Josephson effects in an interaction-asymmetric junction across the BCS-BEC crossover

This paper theoretically investigates the DC and AC Josephson effects in interaction-asymmetric ultracold Fermi gas junctions across the BCS-BEC crossover, revealing a competition between pair spectral weight and chemical potential that leads to a distinct interaction-biased Riedel peak in the tunneling current.

The Gist

Imagine two crowds of people standing on opposite sides of a narrow bridge. In the world of quantum physics, these "people" are atoms, and the "bridge" is a tiny gap where they can jump from one side to the other. This setup is called a Josephson Junction.

Usually, these atoms like to pair up and dance together in perfect sync. This synchronized dancing is what allows them to flow across the bridge without any resistance, creating a Josephson current.

This paper explores what happens when the "dance rules" are different on the two sides of the bridge.

The Two Dance Styles: BCS and BEC

In the world of ultracold atoms, there are two main ways these pairs can behave, depending on how strongly they hold hands:

1. The BCS Style (The Loose Waltz): Imagine two dancers holding hands loosely, gliding far apart but still moving in sync. They are like a couple in a large ballroom, barely touching but perfectly coordinated. This happens when the atoms interact weakly.
2. The BEC Style (The Tight Embrace): Now imagine the dancers hugging so tightly they become a single unit, almost like a new creature. They are stuck together, moving as one solid block. This happens when the atoms interact very strongly.

Between these two extremes lies a Crossover, a smooth transition where the dance changes from a loose waltz to a tight embrace.

The Experiment: A Mismatched Bridge

The researchers in this paper set up a special experiment. They took a Josephson junction (the bridge) and did something unusual:

• On the Left Side, they kept the atoms in the BCS mode (loose waltz).
• On the Right Side, they slowly changed the "dance rules" from BCS (loose) all the way to BEC (tight hug).

They wanted to see: What happens to the flow of atoms when one side is dancing loosely and the other is dancing tightly?

The Big Discoveries

1. The "Sweet Spot" (The DC Current)

First, they looked at what happens when both sides are dancing the same way (both loose, both tight, or both in the middle).

• The Result: They found that the flow of atoms isn't strongest when the atoms are super-tight (BEC) or super-loose (BCS). Instead, the flow peaks right in the middle (the "Unitary Limit").
• The Analogy: Think of it like a relay race. If the runners are too loose, they drop the baton. If they are too tight, they can't move fast enough. The fastest time happens when they are just right—cooperative but not stuck. The researchers found that the "perfect balance" between the atoms' energy and their pairing strength creates the strongest current.

2. The "Riedel Peak" (The AC Current)

Next, they looked at the "mismatched" bridge (Left side loose, Right side changing). They applied a tiny voltage difference to make the atoms oscillate back and forth (like a swing).

• The Result: As they tuned the Right side from loose to tight, the swinging motion suddenly got huge at a specific point. They call this a Riedel Peak.
• The Analogy: Imagine pushing a child on a swing. If you push at the wrong time, nothing happens. But if you push exactly when the child is at the bottom of the arc (the perfect moment), the swing goes incredibly high.
  • In this experiment, the "push" is the voltage difference.
  • The "perfect moment" happens when the energy cost to jump from the loose side matches the energy of the tight side.
  • When these energies align perfectly, the atoms "resonate," and the current spikes dramatically.

Why Does This Matter?

This isn't just about cold atoms in a lab.

• Universal Physics: It shows that these quantum dance rules apply everywhere, from superconductors in your electronics to the dense matter inside neutron stars.
• New Controls: It proves that by simply changing how strongly atoms interact (without changing temperature or pressure), we can control the flow of quantum information. This is like having a new "volume knob" for quantum currents.
• The "Riedel Peak" Discovery: Finding this peak in a gas of atoms (instead of just in solid metal) is a big deal. It means we can use these cold gases to study complex quantum phenomena that are usually hard to see.

In a Nutshell

The researchers built a quantum bridge where one side was a "loose dance" and the other was a "tight hug." They discovered that the flow of atoms is strongest when the dance styles are balanced, and that if you tune the dance just right, you can create a massive surge of current (the Riedel Peak) simply by matching the energy of the two sides. It's a beautiful demonstration of how nature finds harmony even when the rules on either side of the divide are different.

Technical Summary

1. Problem Statement

The paper investigates the Josephson effect in ultracold Fermi gases, specifically focusing on a Josephson junction where the two reservoirs are independently tuned to different interaction regimes across the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein Condensation (BEC) crossover.

While previous studies have examined symmetric junctions (where both sides are tuned synchronously) or standard thermodynamic biases (chemical potential or temperature differences), this work addresses a novel scenario: an interaction-asymmetric junction. In this setup, the interaction strength (scattering length) in one reservoir is fixed (e.g., in the BCS limit), while the other is tuned from the BCS to the BEC regime. The authors aim to understand how this interaction asymmetry, which is distinct from conventional thermodynamic biases, affects DC and AC Josephson currents, particularly regarding the competition between pair spectral weight and chemical potential shifts.

2. Methodology

The authors employ a theoretical framework based on nonequilibrium quantum field theory, specifically combining:

• The Schwinger-Keldysh (Nonequilibrium Green's Function) formalism: This allows for the treatment of many-body aspects of tunneling transport in a non-equilibrium steady state.
• The Tunnel Hamiltonian Formalism: The system is modeled as two reservoirs (Left and Right) connected by a tunneling barrier. The total Hamiltonian includes reservoir terms ($\hat{H}_{L,R}$) and a tunneling term ($\hat{H}_T$).
• Nambu Representation: Green's functions are formulated in the Nambu space to account for particle-hole mixing and superfluid order parameters.
• Perturbative Expansion: The tunneling current is calculated to the leading order in the tunneling amplitude ($T_{k,k'}$).

Key Theoretical Steps:

1. Current Operator Derivation: The tunneling current operator is derived from the Heisenberg equation of motion for the particle number operator.
2. Separation of Currents: The total current is decomposed into a quasiparticle current ($I_q$) and a Josephson current ($I_J$).
   • $I_q$ is time-independent (stationary background).
   • $I_J$ is oscillatory (AC component) or constant (DC component when $\Delta\mu=0$).
3. Asymptotic Analysis: Analytical expressions for the DC current are derived in the deep BCS and deep BEC limits to explain the behavior observed in numerical results.
4. Regularization: A momentum cutoff ($\Lambda$) is introduced in the tunneling coupling to avoid ultraviolet divergences inherent in the contact interaction model.

3. Key Contributions

• Interaction-Asymmetric Junction Model: The paper establishes a theoretical framework for Josephson transport where interaction strengths are independently controlled on either side of the junction, a scenario enabled by modern spatial control of interactions in ultracold gases.
• Distinction from Thermodynamic Bias: The authors clarify that interaction asymmetry is not a standard thermodynamic bias (like $\Delta\mu$ or $\Delta T$) because the interaction operator is not a conserved quantity. However, it acts as a tunable control knob that reshapes the quasiparticle spectrum.
• Identification of the Interaction-Biased Riedel Peak: The study identifies a specific resonance condition in the AC Josephson current arising from the mismatch in interaction regimes, analogous to the Riedel singularity in superconductors but driven by interaction tuning.

4. Key Results

A. DC Josephson Current (Symmetric Junction)

• Setup: Both reservoirs are tuned synchronously from BCS to BEC.
• Observation: The DC critical current exhibits a peak near the unitary limit (strong coupling).
• Mechanism: This peak results from a competition between two factors:
  1. Increasing Pair Spectral Weight: As interaction increases, the pairing gap ($|\Delta|$) grows, enhancing the condensate fraction and tunneling probability.
  2. Decreasing Chemical Potential: As the system moves toward the BEC limit, the chemical potential ($\mu$) decreases (becoming negative), which suppresses the density of states available for tunneling.
• Asymptotic Limits:
  • BCS Limit: The current follows the Ambegaokar-Baratoff formula, scaling linearly with the gap ($I_{DC} \propto |\Delta|$).
  • BEC Limit: The current decreases with increasing interaction strength, dominated by the change in chemical potential rather than the gap.

B. AC Josephson Current (Asymmetric Junction)

• Setup: The Left reservoir is fixed in the BCS limit ($1/a k_F = -1$), while the Right reservoir is tuned from BCS to BEC. A chemical potential bias ($\Delta\mu$) exists due to the different interaction regimes.
• Observation: A distinct peak in the AC Josephson current amplitude is found when the Right reservoir reaches the crossover region ($1/a k_F \approx 0.7 - 0.8$).
• Mechanism (Riedel Peak): This peak corresponds to a resonance condition where the chemical potential bias matches the threshold for quasiparticle excitation across the junction:
  $$\Delta\mu \approx E_{min}^L + E_{min}^R$$
  This is the interaction-biased Riedel peak. It occurs when the energy provided by the bias allows for the resonant creation of quasiparticles, enhancing the tunneling current.
• Separation of Currents: The authors demonstrate that the oscillatory Josephson current ($I_J$) can be unambiguously separated from the quasiparticle current ($I_q$) based on time dependence and noise characteristics (Fano factor). $I_J$ is coherent (low noise), while $I_q$ is incoherent (high noise).

5. Significance

• Fundamental Physics: The work demonstrates that the Riedel singularity, traditionally associated with weak-coupling solid-state superconductors, also emerges in strongly interacting quantum gases when interaction asymmetry is introduced.
• Experimental Relevance: The results provide a theoretical guide for future experiments with ultracold atoms (e.g., $^6$Li) where spatially dependent Feshbach resonances can create such asymmetric junctions.
• Transport Control: It highlights interaction strength as a unique "control knob" for quantum transport, offering a new degree of freedom to manipulate superfluid currents beyond standard chemical potential or temperature gradients.
• Methodological Validation: The study validates the utility of the nonequilibrium Green's function approach for studying transport in the BCS-BEC crossover, complementing the Bogoliubov-de Gennes (BdG) approach by naturally incorporating many-body effects and multi-particle tunneling.

In summary, this paper provides a comprehensive theoretical analysis of how interaction asymmetry modifies Josephson transport, revealing a competition between pairing coherence and chemical potential shifts that leads to a characteristic peak in the critical current and a resonant enhancement in the AC current.