⚛️ quantum physics

Counterdiabatic quasi-Floquet control for the generation of entangled BICs using giant atoms

This paper proposes a robust counterdiabatic quasi-Floquet control protocol using two braided giant atoms to generate high-fidelity entangled bound states in the continuum (BICs) in waveguide quantum electrodynamics, achieving near-perfect fidelity in lossless models and demonstrating resilience against noise and calibration errors.

Original authors: Alexis R. Legón, Pedro Orellana, Ariel Norambuena

Published 2026-09-29

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CC BY 4.0

Original authors: Alexis R. Legón, Pedro Orellana, Ariel Norambuena

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the most valuable resource is entanglement, a state where two particles become so deeply linked that measuring one instantly reveals the state of the other, no matter how far apart they are. This connection is the engine behind future quantum computers and ultra-secure communication. However, keeping this link alive is notoriously difficult. In many setups, the particles interact with their environment, leaking information and causing the entanglement to dissolve into noise. Scientists have long sought a way to trap these particles in a state where they are protected from this decay, effectively hiding them in plain sight within a continuous flow of energy. This elusive state is known as a bound state in the continuum. It is a rare condition where a particle remains localized and stable even though it sits at an energy level where it should theoretically be free to escape.

Researchers have recently turned their attention to "giant atoms," which are artificial quantum systems that do not interact with a field at a single point, but rather at multiple, widely separated locations simultaneously. This unique geometry allows the waves of interaction to interfere with one another, creating opportunities to cancel out decay and lock the system into a stable state. The challenge has been to not only find these stable states but to actively create them with high precision and then keep them there. A team of physicists has now proposed a robust method to generate these protected, entangled states using a combination of rapid frequency modulation and a specific type of control that corrects errors in real time. Their work suggests a pathway to preparing these delicate quantum states with near-perfect accuracy, offering a new tool for preserving quantum information.

The researchers focused on a system containing two giant atoms connected to a waveguide, a structure that guides electromagnetic waves much like a fiber optic cable guides light. In their proposed setup, the atoms are arranged in a braided configuration, meaning their connection points to the waveguide are interleaved. To create the desired entangled state, the team designed a two-stage protocol. First, they use a brief, intense pulse to excite one of the atoms, moving the system from a calm, empty state into a state with a single unit of energy. This initial step is straightforward, but the second stage is where the innovation lies. The team then slowly and smoothly changes how strongly the atoms interact with the waveguide. They do this by rapidly oscillating the frequency of the atoms, a technique that effectively tunes the strength of their connection to the environment without physically moving the atoms.

As the interaction strength changes, the system is guided along a specific path toward the target entangled state. However, moving a quantum system quickly often introduces errors, causing the system to drift away from the intended path. To prevent this, the researchers introduced a counterdiabatic control mechanism. This acts like a corrective steering wheel, applying a precise, independent exchange of energy between the two atoms to cancel out any deviations caused by the speed of the process. By combining this rapid frequency modulation with the corrective exchange, the team simulated a process that drives the system into a highly stable, entangled configuration. In their computer models, which assumed a perfect, loss-free environment, the system reached the target state with a fidelity of 0.99992, meaning it was almost indistinguishable from the perfect theoretical state. The entanglement between the two atoms themselves was also preserved with a fidelity of 0.99588.

A crucial insight from the study is that the target state is not just a pair of entangled atoms; it is a "dressed" state where the atoms are surrounded by a localized cloud of energy in the waveguide. This cloud is an integral part of the stability. The researchers found that if one were to prepare the atoms in an entangled state first and then attach them to the waveguide, the resulting state would be less stable and more prone to leaking energy. By preparing the atoms and their surrounding energy cloud together, the system remains locked in place. The simulations showed that once this dressed state is formed, it can be held for a significant period without the atoms losing their connection to the waveguide or to each other. Furthermore, the process is reversible; by running the control sequence backward, the researchers could retrieve the excitation back into the atoms with high efficiency, demonstrating that the information was truly stored and not lost.

The team also tested how robust this method is against real-world imperfections. They simulated various errors, such as slight miscalibrations in the control signals or small variations in the frequency of the atoms. Even with these imperfections, the system maintained a high level of fidelity, suggesting that the protocol is resilient to the kinds of noise found in actual laboratory equipment. They also modeled the effects of energy loss and dephasing, which are common in real devices, using parameters based on existing superconducting circuits. In these more realistic scenarios, the shaped control pulses provided a distinct advantage over simpler, fixed approaches. Specifically, the method extended the time window during which the entanglement remained useful from about 25 nanoseconds to roughly 91 nanoseconds. While this may seem like a tiny fraction of a second, in the quantum realm, it represents a significant improvement in the ability to store and manipulate information.

The study concludes that this approach offers a practical framework for creating and maintaining entangled states in giant-atom systems. By using a combination of frequency modulation and error-correcting exchange, the researchers have demonstrated a way to navigate the complex landscape of quantum states to reach a stable, protected destination. The work highlights the importance of preparing the entire system, including the surrounding energy field, rather than just the atoms themselves. While the results are currently based on simulations and theoretical models, the use of experimentally motivated parameters suggests that this protocol could be tested in future experiments with superconducting circuits. The findings provide a clear blueprint for how to control these complex quantum systems, potentially paving the way for more reliable quantum memories and processors in the years to come.

Technical Summary: Counterdiabatic Quasi-Floquet Control for Entangled BICs in Giant Atoms

Problem Statement
The generation and preservation of entanglement in waveguide quantum electrodynamics (QED) face challenges from radiative decay and the complexity of preparing complete eigenstates. While Bound States in the Continuum (BICs) offer a mechanism to localize excitations and preserve entanglement, dynamically preparing a complete dressed BIC (including both atomic and photonic components) is non-trivial. A pure atomic Bell state coupled to a waveguide vacuum does not generally constitute a BIC eigenstate. Previous methods for dynamical BIC preparation, such as multi-photon scattering with delayed feedback or time-dependent detuning, often lack robustness or do not explicitly address the preparation of the localized photonic dressing required for the full eigenstate. Furthermore, in degenerate BIC sectors, slow adiabatic evolution alone does not uniquely select a specific trajectory toward a target entangled state.

Methodology
The authors propose a two-stage quantum control protocol combining quasi-Floquet modulation and restricted counterdiabatic (CD) assistance to prepare high-fidelity entangled BICs in a system of two braided giant atoms coupled to a one-dimensional coupled-resonator waveguide.

Quasi-Floquet Modulation: The transition frequencies of the two giant atoms are locally modulated with a common carrier frequency $\nu$ and atom-dependent smooth envelopes $\epsilon_i(t)$ . Using Floquet theory extended to slowly varying envelopes, the authors derive an effective Hamiltonian where the atom-waveguide couplings are renormalized by Bessel function factors, $u_i(t) = J_0[\beta_i(t)]$ . This allows for the coherent control of effective couplings $g_{ik}^{eff}(t) = u_i(t)g_{ik}$ without changing the physical geometry.
State-Selective Counterdiabatic Control: To navigate the degenerate BIC sector and suppress non-adiabatic transitions during finite-time preparation, the protocol introduces an independent, phase-controlled atomic exchange interaction, $H_{cd}(t) = \dot{\theta}(t)\sigma_y$ . This term is derived by minimizing the instantaneous tracking defect for a selected dressed eigenstate path. The control path involves varying the modulation envelopes to define a trajectory in the coupling space ( $u_1, u_2$ ) while the CD exchange suppresses atomic-rotation errors.
Two-Stage Protocol:
- Stage 1 (Loading): A local $\pi$ -pulse prepares the system in the single-excitation product state $|e, g, 0\rangle$ while one atom is effectively decoupled ( $u_1=0$ ).
- Stage 2 (Passage): The system evolves under the time-dependent effective couplings and the CD exchange to reach the target symmetric Bell-type dressed BIC, $|B(\pi/4)\rangle$ , where the atomic component is maximally entangled.
- Stage 3 (Retention/Retrieval): The couplings are held fixed to retain the state, and the process can be reversed to retrieve the excitation.

Key Contributions

Explicit Compact Eigenstate: The authors derive an explicit form for a compact, normalized dressed BIC eigenstate in a braided giant-atom geometry. This allows for the quantitative distinction between the conditional atomic entanglement (assuming one excitation exists) and the unconditional fidelity of the full atom-field state.
Restricted CD Derivation: The paper derives a state-selective counterdiabatic control term restricted to accessible atomic exchange operators. It identifies that while this term perfectly cancels atomic-rotation errors, a residual error remains due to the evolution of the photonic dressing, which cannot be controlled by atomic exchange alone.
Effective vs. Microscopic Validation: The study validates the central-sideband effective model against full microscopic finite-mode simulations, including the fast Floquet oscillations. It demonstrates that the effective description accurately predicts system dynamics within the considered parameter regimes.
Robustness Analysis: The protocol's tolerance to static calibration errors (coupling offsets, modulation depth errors, detunings) and its performance under experimentally motivated relaxation and dephasing rates are quantified.

Results

High Fidelity: In the lossless effective model, the protocol achieves a full-state BIC fidelity of $F_B = 0.99992$ and an unconditional atomic Bell fidelity of $F_{\Psi^+} = 0.99588$ using a fast control protocol ( $\xi T \approx 20$ ).
Photonic Dressing: The difference between the full-state fidelity and unity is primarily attributed to the intended photonic dressing of the BIC, not preparation errors. The unconditional Bell fidelity is lower than the conditional fidelity because the target state includes a localized photonic component.
Microscopic Validation: Microscopic simulations confirm the effective model's accuracy, with a maximum pointwise difference in Bell fidelity of $\approx 1.17 \times 10^{-3}$ between the laboratory-frame dynamics and the slow-frame effective model.
Noise Resilience: In a microscopic write-hold-read simulation with relaxation and dephasing (using parameters from superconducting circuits), shaping the coupling envelopes extends the "useful entanglement window" (where Bell fidelity $\ge 0.90$ ) from approximately 25 ns (fixed envelopes) to 91 ns.
Error Tolerance: The protocol maintains high fidelity ( $>0.99$ ) under static calibration errors of up to 10% in coupling amplitudes and small relative detunings.

Significance and Claims
The paper claims to provide a robust framework for the controlled preparation, retention, and retrieval of dressed entangled states in giant-atom waveguides. The significance lies in:

Distinguishing Atomic vs. Full-State Entanglement: The work explicitly quantifies the "dressing" effect, showing that high atomic entanglement does not guarantee a high-fidelity BIC if the photonic component is not correctly prepared.
Feasibility of Fast Protocols: The results support the feasibility of generating high-fidelity entangled BICs on timescales ( $\sim 22$ units of $1/\xi$ ) significantly shorter than those required by purely adiabatic or weak-drive approaches, provided the control resources (frequency modulation and atomic exchange) are available.
Controlled Robustness: The study demonstrates that the protocol is resilient to static parameter errors and that shaping coupling envelopes offers a tangible benefit in noisy environments by extending the useful entanglement window.

The authors modestly note that the proposed hardware implementation (requiring tunable complex hopping and specific frequency modulation bandwidths) relies on component-level demonstrations from different devices and that a joint experimental validation remains to be performed. The work serves as a theoretical basis and feasibility study for high-fidelity entanglement management in waveguide QED systems.

Technical Summary: Counterdiabatic Quasi-Floquet Control for Entangled BICs in Giant Atoms

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