Supercurrent Growth in Nonequilibrium Superconductors

This paper reveals that in nonequilibrium superconductors, a supercurrent initiated by a probe field can paradoxically grow in time during the cooling phase due to momentum-relaxing scattering of Bogoliubov quasiparticles by impurities and phonons, leading to experimental signatures such as an ultrafast Meissner effect and optical reflectivity exceeding unity.

The Gist

The Big Idea: The "Magic" Current That Grows

Imagine you have a superhighway where cars (electrons) usually drive at a steady speed. In a normal road, if you hit a bump (impurity) or a pothole (phonon), the cars slow down or crash. You expect traffic to get worse.

But in this paper, the researchers discovered a strange, counter-intuitive phenomenon in superconductors (materials that conduct electricity with zero resistance). They found that under specific conditions, the more obstacles you add, the faster the traffic flows.

They call this "Supercurrent Growth." It's like a river that, instead of slowing down when it hits rocks, suddenly speeds up and flows with more power as the water level rises.

---

The Story: The "Pump, Probe, and Cool" Dance

To understand how this happens, imagine a three-step dance performed by the electrons in a superconductor:

Step 1: The "Pump" (The Party Starts)

Imagine a superconductor is a quiet library. Suddenly, a laser pulse (the "pump") hits it like a giant speaker blasting loud music.

• What happens: The electrons get excited and heated up. They stop being a calm, organized group and turn into a chaotic crowd of "quasiparticles" (excited electrons).
• The Result: The superconducting "magic" (the ability to flow without resistance) gets suppressed. The library is now a noisy, chaotic mess.

Step 2: The "Cooling" (The Calm Returns)

After the music stops, the electrons start to calm down. They give their extra energy to the material's atoms (phonons), like people sweating after a workout.

• What happens: The electrons begin to pair up again to form "Cooper pairs" (the special teams that allow superconductivity). The "superfluid density" (the number of these super-teams) starts to grow back.
• The Expectation: Usually, we think that as the system cools, things just return to normal.

Step 3: The "Probe" (The Surprise)

Here is the twist. While the electrons are in the middle of this "cooling down" phase, the researchers hit them with a tiny, quick electric pulse (the "probe").

• The Surprise: Instead of the current staying steady or dying out, the current actually grows stronger over time.
• The Metaphor: Imagine a group of people trying to walk through a crowded hallway. Usually, if the hallway gets crowded (more obstacles), you move slower. But in this experiment, as the crowd organizes itself (cools down), the obstacles (impurities) actually help the organized group move faster than they would have on their own.

---

The Secret Mechanism: Why Do Obstacles Help?

This is the part that breaks our common sense. We usually think impurities (dirt, defects, or atoms that don't fit) are bad because they block the flow.

The Paper's Discovery:
In this specific "cooling down" phase, impurities and vibrations (phonons) act like traffic directors.

1. The Setup: When the laser hits, the electrons are moving in all directions chaotically.
2. The Collision: As the electrons try to pair up and form a supercurrent, they hit impurities.
3. The Magic: In a normal metal, hitting an impurity stops you. But in this superconducting state, when an electron hits an impurity, it doesn't just stop; it gets "scattered" in a way that helps it align with the flow of the other electrons.
4. The Result: The scattering process actually removes the "backward" moving electrons and pushes the "forward" moving ones into a tighter, faster stream.

Analogy: Think of a group of dancers trying to form a line. If they are all running in different directions, they crash. But if they bump into a wall (an impurity) at the right moment, it forces them to turn and join the line. The "bump" didn't stop the dance; it helped organize the dance, making the line move faster.

---

Why Does This Matter? (The Real-World Magic)

The paper predicts two cool things that could happen because of this "growing current":

1. The "Ultrafast Meissner Effect":

   • Superconductors usually push magnetic fields away (like a magnet floating over a superconductor).
   • This paper suggests that during the cooling phase, the superconductor pushes the magnetic field away so fast and so hard that it creates a temporary, super-strong shield. It's like a shield that gets stronger the moment you try to break it.

2. Light Amplification (Reflectivity > 100%):

   • Normally, if you shine a light on a mirror, you get back less light than you sent in (some is absorbed).
   • Because this "supercurrent growth" acts like a gain medium (an amplifier), the paper predicts that if you shine a specific type of light on this cooling superconductor, you could get back more light than you sent in.
   • Analogy: It's like whispering into a microphone that is connected to a speaker, but instead of just making your voice louder, the room itself starts singing along, making the sound louder than your original whisper.

Summary

The paper solves a puzzle: How can a current grow when it's supposed to be dying out?

• Old Idea: Impurities always slow things down.
• New Idea: In a superconductor that is cooling down, impurities act like coaches, helping the electrons organize into a super-fast team.
• The Result: A temporary state where electricity flows with increasing power, potentially allowing us to amplify light and create ultra-fast magnetic shields.

It turns the idea of "obstacles" on its head, showing that sometimes, a little bit of friction is exactly what you need to get moving faster.

Technical Summary

1. Problem Statement

In ultrafast pump-probe experiments on superconductors, a high-energy laser pulse heats the electronic system, suppressing the superconducting order parameter (gap) and creating a high density of quasiparticles. As the system cools via electron-phonon thermalization, the superfluid density ($n_s$) recovers.

• The Paradox: Standard Time-Dependent Ginzburg-Landau (TDGL) theory predicts that if a probe electric field is applied during this recovery, the supercurrent ($j_s = n_s v$) should grow exponentially in time as $n_s$ increases, even though the velocity $v$ (induced by the probe) remains constant.
• The Conflict: This "supercurrent growth" appears to violate momentum conservation. In a Galilean invariant system, the total current is proportional to the total momentum, which should be conserved. If impurities and phonons are present, the conventional wisdom is that they only attenuate currents (dissipation), not amplify them.
• The Question: Does this supercurrent growth actually occur in real materials, and if so, what is the microscopic mechanism that allows current amplification without violating fundamental conservation laws?

2. Methodology

The authors move beyond the phenomenological TDGL theory to a microscopic approach to resolve the paradox.

• Hamiltonian: They start with the Bardeen-Cooper-Schrieffer (BCS) mean-field Hamiltonian coupled with electron-impurity, electron-phonon, and electron-electron interactions.
• Kinetic Theory: They employ the Boltzmann kinetic equation for the distribution function of Bogoliubov quasiparticles ($f_k$). This equation is coupled with the self-consistent gap equation to describe the incoherent dynamics of the system on time scales longer than the inverse gap ($1/\Delta$).
• Perturbative Analysis: They analyze the linear response to a probe electric field (represented by a transverse vector potential $A$) applied during the cooling phase. The distribution function is decomposed into:
  1. An equilibrium-like part ($f^0_k$).
  2. A symmetric "energy mode" deviation ($\delta f^s_k$) driven by heating/cooling.
  3. An asymmetric "momentum mode" deviation ($\delta f^a_k$) driven by the probe field and scattering.
• Scattering Rates: The model explicitly includes momentum-relaxing scattering rates ($\gamma_i$) due to impurities and phonons, and energy-relaxation rates ($\gamma_E$).

3. Key Contributions and Mechanism

The paper's central contribution is identifying the microscopic origin of supercurrent growth, which fundamentally challenges the intuition that impurities always dampen currents.

• Mechanism of Amplification:

  • When a vector potential $A$ is applied, it shifts the quasiparticle energy dispersion, creating an asymmetry in momentum space.
  • Impurity and Phonon Scattering: These momentum-relaxing processes scatter quasiparticles to restore the distribution toward the shifted equilibrium. Crucially, as the superconducting gap $\Delta(t)$ recovers during cooling, the number of quasiparticles decreases (via pair recombination).
  • The "Gain" Process: The recovery of the gap reduces the population of quasiparticles that carry the paramagnetic current ($j_p$), which flows in the direction opposite to the applied field (canceling the diamagnetic current).
  • Momentum Relaxation's Role: Impurity scattering relaxes the momentum imbalance caused by the probe. As the gap grows and quasiparticles annihilate, the paramagnetic current (which opposes the supercurrent) decays faster than the diamagnetic current. Consequently, the net current increases.
  • Resolution of Paradox: The extra momentum required for the current growth is supplied by the impurities and phonons (the lattice), which act as a momentum sink/source. This breaks Galilean invariance locally, allowing the current to grow without violating global conservation laws.

• Nonequilibrium Two-Fluid Model:
  The authors derive a generalized optical conductivity for the nonequilibrium state:
  $$j(t) = -[n_s(t) + n_n(t)e^{-\gamma_i t}] \frac{e^2}{m} A(t)$$
  where $n_s(t)$ is the growing superfluid density and $n_n(t)$ is the transient normal fluid density. The term $n_n(t)e^{-\gamma_i t}$ represents the decaying paramagnetic current. As $n_s(t)$ grows and the paramagnetic term decays, the total current $j(t)$ increases.

4. Key Results

1. Confirmation of Supercurrent Growth: The Boltzmann equation confirms the TDGL prediction: the supercurrent grows exponentially during the cooling phase, proportional to the recovery of the superfluid density.
2. Role of Impurities: Contrary to the belief that impurities only cause resistance, the paper proves that momentum-relaxing scattering is essential for supercurrent growth. In a perfectly clean system (without momentum relaxation), the paramagnetic current would not decay, and the net current would not amplify.
3. Transient Meissner Effect: The growth of the supercurrent implies a transient enhancement of the Meissner effect. As the supercurrent grows, it expels magnetic flux more vigorously than in equilibrium, a phenomenon observable in ultrafast experiments.
4. Reflectivity Exceeding Unity:
   • The system acts as a gain medium for electromagnetic waves during the recovery phase.
   • Numerical simulations of optical reflectivity show that for frequencies below the gap, the reflectivity can exceed unity ($R > 1$).
   • Physical Interpretation: A probe pulse penetrates the surface where the plasma frequency is initially low. As the pulse propagates, the superfluid density (and thus the plasma frequency) increases, adiabatically amplifying the wave. Since the wave cannot penetrate deeper (where the plasma frequency is even higher) and the energy cannot be dissipated (due to the gain nature), it is reflected back with amplified energy.

5. Significance

• Theoretical Correction: The work corrects a fundamental misunderstanding in nonequilibrium superconductivity: impurities are not merely dissipative but can facilitate current amplification in dynamic regimes.
• Experimental Predictions:
  • It provides a microscopic explanation for recent experimental observations of magnetic field expulsion in optically driven cuprates (e.g., YBCO).
  • It predicts that optical reflectivity > 1 is a signature of supercurrent growth, offering a direct experimental probe for this phenomenon in the terahertz (THz) range.
• New Physics: It highlights the unique non-equilibrium physics where the interplay between order parameter recovery and scattering leads to transient states with properties (like gain) impossible in equilibrium. This opens avenues for studying amplification of collective modes (e.g., superfluid plasmons) and potential applications in ultrafast THz amplification.

In summary, the paper establishes that supercurrent growth is a robust microscopic phenomenon driven by the interplay of gap recovery and momentum-relaxing scattering, leading to observable effects like transient Meissner enhancement and optical gain.