Hybrid light-matter excitations and spontaneous time-reversal symmetry breaking in two-dimensional Josephson Junctions

This paper investigates the inductive coupling between a quantum LC resonator and a graphene-based Josephson junction, revealing that the system's current-phase relation can exhibit spontaneous time-reversal symmetry breaking and predicting the low-energy spectrum of hybridized light-matter excitations.

The Gist

Imagine you have a tiny, super-fast highway for electrons, made of a material called graphene (which is just a single layer of carbon atoms, like chicken wire). Now, imagine you connect the two ends of this highway with a superconducting bridge. This setup is called a Josephson Junction.

Normally, electrons flow across this bridge in a very predictable, rhythmic way, like a pendulum swinging back and forth. The paper you shared investigates what happens when you put this bridge inside a special "cage" made of a superconducting loop, and then connect that loop to a tiny, vibrating quantum drum (a resonator).

Here is the story of what the scientists discovered, broken down into simple concepts:

1. The Setup: A Dance Between Light and Matter

Think of the Josephson Junction (the graphene bridge) as a dancer (Matter). Think of the LC Resonator (the quantum drum) as a musician playing a specific note (Light).

Usually, the dancer and the musician are in separate rooms. But in this experiment, they are in the same room, and they are holding hands. The "hand-holding" is called inductive coupling. When the musician plays a note, the dancer feels it and changes their steps. When the dancer moves, it changes the rhythm of the music.

The scientists wanted to see: If we make them hold hands tightly, does the dancer start doing something completely unexpected?

2. The Surprise: Breaking the Rules of Symmetry

In the world of physics, there is a rule called Time-Reversal Symmetry. Imagine watching a movie of a pendulum swinging. If you play the movie backward, it looks exactly the same. The physics works the same way forward and backward in time.

For a long time, scientists thought that if you had a simple, clean graphene bridge, the current flowing through it would always respect this rule. If you reversed time, the current would just flow the other way, but the system would look balanced.

The Discovery:
The scientists found that when the graphene bridge is connected to the quantum drum, and they are tuned just right, the system spontaneously breaks this symmetry.

• The Analogy: Imagine a perfectly balanced seesaw. If you sit on one side, it goes down; if you sit on the other, it goes up. It's symmetric.
• The Twist: In this experiment, the "seesaw" suddenly decides to tilt to the left all by itself, even though no one pushed it. It creates a "super-current" flowing in one direction even when the system is supposed to be perfectly balanced (at a specific angle called $\pi$).

This is like a clock that, instead of ticking back and forth, suddenly decides to spin only clockwise, breaking the symmetry of time. This happens because the interaction between the "light" (the drum) and the "matter" (the electrons) creates a new, stable state that prefers one direction over the other.

3. The "Ghost" Currents and the Tunneling Effect

Why does this happen? The graphene bridge is special. It has "highways" where electrons can pass through with 100% efficiency (called perfect transmission). It's like a tunnel where no cars ever get stuck.

When the quantum drum starts vibrating, it interacts with these perfect highways. The scientists found that the "noise" from the drum pushes the electrons into a state where they form pairs that move in a specific direction.

• The Analogy: Imagine a crowd of people trying to walk through a door. Usually, they walk back and forth equally. But if a loudspeaker (the resonator) starts playing a specific beat, the crowd suddenly organizes itself into a conga line moving only to the right. The beat didn't push them; it just gave them the rhythm to break the balance.

4. Temperature: The "Hot Coffee" Problem

The scientists also asked: What if we make the system hot?

In physics, heat is like shaking the table. If you shake the table too much, the delicate dance between the light and the matter gets ruined. The electrons get jumbled up, and the special "one-way" current disappears.

They calculated a Critical Temperature. Below this temperature, the special "tilted seesaw" state exists. Above it, the system goes back to being a normal, balanced seesaw.

• The Analogy: Think of a house of cards. It can stand perfectly still (the special state). But if you blow on it (heat), it collapses. The paper shows exactly how hard you can blow before the house of cards falls.

5. Why Should We Care?

This isn't just a cool trick; it has big implications for the future of technology:

• New Quantum Computers: Quantum computers need to be very stable. This "spontaneous symmetry breaking" creates a new type of state that could be used to build qubits (the bits of a quantum computer) that are harder to mess up by noise.
• Super-Sensitive Sensors: Because the system is so sensitive to the connection between light and matter, it could be used to build detectors that can hear the faintest whispers of the universe (like detecting dark matter or gravitational waves).
• Understanding the Universe: It's a small-scale model of how complex systems (like magnets or even the early universe) can spontaneously choose a direction or state, a concept called "phase transition."

Summary

In simple terms: The paper describes a tiny experiment where a graphene bridge and a vibrating quantum drum are glued together. When they interact, they spontaneously decide to break the laws of "time symmetry," creating a one-way super-current that shouldn't exist. This happens only when it's very cold and the connection is just right. It's like finding a way to make a pendulum swing in only one direction forever, opening the door to new types of quantum machines.

Technical Summary

Here is a detailed technical summary of the paper "Hybrid light-matter excitations and spontaneous time-reversal symmetry breaking in two-dimensional Josephson Junctions" by Varrica et al.

1. Problem Statement

The paper investigates the quantum mechanical interaction between a superconducting loop containing a two-dimensional (2D) material-based Josephson Junction (JJ) and a superconducting LC resonator. Specifically, the authors focus on Graphene Josephson Junctions (GJJs) in the "wide and short" limit (where the junction width $W$ is much larger than its length $L$, and $L$ is much smaller than the superconducting coherence length $\xi$).

The central problem addressed is how the inductive coupling between the JJ and a quantum LC resonator modifies the equilibrium properties of the system. While previous studies often treated this interaction as a weak perturbation (valid for few-channel junctions), the authors argue that for wide 2D junctions hosting a continuum of Andreev Bound States (ABSs), the coupling can be strong enough to induce non-perturbative effects. The primary goal is to determine if this light-matter interaction can induce spontaneous time-reversal symmetry breaking (TRB), manifested as a finite supercurrent at a phase difference of $\phi = \pi$, and to characterize the resulting hybridized light-matter excitation spectrum.

2. Methodology

The authors employ a mean-field formalism combined with linear response theory to analyze the system.

• Model Hamiltonian:

  • The system consists of a quantum LC resonator (bosonic mode $\hat{a}$) inductively coupled to a superconducting loop with a GJJ (fermionic ABSs).
  • The GJJ is modeled using the Bogoliubov-de Gennes (BdG) approach within the Dirac-BdG framework for graphene.
  • In the wide-short limit ($W/L \gg 1$), the ABSs form a continuous spectrum characterized by transmission probabilities $\tau(k)$.
  • The total Hamiltonian includes the resonator energy, the Andreev Hamiltonian, and interaction terms derived from the flux fluctuations of the resonator. The interaction is non-quadratic, involving terms proportional to $(\hat{a} + \hat{a}^\dagger)$.

• Mean-Field Approximation:

  • Due to the complexity of the many-body Hamiltonian, a mean-field approach is used to decouple light and matter degrees of freedom.
  • The interaction is replaced by effective mean fields: a photonic shift $\alpha$ (related to the resonator coordinate) and fermionic currents/inductances ($P$ and $D$).
  • A self-consistent procedure is established to solve for $\alpha$, $P$, and $D$ at equilibrium. The ground state is treated as a product of a fermionic state (filled quasiparticle sea) and a bosonic coherent state.

• Linear Response & Fluctuations:

  • To study the excitation spectrum, the authors analyze Gaussian fluctuations around the mean-field solution.
  • They calculate the current-current susceptibility and the linear response function $\tilde{\Pi}(\Omega)$ to determine the poles of the system, which correspond to the hybridized light-matter eigenenergies.

3. Key Contributions

1. Non-Perturbative Treatment: The paper moves beyond perturbative treatments used for few-channel junctions, applying a mean-field theory suitable for the continuum of states in wide 2D junctions.
2. Spontaneous TRB Mechanism: The authors demonstrate that the inductive coupling can spontaneously break time-reversal symmetry in a system that is otherwise time-reversal invariant. This results in a finite supercurrent at $\phi = \pi$, a phenomenon typically associated with magnetic fields or ferromagnetic links.
3. Role of Transparency: The study identifies that the instability is driven by highly transparent modes (specifically those with $\tau \approx 1$, such as Klein tunneling and stationary waves). The density of states (DOS) exhibits a square-root divergence at $\tau=1$, which is crucial for the instability.
4. Analytical Critical Conditions: The authors derive analytical expressions for the critical coupling constant ($g_c$) and critical temperature ($T_c$) required to trigger the TRB phase. They show that $g_c$ is infinitesimally small for graphene due to the presence of perfect transmission channels.

4. Key Results

• Current-Phase Relation (CPR) and TRB:

  • In the absence of coupling ($g=0$), the CPR is continuous, and the supercurrent at $\phi=\pi$ is zero (as expected by symmetry).
  • When coupling is turned on ($g > 0$), the CPR develops a discontinuity at $\phi = \pi$. The system spontaneously selects one of two degenerate ground states with finite supercurrents of opposite signs ($I(\pi^-) = -I(\pi^+)$).
  • This is interpreted as a photon condensation instability analogous to magnetostatic instability in Van Vleck paramagnets.

• Dependence on Fermi Level and Temperature:

  • Fermi Level ($\mu_0$): The magnitude of the spontaneous supercurrent oscillates with $\mu_0$. Maxima occur when $\mu_0$ aligns with conditions that maximize the number of perfectly transmitting channels (stationary wave conditions, $k \approx n\pi$).
  • Temperature: Thermal fluctuations suppress the supercurrent. The authors derive a critical temperature $T_c$ below which the TRB phase exists:
    $$k_B T_c \approx \Delta_0 \exp\left[ -\frac{\hbar\omega_r}{g^2 n_v \rho_0 \Delta_0} \right]$$
    where $\rho_0$ is the weight of the transparent modes in the DOS. The instability is fragile at finite temperatures but robust if the Fermi level is tuned to maximize transparent modes.

• Hybrid Excitation Spectrum:

  • The coupling leads to the formation of hybrid light-matter excitations (polaritons).
  • The spectrum shows vacuum Rabi splitting behavior. As coupling $g$ increases, the lower-energy hybrid mode shifts upward and broadens, while the higher-energy mode remains distinct.
  • The minigap (energy gap for quasiparticle excitations) remains finite at $\phi=\pi$ in the TRB phase, unlike the isolated junction where it closes.

• Phase Diagram:

  • A phase diagram in the $(g, T)$ plane is constructed, showing a transition from a normal phase (no supercurrent at $\pi$) to a TRB phase (finite supercurrent). The boundary is defined by the critical temperature $T_c$.

5. Significance

• Fundamental Physics: The work provides a theoretical framework for understanding spontaneous symmetry breaking in hybrid light-matter systems without external magnetic fields. It connects the physics of Josephson junctions with concepts from quantum optics (photon condensation) and magnetism.
• Device Applications:
  • Qubits and Sensors: The ability to tune the symmetry breaking via gate voltage (Fermi level) and temperature offers new pathways for designing noise-protected qubits or highly sensitive microwave bolometers and parametric amplifiers.
  • Quantum Simulation: The system serves as a tunable platform to study phase transitions and critical phenomena in hybrid quantum circuits.
• Graphene Specifics: The results highlight the unique role of graphene's Dirac nature (Klein tunneling) in facilitating these instabilities due to the abundance of perfectly transmitting channels, suggesting that 2D materials are superior candidates for observing these effects compared to traditional nanowires.

In summary, the paper establishes that inductive coupling in wide 2D Josephson junctions can drive a spontaneous transition to a time-reversal symmetry broken state, characterized by a finite supercurrent at $\pi$ and distinct hybrid excitation spectra, with the effect being highly tunable by gate voltage and temperature.