerically evaluate the relevant nonlinear susceptibilities for a square lattice model with random disorder. The SCBA is employed to treat disorder effects, allowing for the satisfaction of the required scale separation ($v_F/W \ll v_F/\gamma \ll v_F/2\Delta \ll L$) without the computational cost of large real-space simulations. The calculations decompose the total susceptibility into quasiparticle (QP) and Higgs-mode (H) contributions, further classified by photon vertex couplings (diamagnetic vs. paramagnetic).

Key Contributions

1. General Formulation: The paper provides a general derivation linking 2DCS signals to third-order nonlinear susceptibilities for arbitrary pulse shapes.
2. Identification of New Susceptibility: A primary finding is that while the narrow-band limit 2DCS signal is related to the ac Kerr susceptibility $\chi^{(3)}(\Omega; \Omega, \Omega, -\Omega)$ and third harmonic generation (THG), the broad-band limit signal along diagonal and horizontal lines in the 2D frequency space is directly related to the dc Kerr susceptibility $\chi^{(3)}(\Omega; \Omega, 0, 0)$.
3. Physical Mechanism Differentiation: The study clarifies the physical origins of the signals in the dirty regime:
   • ac Kerr (Narrow-band): Shows a threshold behavior at $\Omega = 2\Delta$. This is mediated by the Higgs mode carrying zero frequency (off-resonant), effectively picking up the tail of the Higgs resonance.
   • dc Kerr (Broad-band): Exhibits a resonance peak at $\Omega = 2\Delta$. This is mediated by the Higgs mode carrying the fundamental frequency $\Omega$ (resonant process).
4. Disorder Dependence: The paper investigates the competition between Higgs mode and quasiparticle contributions as a function of disorder strength ($\gamma$), showing that the dominant mechanism for the resonance at $2\Delta$ shifts depending on whether the system is in the dirty or clean regime.

Results

• Narrow-Band Limit: The calculated ac Kerr susceptibility $|\chi^{(3)}(\Omega; \Omega, \Omega, -\Omega)|^2$ demonstrates a threshold behavior where the signal intensity grows only when $\Omega \geq 2\Delta$. In the dirty regime, this signal is dominated by the Higgs mode diagram (H3) with paramagnetic coupling, but the Higgs mode is far off-resonant (carrying zero frequency). A minor resonance peak at $\Omega = \Delta$ is observed, attributed to two-photon absorption by quasiparticles.
• Broad-Band Limit: The dc Kerr susceptibility $|\chi^{(3)}(\Omega; \Omega, 0, 0)|^2$ shows a distinct resonance peak at $\Omega = 2\Delta$. In the dirty regime, this is dominated by the Higgs mode (H3) where the propagator carries the driving frequency $\Omega$, leading to resonance with the Higgs eigenfrequency.
• Temperature Dependence:
  • For the ac Kerr susceptibility, the temperature profile shows a peak near the critical temperature $T_c$, independent of the fixed frequency $\Omega$. This reflects the threshold behavior where the spectral weight builds up as the gap opens.
  • For the dc Kerr susceptibility, the peak position in the temperature profile shifts to satisfy the resonance condition $\Omega = 2\Delta(T)$.
• Clean vs. Dirty Regimes: In the dirty regime, the Higgs mode dominates the resonance at $2\Delta$ for the dc Kerr effect. In the clean limit, paramagnetic coupling vanishes, and the resonance at $2\Delta$ becomes dominated by quasiparticle contributions (diamagnetic coupling). The ac Kerr susceptibility remains Higgs-dominated in the clean regime but decays rapidly at high frequencies.

Significance and Claims
The paper claims that the 2DCS signal is not universally described by a single susceptibility but depends critically on the pulse bandwidth. The authors argue that the experimentally observed resonance peak at $\Omega = 2\Delta$ in NbN superconductors (previously attributed to the ac Kerr effect) may actually contain significant contributions from the dc Kerr susceptibility, given that experimental pulses have finite bandwidths intermediate between the idealized limits.

The study emphasizes that 2DCS alone cannot unambiguously distinguish between Higgs mode and quasiparticle contributions, as both can contribute to the resonance at $2\Delta$ depending on the disorder strength and the specific susceptibility probed. Consequently, the authors assert that a rigorous comparison between experimental data and theoretical calculations of the full frequency-dependent susceptibilities is necessary to determine the physical origin of 2DCS signals. The work provides a framework for interpreting 2DCS data in disordered superconductors and highlights the utility of the broad-band limit in accessing the dc Kerr susceptibility, which offers a resonant probe of the Higgs mode in the dirty regime.