Giant atoms coupled to waveguide: Continuous coupling and multiple excitations

1. Problem Statement

The paper addresses two critical gaps in the existing theoretical framework for giant atoms (artificial atoms with sizes comparable to the wavelength of the field) coupled to waveguides:

• Gap 1: Continuous Coupling: Most existing studies model giant atoms as coupling to waveguides at a few discrete points (e.g., two points). This ignores the physics of continuous coupling, where the atom interacts with the field over a spatial region. The authors hypothesize that continuous coupling breaks the "constant phase difference" condition essential for strong interference effects, potentially altering system dynamics significantly.
• Gap 2: Multiple Excitations: Conventional methods (like the dressed state approach) typically restrict the Hilbert space to a single-excitation subspace to maintain mathematical tractability. However, this assumption fails in realistic scenarios, particularly at finite temperatures where waveguide modes are thermally populated, or when studying non-classical initial states (e.g., squeezed states). Existing methods become computationally intractable as the number of excitations increases.

2. Methodology: Stochastic Schrödinger Equation (SSE)

To overcome these limitations, the authors propose a Stochastic Schrödinger Equation (SSE) approach based on non-Markovian quantum state diffusion.

• Theoretical Framework:

  • Instead of solving the master equation directly or expanding the Hilbert space for multi-excitation states, the authors expand the total wavefunction in a coherent state basis of the waveguide.
  • They derive a stochastic state vector $|\psi(t, z^*)\rangle$ governed by an SSE. The reduced density operator $\rho(t)$ is recovered by averaging over many stochastic trajectories: $\rho(t) = \mathcal{M}[|\psi(t, z^*)\rangle\langle\psi(t, z)|]$.
  • The interaction is encoded in correlation functions (auto-correlation $\alpha_{\mu\mu}$ and cross-correlation $\alpha_{\mu\nu}$) which depend solely on the spatial coupling distribution $g_\mu(x)$ and are independent of the atomic state.

• Handling Complexity:

  • The SSE approach introduces time-dependent operators ($O_\mu$) to handle functional derivatives arising from the non-Markovian noise.
  • Crucially, the form of the dynamical equation remains unchanged regardless of the number of excitations in the initial state. This contrasts with the dressed state method, where the number of coupled differential equations grows exponentially with the excitation number.

• Extension to Non-Vacuum States:

  • The framework is extended to handle thermal states (via fictitious negative-frequency modes and Bogoliubov transformations) and squeezed states (via normalized trajectories and additional noise terms), allowing for the study of realistic, multi-photon initial conditions.

3. Key Contributions

1. Unified Framework for Continuous Coupling: The paper provides the first systematic derivation of dynamics for giant atoms with continuous spatial coupling distributions, moving beyond the discrete point-coupling approximation.
2. Scalability for Multi-Excitation: The SSE method is demonstrated to be computationally efficient for multi-excitation systems (including thermal and squeezed states) without increasing the complexity of the governing equations.
3. Physical Insight via Correlation Functions: The authors show that auto- and cross-correlation functions directly reflect the photon emission/absorption paths and time-delay effects, offering a new tool to analyze interference without solving the full dynamics first.
4. Extension of SSE Formalism: The work extends the standard SSE formalism to include two types of noise (for two giant atoms) and handles cross-correlations between them, broadening the applicability of the method.

4. Key Results

A. Impact of Continuous Coupling on Interference

• Discrete vs. Continuous: In discrete coupling (e.g., two points), photons travel fixed paths, maintaining a constant phase difference that leads to strong constructive/destructive interference.
• Phase Breaking: In continuous coupling (modeled as Gaussian or double-peak distributions), photons can be emitted and absorbed at arbitrary points within the coupling region. This results in a broad distribution of propagation times, breaking the constant phase difference condition.
• Weakened Interference: Numerical simulations show that as the coupling region broadens, the characteristic interference fringes in entanglement generation are washed out. The system transitions from exhibiting strong interference patterns to a more averaged, less oscillatory behavior.

B. Entanglement Dynamics

• Discrete Coupling: Entanglement generation is transient and highly sensitive to the distance between atoms.
• Continuous Coupling:
  • Localized Regime: Similar to discrete coupling, entanglement is sensitive to atomic separation.
  • Delocalized Regime: As the coupling width increases, the system supports a vast number of photon emission/absorption pathways. This leads to stronger and more robust entanglement generation that is less sensitive to the precise spatial separation of the atoms. The "smearing" of phase differences actually enhances the average indirect coupling strength.

C. Multi-Excitation Dynamics

• The authors successfully simulated double-excitation states ($|ee\rangle$) and demonstrated that the SSE approach handles these without the exponential scaling of equations seen in conventional methods.
• Thermal and Squeezed States: The paper proves the compatibility of the SSE with thermal initial states (finite temperature) and squeezed vacuum states.
• Enhanced Interactions: For squeezed initial states, the cross-correlation functions are shown to be exponentially enhanced, suggesting a mechanism for exponentially enhanced indirect interactions between giant atoms mediated by the waveguide.

5. Significance

• Theoretical Advancement: This work bridges the gap between idealized discrete models and realistic continuous coupling scenarios, providing a more accurate description of experimental giant atom systems (e.g., superconducting circuits).
• Practical Utility: By solving the multi-excitation problem efficiently, the method enables the study of non-Markovian dynamics in realistic environments (finite temperature, non-vacuum fields), which was previously computationally prohibitive.
• Control Mechanisms: The findings suggest that while continuous coupling may weaken specific interference-based phenomena, it offers a route to robust entanglement and enhanced interactions via spatial delocalization and non-classical reservoir engineering (squeezing).
• Tool Development: The proposed SSE framework serves as a versatile tool for future research in waveguide QED, particularly for exploring complex scenarios involving continuous coupling, multiple excitations, and non-classical initial states.