Josephson spectroscopy in a circular atomic tunnel junction with acceleration-induced symmetry breaking

This paper demonstrates that linear acceleration breaks the axial symmetry of a dual-ring Bose-Einstein condensate junction, activating multimode Josephson dynamics and enabling a controllable spectroscopic protocol to probe the angular structure and eigenfrequencies of collective Bogoliubov modes.

The Gist

Imagine you have two perfectly round, hula-hoop-shaped tracks floating in space. Inside these tracks, you have a special kind of super-cold gas called a Bose-Einstein Condensate (BEC). Think of this gas not as individual particles, but as a single, giant "super-fluid" wave that flows without any friction.

Now, imagine these two hoops are sitting side-by-side, separated by a tiny, invisible wall. Because they are quantum objects, the gas can "tunnel" through this wall, hopping back and forth between the inner hoop and the outer hoop. This hopping creates a rhythmic back-and-forth flow, much like water sloshing between two connected buckets. In physics, this is called the Josephson effect.

The Experiment: Pushing the System

In a normal, perfectly balanced setup, this sloshing happens at one single, steady beat. It's like a metronome ticking perfectly in time.

However, the researchers in this paper decided to shake things up. They applied a gentle, constant "push" (acceleration) to the entire system, like tilting the table the hoops are sitting on.

What happened?
When they pushed the system, the perfect, single beat broke apart. Instead of one steady rhythm, the gas started sloshing in a complex, multi-layered pattern. It was as if the metronome suddenly started playing a chord of different notes at once, creating a "beating" sound where the volume swells and fades.

The Secret: Symmetry Breaking

Why did this happen? The paper explains it using the concept of symmetry.

• Before the push: The system was perfectly symmetrical. It looked the same no matter which way you spun it. Because of this perfect balance, only one specific type of movement (the main Josephson slosh) was allowed to show up in the population count. All other possible movements were "silent" or hidden because they canceled each other out.
• After the push: The push broke the symmetry. It's like taking a perfectly round balloon and squishing it slightly on one side. Now, the rules change. The "hidden" movements, which were previously silent, are now allowed to speak up. They acquire a "voice" and start contributing to the sloshing.

The researchers found that only movements that matched the direction of the push (symmetric) became active, while those that fought against the push (antisymmetric) stayed silent.

The New Tool: Josephson Spectroscopy

The paper doesn't just observe this; it proposes a new way to "listen" to the gas to understand its internal structure. They call this Josephson Spectroscopy.

Imagine you are trying to figure out the shape of a hidden object inside a dark room. You can't see it, but you can tap on the walls.

1. Frequency Scans: The researchers suggest tapping the gas at different speeds (frequencies). When they tap at a speed that matches one of the gas's natural internal vibrations (called Bogoliubov modes), the gas responds strongly, like a swing being pushed at just the right time. This tells them the "pitch" or frequency of that specific vibration.
2. Angular Scans: Once they find a vibration, they can move their "tap" to different spots around the ring. By seeing how the response changes as they move the tap, they can map out the shape of that vibration. It's like tracing the outline of the hidden object by feeling where the wall is thickest.

The Bottom Line

The paper demonstrates that by simply accelerating a dual-ring system of super-cold gas, you can turn a simple, single-note system into a complex, multi-note orchestra.

They proved that:

1. Acceleration breaks the symmetry, "waking up" hidden vibrations that were previously silent.
2. These vibrations can be detected by listening to how the gas sloshes between the rings.
3. By carefully tuning their "tap" (the driving force), they can not only identify which vibrations are active but also map out their physical shapes.

In short, they turned a simple quantum tunneling experiment into a sophisticated tool for "listening" to the hidden internal music of a quantum fluid, using a gentle push to reveal its secrets.

Technical Summary

Technical Summary: Josephson Spectroscopy in a Circular Atomic Tunnel Junction with Acceleration-Induced Symmetry Breaking

Problem Statement
The paper investigates the dynamics of a long atomic Bose-Josephson junction (BJJ) formed by two tunnel-coupled, coplanar Bose-Einstein condensate (BEC) rings. While the Josephson effect in simple geometries (e.g., double-well potentials) is well-described by a two-mode model yielding a single plasma oscillation frequency, extended geometries like ring-shaped traps introduce additional collective degrees of freedom. The authors address the specific problem of how external linear acceleration, which breaks the axial symmetry of the trap, modifies the Josephson population-imbalance dynamics. Specifically, they seek to understand the transition from a single-mode Josephson oscillation to a multimode response and to develop a protocol for spectroscopically probing the underlying collective Bogoliubov modes in this symmetry-broken regime.

Methodology
The study employs a combination of numerical simulations and analytical theory:

1. System Model: The system is modeled as two coaxial toroidal condensates of $^{87}\text{Rb}$ atoms separated by a repulsive potential barrier. The dynamics are described by the two-dimensional Gross-Pitaevskii equation (GPE) in the $(r, \phi)$ plane, assuming the $z$-dimension is frozen.
2. Symmetry Breaking: An in-plane linear acceleration is introduced as an inertial potential $V_a = M(\mathbf{a} \cdot \mathbf{r})$, explicitly breaking the axial rotational symmetry while preserving reflection symmetry about the acceleration axis.
3. Numerical Simulations:
   • Free Evolution: A small quench potential creates an initial population imbalance, after which the system evolves freely. The time evolution of the population imbalance $Z(t)$ is analyzed via Fourier transforms to identify oscillation frequencies.
   • Driven Response: A weak, localized, time-periodic perturbation is applied to the system. The response is analyzed using a dissipative time-dependent GPE to stabilize the oscillations and extract amplitude and phase shifts.
4. Analytical Framework:
   • Bogoliubov-de Gennes (BdG) Analysis: The BdG equations are solved to determine the spectrum of elementary excitations (modes) and their spatial structures ($u_n, v_n$) under varying acceleration.
   • Mode Activity Definition: A "mode activity" metric $D_n = \langle \psi_0 | \hat{Z} | u_n + v_n \rangle$ is defined to quantify the overlap between a specific Bogoliubov mode and the population-imbalance operator.
   • Dissipative Response Theory: A linearized, dissipative BdG theory is derived to provide analytical expressions for the frequency response (amplitude and phase) of the population imbalance, incorporating mode activity, drive-mode overlap, and damping.

Key Contributions and Results

• Acceleration-Induced Multimode Dynamics:
  The authors demonstrate that in the axially symmetric case ($a=0$), the population imbalance exhibits a single dominant Josephson plasma frequency. When linear acceleration is applied ($a > 0$), the single oscillation transforms into a multimode response characterized by beating patterns in the time domain and a multiplet structure in the frequency spectrum.

• Mechanism of Mode Activation:
  The paper clarifies that this spectral multiplet is not a simple splitting of the primary Josephson mode. Instead, it results from the activation of specific Bogoliubov modes that acquire a finite overlap with the population-imbalance operator due to symmetry breaking.

  • In the symmetric case, modes with certain angular symmetries (e.g., in-phase oscillations or out-of-phase oscillations with non-zero angular momentum) have zero net overlap with the imbalance operator due to cancellation effects.
  • Under acceleration, the reduction of symmetry from rotational to reflection symmetry lifts degeneracies and distorts mode wavefunctions. Consequently, S-type modes (symmetric with respect to the acceleration direction) become active in the imbalance dynamics, while A-type modes (antisymmetric) remain essentially inactive.

• Mode-Resolved Josephson Spectroscopy Protocol:
  The authors propose and validate a protocol to probe the spatial structure of these active modes:

  1. Frequency Scans: Tuning the drive frequency reveals resonant peaks in the population-imbalance amplitude and characteristic phase shifts at the eigenfrequencies of the active Bogoliubov modes.
  2. Angular Scans: By fixing the drive frequency near a resonance and scanning the angular position of the localized perturbation, the response amplitude maps the angular structure of the corresponding mode's density perturbation.
  • The analytical response functions derived from the dissipative BdG theory show quantitative agreement with full GPE simulations in the linear regime, confirming that the resonance lineshapes are controlled by mode activity, drive-mode overlap, and damping.

• Quantitative Agreement:
  The study establishes that the observed spectral components in the Josephson dynamics correspond directly to the eigenfrequencies of the active S-type Bogoliubov modes. The "activity" $D_n$ serves as a predictor for which modes will appear in the Josephson spectrum.

Significance and Claims
The paper claims to demonstrate that accelerated dual-ring condensates provide a controllable platform for symmetry-selected Josephson dynamics. The primary significance lies in:

1. Fundamental Understanding: Providing a clear physical mechanism for how symmetry breaking (via acceleration) redistributes oscillator strength among collective modes, transforming a single-mode Josephson system into a multimode one.
2. Spectroscopic Tool: Establishing Josephson spectroscopy as a sensitive probe for detecting and characterizing collective excitations in ring-shaped condensates. The proposed protocol allows for the identification of active modes and the mapping of their spatial structure without requiring direct imaging of the density perturbations.
3. Inertial Sensing Potential: The strong dependence of the active-mode spectrum on acceleration suggests a route toward symmetry-sensitive inertial probing, extending the utility of atomtronic circuits for sensing applications.

The authors remain modest in their claims, noting that the protocol relies on linear response theory and that the resolution of angular scans is limited by the drive profile and contributions from nearby modes. They do not propose specific experimental implementations beyond the theoretical framework but highlight the compatibility of their findings with existing experimental capabilities in ring-condensate systems.