Nonequilibrium phase transition of dissipative fermionic superfluids: Case study of multi-terminal Josephson junctions

This paper investigates the nonequilibrium dynamics of a triad of fermionic superfluids in a multi-terminal Josephson junction under sudden two-body loss, revealing that dissipation induces a two-step or simultaneous nonequilibrium dynamical phase transition characterized by the vanishing of dc Josephson currents depending on the inter-superfluid tunneling strength.

The Gist

Imagine a trio of dancers, each representing a superfluid (a special state of matter where atoms flow without any friction). These three dancers are holding hands, connected by invisible elastic bands called Josephson junctions. Because they are "superfluids," they move in perfect unison, like a single, synchronized team.

In this paper, the researchers ask: What happens if we suddenly start tripping one of the dancers?

The Setup: The Triad and the Trip

1. The Trio: We have three superfluids (let's call them Dancer 1, Dancer 2, and Dancer 3). They are all connected to each other.
2. The Trip (Dissipation): Suddenly, we introduce "friction" or "loss" to Dancer 2. In the real world, this is like shining a laser that causes pairs of atoms in Dancer 2 to disappear. This is called "two-body loss."
3. The Goal: The scientists want to see how this sudden trip affects the rhythm and movement of the whole group. Do they keep dancing together? Do they fall apart? Do they change their steps?

The Discovery: The "Phase Rotation"

When Dancer 2 starts losing atoms, something strange happens. The "rhythm" (or phase) of the superfluids starts to rotate. It's as if the dancers suddenly decide to spin around a different axis.

Because of this spinning, the invisible elastic bands between them start to pull and push, creating electric currents (specifically, Josephson currents) flowing between the dancers. The researchers found that depending on how strong the connection is between Dancer 1 and Dancer 3 (the ones not being tripped), the group reacts in two very different ways.

The Two Scenarios

Scenario A: The "Weak Link" (Two-Step Collapse)

Imagine Dancer 1 and Dancer 3 are holding hands loosely. They aren't very connected to each other.

• Step 1 (The First Trip): As the friction on Dancer 2 increases, the group doesn't fall apart all at once. First, the connection between Dancer 1 and Dancer 2 breaks. The current between them stops flowing. But, Dancer 2 and Dancer 3 are still dancing together, and a current flows between them.
• Step 2 (The Second Trip): If we increase the friction even more, the last remaining connection (between Dancer 2 and Dancer 3) also breaks. Now, all currents stop. The group has completely lost its synchronized flow.

The Analogy: It's like a relay race where the first runner drops the baton, but the second runner keeps running. Then, the second runner also drops the baton, and the race is over. It happens in two distinct stages.

Scenario B: The "Strong Link" (The Sudden Snap)

Now, imagine Dancer 1 and Dancer 3 are holding hands very tightly. They are practically fused into a single unit.

• The Reaction: When we start tripping Dancer 2, the tight bond between Dancer 1 and Dancer 3 keeps them moving as one big block.
• The Result: As friction increases, the whole group doesn't fall apart in steps. Instead, at a specific point of friction, all the currents stop flowing simultaneously. The entire synchronized dance collapses in one instant.

The Analogy: It's like a tightrope walker holding a long pole. If the pole is too heavy (strong connection), the walker doesn't wobble side-to-side first; they just tip over all at once when the balance is lost.

Why Does This Matter?

This isn't just about dancing atoms. This research helps us understand Non-Equilibrium Phase Transitions.

In physics, we usually study things that are calm and stable (equilibrium). But the real world is messy and changing (non-equilibrium). This paper shows that when you push a quantum system hard (by making atoms disappear), it doesn't just fade away smoothly. It can undergo sudden, dramatic shifts in behavior, switching from "flowing" to "stopped" in specific, predictable ways.

The "Secret Sauce": The Simplified Model

To prove this wasn't just a computer glitch, the authors created a simplified mathematical model. They imagined the dancers as a single particle rolling on a washboard potential (like a corrugated metal roof).

• Low Friction: The particle gets stuck in the valleys of the washboard (it can't roll away). This represents the currents flowing.
• Medium Friction: The particle gets stuck in one valley but rolls out of another. This represents the "two-step" transition.
• High Friction: The particle rolls freely everywhere. This represents the currents stopping completely.

The Takeaway

This paper reveals that dissipation (loss) isn't just a destructive force; it's a control knob. By tuning how much "friction" we apply and how strongly the parts of the system are connected, we can force a quantum system to switch between different states of motion.

This is a huge step forward for ultracold atom experiments. Scientists can now use these "lossy" systems to build new types of quantum switches or sensors, knowing exactly how the system will react when things start to go wrong. It turns a potential disaster (atoms disappearing) into a tool for engineering new quantum behaviors.

Technical Summary

1. Problem Statement

The paper investigates the nonequilibrium dynamics of a system consisting of three fermionic superfluids (a triad) connected via Josephson junctions. The specific scenario involves the sudden switch-on of two-body loss (dissipation) in one of the three superfluids (System 2).

While equilibrium properties of superfluids and single-junction dissipative dynamics are well-studied, the interplay between dissipation and multiple coherence in multi-terminal Josephson junctions remains an open question. The authors aim to understand how dissipation affects the superfluid order parameters, phase synchronization, and Josephson currents in this complex geometry, specifically looking for Nonequilibrium Dynamical Phase Transitions (NDPTs) characterized by the vanishing of DC Josephson currents.

2. Methodology

The authors employ a combination of analytical theory and numerical simulation:

• Theoretical Framework:

  • Dissipative BCS Theory: They formulate a dissipative Bardeen-Cooper-Schrieffer (BCS) theory for the Lindblad equation. The system is described by a pairing Hamiltonian with Cooper-pair tunneling between the three systems.
  • Lindblad Equation: The time evolution of the density matrix $\rho$ is governed by $\dot{\rho} = -i[H_{sys}, \rho] - \frac{\gamma}{2}\{L^\dagger L, \rho\} + \gamma L \rho L^\dagger$, where $L$ represents the two-body loss operator acting on System 2.
  • Anderson's Pseudospin Representation: To handle the dynamics, they map the problem to the Bloch equation of pseudospins. This allows them to derive equations of motion for the superfluid order parameters ($\Delta_\nu$) and the Josephson currents ($J_{\nu\mu}$).
  • Simplified Model: For analytical insight, they derive a simplified equation of motion for the phase differences, mapping the system to a particle moving in a "washboard potential."

• Numerical Simulation:

  • They solve the coupled Bloch equations numerically using a 4th-order Runge-Kutta method.
  • The system is initialized in the BCS ground state, followed by a sudden quench (switch-on) of the loss rate $\gamma$ and tunneling amplitudes ($V_{12}, V_{23}, V_{31}$).
  • They analyze the time evolution of order parameters, phase differences, and DC/AC components of Josephson currents.

3. Key Contributions

1. Formulation of Dissipative Multi-Terminal BCS Theory: The authors successfully extend the dissipative BCS theory to a multi-terminal Josephson junction setup, deriving analytical expressions for the time-evolution of order parameters and Josephson currents under two-body loss.
2. Discovery of Phase Rotation Mechanism: They demonstrate that dissipation induces a phase rotation of the superfluid order parameter. This rotation is the fundamental driver for the emergence of DC Josephson currents in a system that would otherwise exhibit only AC oscillations or decay.
3. Identification of Two Distinct NDPT Scenarios: The paper reveals that the nature of the nonequilibrium dynamical phase transition depends critically on the tunneling amplitude ($V_{31}$) between the two loss-free superfluids (Systems 1 and 3).

4. Key Results

A. Weak Tunneling Case ($V_{31}$ is small)

When the coupling between the two loss-free superfluids is weak, the system exhibits a two-step NDPT as the dissipation strength $\gamma$ increases:

1. Step 1 (Weak to Medium Dissipation): Initially, all three junctions ($J_{12}, J_{23}, J_{31}$) exhibit finite DC currents. As dissipation increases past a critical value $\gamma_{c1}$, the DC currents for $J_{12}$ and $J_{31}$ vanish, while $J_{23}$ (the junction involving the lossy system) remains finite.
   • Physical Interpretation: The particle number in System 1 decays very slowly, reminiscent of the continuous quantum Zeno effect, as the flow is suppressed by the vanishing DC currents.
2. Step 2 (Strong Dissipation): As $\gamma$ increases further past a second critical value $\gamma_{c2}$, the remaining DC current ($J_{23}$) also vanishes. All DC currents disappear, and the system enters a state where only AC oscillations persist.

B. Strong Tunneling Case ($V_{31}$ is large)

When the coupling between Systems 1 and 3 is strong:

• Systems 1 and 3 effectively merge into a single coherent superfluid entity.
• The system behaves like two superfluids connected by a junction.
• Single-Step NDPT: As dissipation increases, all DC Josephson currents vanish simultaneously at a single critical dissipation strength. The two-step transition observed in the weak coupling case disappears.

C. Analytical Confirmation

Using the simplified washboard potential model, the authors analytically derive the critical dissipation strengths ($\gamma_{c1} \propto V_{12}$ and $\gamma_{c2} \propto V_{23}$). They show that the transitions correspond to the particle in the potential escaping confinement in specific directions (phase differences), confirming the numerical findings.

5. Significance

• Control of Nonequilibrium Phases: The study demonstrates that the interplay between dissipation and multi-terminal coherence can be tuned to induce distinct dynamical phases. This offers a new mechanism for controlling superfluid transport in open quantum systems.
• Experimental Relevance: The proposed setup is realizable with current ultracold atom technologies (e.g., using $^6$Li atoms and photoassociation techniques to induce two-body loss). The observation of such NDPTs provides a direct probe of the relative phase modes (Leggett modes) and macroscopic quantum coherence in degenerate Fermi gases.
• Fundamental Physics: The work highlights the role of the quantum Zeno effect in suppressing particle transport in dissipative superfluids and clarifies how non-Hermitian dynamics (loss) can induce phase transitions that are distinct from equilibrium phase transitions.
• Multi-terminal Complexity: It underscores that multi-terminal Josephson junctions exhibit richer physics than simple two-terminal junctions, where the connectivity between loss-free components fundamentally alters the system's response to dissipation.

In conclusion, the paper establishes a theoretical foundation for understanding how dissipation drives dynamical phase transitions in multi-terminal fermionic superfluids, revealing a tunable transition between single-step and two-step NDPTs governed by inter-terminal coupling.