Properties of two level systems in current-carrying superconductors

The paper demonstrates that in disordered superconductors, a dc supercurrent causes a dramatic, parametric enhancement of the coupling between two-level systems and external ac electric fields, leading to increased ac conductivity and 1/f noise in equilibrium current fluctuations.

The Gist

Imagine a superconductor as a super-highway where electrons travel together in a perfectly synchronized dance, creating a frictionless flow of electricity. Usually, this dance is so smooth that it ignores tiny bumps on the road. However, this paper reveals a surprising twist: if you push this super-highway hard enough to create a steady "super-current," tiny, hidden defects in the material suddenly become extremely sensitive to outside vibrations.

Here is the breakdown of what the authors, Liu, Andreev, and Spivak, discovered, using simple analogies:

1. The Hidden "Two-Level" Switches (TLS)

Inside almost all materials, especially those that aren't perfectly pure, there are tiny atomic defects called Two-Level Systems (TLS).

• The Analogy: Think of these as tiny, wobbly seesaws buried deep inside the material. An atom can sit on the left side or the right side. It can occasionally "tunnel" (jump) from one side to the other.
• The Problem: In normal metals, these seesaws are mostly quiet. But in superconductors, they are the main source of "noise" and energy loss, which is bad for sensitive quantum computers.

2. The "Super-Current" Effect

The paper asks: What happens if we run a steady super-current through the material?

• The Discovery: When a steady current flows, the "seesaws" (TLS) become hypersensitive to any new electrical signal (like a radio wave or AC current) hitting them.
• The Metaphor: Imagine a tightrope walker (the super-current) balancing perfectly. If you gently tap the tightrope (apply a small AC electric field), the walker wobbles. Now, imagine the seesaws are tiny acrobats standing on that tightrope. Because the tightrope is already under tension from the walker, the acrobats react massively to even the tiniest tap. The paper calls this a "giant enhancement."

3. Why Does This Happen? (The Friedel Oscillation)

The authors explain that the electrons in the superconductor create a complex interference pattern (like ripples in a pond) around every impurity.

• The Mechanism: When the super-current flows, it changes the speed and direction of the electron "dance." This shifts the ripples (ripples are called Friedel oscillations).
• The Connection: The tiny seesaws (TLS) are sitting right in the middle of these ripples. When the current changes the ripples, it physically pushes or pulls on the seesaws, making it much easier for them to flip from one side to the other.
• The Result: The material becomes incredibly good at absorbing energy from the outside world, but only if the outside signal is slow (low frequency) and aligned with the direction of the current.

4. The "1/f Noise" Mystery

One of the most famous mysteries in physics is 1/f noise (also called pink noise). It's a type of static where the noise gets louder as the frequency gets lower. It's found everywhere, from electronics to stock markets, but nobody fully understands why it happens in superconductors.

• The Paper's Claim: The authors show that this "giant enhancement" of the seesaws explains the 1/f noise perfectly.
• The Analogy: If you have a crowd of people (TLS) flipping switches at random times, and the crowd is huge and varied, their combined flipping creates a specific hum. The paper shows that when a super-current is flowing, this hum becomes deafeningly loud at low frequencies.
• The Key Difference: In normal metals, this noise only happens when you force a current through them. In these superconductors, the noise happens even when the system is in a state of "equilibrium" (balanced), simply because the super-current is flowing.

5. What This Means for the Material

• Direction Matters: This effect only happens if the new electrical signal is moving in the same direction as the super-current. If you hit it from the side, the seesaws don't react as strongly.
• Frequency Matters: The effect is strongest at very low frequencies. As the frequency gets higher, the effect fades away.
• The Bottom Line: The presence of a super-current turns the material into a giant amplifier for low-frequency electrical noise and energy loss.

Summary

The paper argues that in disordered superconductors, a steady super-current acts like a "tuning fork" that makes tiny atomic defects (TLS) scream when exposed to low-frequency electrical signals. This explains why these materials generate a lot of "1/f noise" (static) and lose energy in specific ways, a phenomenon that is much stronger than anyone previously realized. This is purely a theoretical explanation of how these materials behave, not a guide for building new devices yet.

Technical Summary

1. Problem Statement

Two-level systems (TLS) are ubiquitous defects in disordered materials (glasses, amorphous solids) where atoms or groups of atoms tunnel between two quasi-stable configurations. In superconducting devices, TLS are a primary source of dissipation, decoherence, and $1/f$ noise.

• The Gap: While TLS behavior in normal metals and insulators is well-studied, the specific effect of a dc supercurrent on the coupling between TLS and external ac electric fields in disordered $s$-wave superconductors had not been theoretically established.
• The Question: How does the presence of a superfluid momentum ($\bar{p}_s$) modify the interaction between TLS and electromagnetic fields, particularly at low frequencies?

2. Methodology

The authors employ a combination of phenomenological modeling and microscopic diagrammatic techniques within the framework of disordered $s$-wave superconductors in the diffusive regime ($\xi > \ell > \lambda_F$).

• Hamiltonian Formulation:

  • The TLS is modeled by a standard $2 \times 2$ Hamiltonian with energy splitting $2\epsilon$ and tunneling amplitude $t$.
  • The coupling to external fields is introduced via the modulation of the inter-well energy splitting $\epsilon(t)$.
  • Key Insight: The authors propose that in a current-carrying superconductor, the energy splitting acquires a correction $\delta\epsilon_s$ dependent on the superfluid momentum $p_s(t)$ due to the interaction between the TLS atoms and the electron density (specifically Friedel oscillations).
  • Since TLS preserve time-reversal symmetry, this correction must be even in $p_s$: $\delta\epsilon_s \propto p_s^2$.

• Microscopic Calculation:

  • The authors calculate the sensitivity of the electron density $\delta n(r, p_s)$ to changes in superfluid momentum using Feynman diagrams (disorder averaging over random impurities).
  • They evaluate the variance of the electron density fluctuations $\langle (\delta n)^2 \rangle$ and the resulting variance of the TLS energy splitting parameter $\alpha_s$.
  • The calculation utilizes the Gorkov-Nambu Green's functions and accounts for the interference of electron waves scattered by impurities over distances comparable to the coherence length $\xi$.

• Transport and Noise Analysis:

  • The derived coupling is used to calculate the ac conductivity tensor ($\sigma_{\parallel}$ and $\sigma_{\perp}$) and the spectral density of equilibrium current fluctuations using the Fluctuation-Dissipation Theorem (FDT).

3. Key Contributions and Physical Mechanism

The central contribution is the identification of a giant parametric enhancement of TLS coupling to ac electric fields in the presence of a dc supercurrent.

• Mechanism of Enhancement:

  • In the absence of current ($\bar{p}_s = 0$), TLS couple to electric fields primarily through their intrinsic dipole moment ($d$).
  • In the presence of a dc supercurrent ($\bar{p}_s \neq 0$), the time-dependent superfluid momentum $p_s(t) = \bar{p}_s + \delta p_s(t)$ induces a time-varying correction to the TLS energy splitting.
  • The effective coupling term becomes:
    $$\delta\epsilon_\omega = -\left( d - i \frac{e}{\omega} \alpha_s \bar{p}_s \right) \cdot E_\omega$$
  • The second term (mediated by superconducting pairing) dominates when the frequency $\omega$ is below a characteristic scale $\omega^* \sim \Delta (\bar{p}_s/p_d) \sqrt{\xi_0/L_z}$.
  • Physically, the supercurrent makes the electron density (Friedel oscillations) extremely sensitive to the TLS state. As the TLS fluctuates, it modulates the local electron density, which in turn modulates the superfluid density and the supercurrent, creating a strong feedback loop.

• Anisotropy:

  • This enhancement is strictly longitudinal. It occurs only when the ac electric field is parallel to the dc supercurrent ($\sigma_{\parallel}$). The transverse component ($\sigma_{\perp}$) remains unaffected.

4. Key Results

A. Conductivity Enhancement

• Low Frequency Regime ($\omega < \omega^*$): The dissipative longitudinal conductivity $\sigma_{\parallel}(\omega)$ is enhanced by a factor proportional to $(\omega^*/\omega)^2$.
• Frequency Dependence:
  • If the TLS relaxation time distribution $f(\tau)$ is narrow, $\sigma_{\parallel}$ becomes frequency-independent (constant) as $\omega \to 0$.
  • If $f(\tau)$ is broad (typical of glasses, $f(\tau) \sim 1/\tau$), the conductivity becomes inversely proportional to frequency:
    $$\sigma_{\parallel}(\omega) \sim \frac{1}{\omega}$$
  • This contrasts sharply with normal metals where conductivity is frequency-independent at low $\omega$.

B. $1/f$ Noise Generation

• Using the Fluctuation-Dissipation Theorem, the authors relate the enhanced conductivity to equilibrium current fluctuations.
• In systems with a broad relaxation time distribution, the spectral density of current fluctuations $S_s(\omega)$ exhibits a $1/f$ noise behavior:
  $$S_s(\omega) \propto \frac{I_s^2}{V \omega}$$
  where $I_s$ is the bias supercurrent and $V$ is the sample volume.
• Significance: Unlike $1/f$ noise in normal metals (which is a non-equilibrium phenomenon driven by bias current), this noise is an equilibrium phenomenon in the superconducting state, driven by the coupling of TLS to the superfluid flow.

C. Microscopic Origin of Noise

• The paper provides a microscopic derivation showing that TLS transitions induce temporal fluctuations in the superfluid density $\delta N_s(r,t)$.
• These fluctuations propagate through the sample via the continuity equation, resulting in global current fluctuations that scale with the square of the supercurrent.

5. Significance and Implications

• Decoherence in Quantum Devices: The results suggest that supercurrents in superconducting qubits and resonators can drastically increase the coupling to TLS, leading to enhanced dissipation and $1/f$ noise. This is critical for understanding decoherence limits in current-carrying quantum circuits.
• New Noise Mechanism: It identifies a distinct source of equilibrium $1/f$ noise in superconductors that is absent in normal metals, characterized by its dependence on the supercurrent and its divergence at low frequencies.
• Experimental Predictions:
  • A giant increase in microwave absorption for current-carrying superconducting films.
  • Anisotropic dissipation (parallel vs. perpendicular to current).
  • A transition from standard noise spectra to $1/f$ noise below a characteristic frequency $\omega^*$ determined by the supercurrent density.
• Relevance to Junctions: The findings may explain excess dissipation in Superconductor-Insulator-Superconductor (SIS) junctions, where oscillating superfluid velocities in the leads could couple strongly to bulk TLS.

In summary, the paper establishes that a dc supercurrent acts as a "parametric amplifier" for the interaction between TLS and electromagnetic fields, fundamentally altering the low-frequency transport and noise properties of disordered superconductors.