Bound state in the continuum and dynamics via phase modulation in giant-atom waveguide setups

This paper demonstrates that in a two-giant-atom waveguide-QED system, engineering the coupling phase between atoms serves as a powerful tool to control the number and profiles of bound states in the continuum, thereby enabling the precise tailoring of quantum-state evolution and interference dynamics.

The Gist

Imagine a world where light and matter play a game of hide-and-seek, but with a twist: sometimes, the light gets trapped in a "ghost room" that doesn't exist on the map. This paper explores how scientists can build these ghost rooms and control them using a specific type of quantum actor called a "giant atom."

Here is a simple breakdown of what the researchers discovered:

1. The Characters: "Giant Atoms" and the "Highway"

In normal physics, we usually think of atoms as tiny dots. But in this experiment, the researchers use "giant atoms." Think of a giant atom not as a single dot, but as a giant octopus. Instead of touching a surface at one point, this octopus has multiple tentacles (connection points) reaching out to touch a "highway" (a waveguide) at different spots simultaneously.

Because the octopus touches the highway in several places at once, the waves traveling down the highway can bounce back and forth between these tentacles, creating interference patterns—like ripples in a pond meeting and canceling each other out.

2. The Magic Trick: The "Ghost Room" (Bound State in the Continuum)

Usually, if you put energy (like a photon) into this highway, it eventually leaks away or travels off into the distance. However, under the right conditions, the interference from the octopus's multiple tentacles can create a "Bound State in the Continuum" (BIC).

Think of a BIC as a perfectly soundproof room inside a noisy stadium. Even though the stadium is full of noise (the continuous energy spectrum), the sound inside that specific room is trapped and cannot escape. The light gets stuck there, and the atoms holding it don't lose their energy. They are "frozen" in a state of perfect balance.

3. The Remote Control: The "Phase Knob"

The most exciting part of this paper is how the researchers control these ghost rooms. They found that by adjusting a specific setting called the "coupling phase" (let's call it the Phase Knob), they can change the rules of the game.

• Turning the Knob: By twisting this Phase Knob, the researchers can decide:
  • How many ghost rooms exist: They can create zero, one, or even two of these trapped states.
  • Where the light hides: They can change exactly where the light gets trapped between the atoms.
  • How the atoms dance: They can change how the two giant atoms interact with each other.

4. The Dance Moves: What Happens When You Turn the Knob?

The paper shows that changing this Phase Knob leads to three very different "dance routines" for the atoms:

• The Trapped Dancer: When the knob is set to create a ghost room, the atoms get excited and then stop decaying. They hold onto their energy forever (or for a very long time), sharing it between them. It's like two dancers holding a pose that never ends.
• The Vanishing Act: If the knob is set to a different position where no ghost room exists, the atoms quickly lose their energy and return to a calm, resting state. The light escapes down the highway.
• The Endless Swing: In some setups, the atoms don't just hold the energy; they swap it back and forth in a rhythmic, long-lasting swing (Rabi oscillations). It's like a pendulum that never slows down.

5. The Entanglement: A Secret Handshake

When the light is trapped in these ghost rooms, the two giant atoms become deeply connected, a phenomenon called entanglement. The paper shows that by tuning the Phase Knob, the researchers can make the atoms share a "secret handshake" (quantum entanglement) that is much stronger in some settings than others. For example, in one setting, the atoms are almost perfectly synchronized (98% entangled), while in another, they are only partially connected.

Summary

In short, this paper demonstrates that by using "giant atoms" with multiple connection points and turning a specific "Phase Knob," scientists can:

1. Create or destroy invisible traps for light.
2. Control exactly where that light hides.
3. Direct the behavior of the atoms, making them either hold onto energy, lose it, or swing back and forth forever.

The researchers suggest this could be a powerful tool for future quantum computers, allowing us to store information (quantum states) without it leaking away, simply by adjusting the phase of the connections. They note that this setup is realistic and could be built using current superconducting circuit technology.

Technical Summary

Technical Summary: Bound State in the Continuum and Dynamics via Phase Modulation in Giant-Atom Waveguide Setups

Problem Statement
Giant atoms, which couple to waveguides through multiple spatially separated points beyond the dipole approximation, offer a versatile platform for generating interference-induced bound states in the continuum (BICs). While single-giant-atom systems and multi-giant-atom networks are being explored, the specific role of direct coupling between giant atoms—particularly the influence of the coupling phase—on atomic dynamics and BIC formation remains insufficiently understood. The central question addressed is whether BICs can emerge and be controlled in a system where two directly coupled giant atoms are further coupled to a waveguide, and how the coupling phase affects the number, spatial profiles, and dynamical consequences of these states.

Methodology
The authors investigate a theoretical model consisting of two directly coupled two-level giant atoms interacting with a one-dimensional coupled-resonator waveguide (CRW).

• Hamiltonian Formulation: The system is described by a total Hamiltonian $H = H_a + H_c + H_I$, where $H_a$ represents the two directly coupled atoms (with transition frequency $\Omega$, coupling strength $\lambda$, and coupling phase $\phi$), $H_c$ describes the CRW (with resonator frequency $\omega_c$ and hopping strength $\xi$), and $H_I$ characterizes the nonlocal coupling between the atoms and the waveguide at four distinct sites ($n_1, n_2$ for atom 1; $m_1, m_2$ for atom 2).
• Dynamical Analysis: Working within the single-excitation subspace, the authors derive coupled amplitude equations for the atomic excitations and photon modes. By formally solving the photon equations and applying the Markov approximation, they reduce the dynamics to a closed equation governed by an effective non-Hermitian matrix $M$.
• BIC Criterion: Following established criteria (Ref. [24]), the existence of a BIC is identified by the vanishing of the imaginary part of the eigenvalues of the matrix $M$. A zero imaginary part implies a non-decaying component in the system.
• Characterization: The authors analyze the eigenvalues of $M$ as functions of the coupling phase $\phi$ and geometric parameters (atom size $N$ and separation $\Delta$). They further characterize the BICs by calculating the spatial distribution of photons ($|\beta_i|^2$) and performing atomic-state tomography (density matrices and concurrence) to determine entanglement properties.

Key Contributions and Results

1. Phase-Controlled BIC Formation: The study demonstrates that the coupling phase $\phi$ acts as a sensitive control parameter for the number of BICs. Depending on the geometric configuration (specifically the atom size $N$ and separation $\Delta$):

   • In certain configurations (e.g., $N=12, \Delta=4$), the system supports exactly one BIC when $\phi = m\pi$ ($m$ is an integer), while no BICs exist for other phases.
   • In other configurations (e.g., $N=14, \Delta=4$), the system supports two BICs regardless of the coupling phase.
   • In specific regimes (e.g., $N=17, \Delta=5$), BICs can be completely suppressed ($I_1 = I_2 < 0$).

2. Modulation of Atomic Dynamics: The presence or absence of BICs, controlled by $\phi$, leads to distinct dynamical behaviors:

   • Fractional Population Trapping: When a single BIC exists, the initially excited atom relaxes to a finite, non-zero steady-state population, and the unexcited atom acquires a similar long-time population.
   • Complete Decay: In the absence of BICs, both atoms eventually decay to their ground states, though the transient dynamics (oscillations and decay rates) are modulated by the coupling phase.
   • Long-Lived Oscillations: When two BICs are present, the system exhibits persistent, long-lived Rabi-like oscillations between the two atoms, independent of the coupling phase.

3. Spatial Profiles and Entanglement: The coupling phase significantly alters the spatial profiles of the BICs and the resulting atomic entanglement:

   • For a single BIC, the photon distribution is primarily confined between the coupling points of the two atoms. The phase $\phi$ determines the interference pattern; for $\phi = \pi$, the intermediate region between the atoms is completely dark, whereas for $\phi = 0$, a weak excitation exists there.
   • Entanglement: The atomic state tomography reveals that the coupling phase tunes the degree of entanglement. For a single BIC with $\phi = \pi$, the atoms form a highly entangled Bell-like state (concurrence $C \approx 0.98$), whereas for $\phi = 0$, the entanglement is significantly lower ($C \approx 0.49$). In the two-BIC regime, one state yields high entanglement ($C \approx 0.98$), while the other yields moderate entanglement ($C \approx 0.76$).

Significance and Claims
The paper claims to provide a unified physical picture of phase-controlled interference and bound-state formation in giant-atom waveguide-QED systems. The authors highlight that coupling-phase engineering serves as a useful tool for:

• Controlling the number and spatial profiles of BICs.
• Tailoring quantum-state evolution, including the ability to trap fractional populations or induce long-lived oscillations.
• Generating strong atomic entanglement.

The authors assert that their proposal is experimentally relevant to superconducting quantum circuits (specifically transmon qubits coupled to CRWs). They estimate that the dynamical time scales (microseconds) are shorter than typical decoherence times ($T_1 \approx 10$ $\mu$s), suggesting that phase-controlled BIC formation and the associated dynamics should be observable with current technologies. The work is positioned as a step toward developing giant-atom quantum networks and devices for quantum information processing, quantum state storage, and dissipation-free state engineering.