Quantum dynamics of coupled quasinormal modes and quantum emitters interacting via finite-delay propagating photons

This paper presents a time-dependent theory describing the retarded interactions between spatially separated lossy cavities and quantum emitters mediated by finite-delay propagating photons, utilizing quantized quasinormal modes and non-bosonic bath operators to model both bath-mediated and QNM-mediated coupling.

The Gist

The Big Picture: A Quantum Game of "Telephone" with a Twist

Imagine you have two tiny, isolated rooms (these are your cavities). Inside each room, there is a special, vibrating bell (a Quantum Emitter, like an atom or a quantum dot). These rooms are made of a material that absorbs some sound and lets some leak out (they are lossy).

In the old days, physicists tried to describe how these rooms interacted by pretending the walls were perfect and the sound never got lost. They used a simple rule: "If Room A rings, Room B hears it instantly."

But in reality, sound takes time to travel. If Room A is far away from Room B, there is a delay. The sound has to travel through the air (the background medium) to get there. This paper is about creating a new, super-accurate rulebook for how these "rooms" and "bells" talk to each other when they are separated by distance and when the "air" between them is messy and lossy.

The Characters in Our Story

1. The Quasinormal Modes (QNMs): Think of these as the natural "humming" notes of the rooms. Unlike a perfect bell that rings forever, these rooms are leaky. The "hum" eventually fades away because energy leaks out into the surrounding space. The paper treats these "fading hums" as real, quantized particles of light (photons) that can be counted and manipulated.
2. The Quantum Emitters (TLS): These are the actors inside the rooms. They can be excited (jump up) or relaxed (jump down). When they jump down, they shout a message (emit a photon).
3. The Photonic Bath: This is the ocean of air surrounding the rooms. It's not empty; it's filled with potential sound waves. When a room leaks energy, it dumps it into this ocean. The ocean then carries that energy to the other room.

The Problem: The "Instant" vs. The "Delayed"

Previous theories had a blind spot. They assumed that if Room A and Room B were close, they talked instantly. If they were far apart, they just ignored each other.

But in the quantum world, distance matters.

• The "Direct" Zone: If an actor is standing right next to the bell in their own room, they feel the vibration immediately. The paper calls this the "Area of Direct Influence." It's like standing right next to a speaker; you hear the sound instantly.
• The "Delayed" Zone: If an actor is far away, or if Room A is trying to talk to Room B, the message has to travel through the "ocean" (the bath). This takes time. The paper introduces a way to calculate exactly how long that delay is and how the message changes while traveling.

The New Theory: How It Works

The authors built a mathematical engine that splits the interaction into two parts:

1. The "Local" Chat (Non-Retarded Coupling)
If the actor is inside the room, they interact directly with the room's "hum." The paper shows that you can calculate this interaction very precisely using the shape of the room and the material it's made of. It's like knowing exactly how loud your own voice sounds in your own bathroom.

2. The "Long-Distance" Chat (Retarded Coupling)
This is the big innovation. When Room A wants to talk to Room B:

• Room A shouts into the "ocean" (the bath).
• The ocean carries the sound wave.
• The wave travels across the gap.
• Room B hears it later.

The paper creates a set of "Correlation Functions." Think of these as weather maps for sound. They don't just say "it's raining"; they say, "It started raining in Room A at 2:00 PM, and it will reach Room B at 2:05 PM, and it will be slightly weaker because of the wind."

These maps allow scientists to predict exactly how the quantum state of one room affects the other, accounting for the time it takes for the "message" to travel.

The Real-World Example: The Metal Dimer Duo

To prove their theory works, the authors simulated a specific setup:

• Imagine two pairs of tiny gold cylinders (like dumbbells) floating in space. These are the cavities.
• Inside the gap of each dumbbell, they placed a tiny "atom" (the emitter).
• The two dumbbells are separated by a distance of about 2 micrometers (roughly 20 times the width of a human hair).

They ran the numbers and found:

• The "hum" inside one dumbbell doesn't just stay there; it leaks out.
• The "ocean" carries this leak to the other dumbbell.
• Because of the distance, the second dumbbell doesn't feel the first one instantly. There is a tiny, measurable delay.
• The paper's new math successfully predicted exactly how strong this delayed connection is, matching complex computer simulations perfectly.

Why Does This Matter?

This isn't just about math; it's about building the future of Quantum Internet.

Imagine you want to send a secret quantum message from a satellite to a ground station, or from one chip to another in a quantum computer.

• If you ignore the time delay, your message arrives garbled or at the wrong time.
• If you ignore the loss (the fact that the signal fades as it travels), your message disappears.

This paper provides the blueprint for engineers to build these systems. It tells them exactly how to design the "rooms" (cavities) and how to account for the "travel time" of the photons so that quantum information can be transferred reliably across distances.

Summary in a Nutshell

• Old Way: "Everything happens instantly, and we pretend the walls are perfect."
• New Way: "Everything takes time to travel, and the walls are leaky. We have a new map (Correlation Functions) that tracks exactly how the energy moves from one leaky room to another, accounting for the delay and the fading."

It's like upgrading from a walkie-talkie that assumes instant communication to a sophisticated system that knows exactly how long the radio waves take to bounce off the atmosphere and arrive at the other end, ensuring the message is received clearly.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the theoretical description of open quantum systems, specifically those involving spatially separated lossy optical cavities (resonators) and quantum emitters (e.g., atoms, quantum dots).

• Limitations of Existing Models: Traditional approaches often rely on the Jaynes-Cummings model with a Rotating Wave Approximation (RWA) for closed cavities, adding losses phenomenologically (e.g., via Lindblad superoperators). These methods assume instantaneous coupling and neglect the non-Hermitian nature of open cavities.
• The Challenge of Retardation: In quantum networks and distributed systems, the finite speed of light introduces time delays (retardation) in photon propagation between subsystems. Standard Markovian approximations fail here.
• The QNM Quantization Issue: While Quasinormal Modes (QNMs) rigorously describe lossy open cavities, previous quantization schemes treated QNMs as quasibound states localized at a specific cavity. The information regarding causality and photon propagation was relegated to a "bath" but not fully integrated into the time-dependent dynamics of coupled, separated systems. There was a lack of a rigorous theory that combines quantized QNMs with non-bosonic bath operators to describe finite-delay interactions.

2. Methodology

The authors develop a comprehensive time-dependent theory for a system of multiple QNM cavities and quantum emitters (modeled as Two-Level Systems, TLSs) in a homogeneous background medium.

• System-Bath Decomposition:
  • The electromagnetic field is decomposed into Quantized Quasinormal Modes (QNMs) and a Photonic Bath.
  • The QNM operators ($\hat{a}_{i\mu}$) are defined via projectors onto the subspace of symmetrized QNMs. These operators create/annihilate quasibound states localized at specific cavities.
  • The Photonic Bath operators ($\hat{c}(\mathbf{r}, \omega)$) are defined as orthogonal to the QNM projectors. Crucially, these bath operators are non-bosonic due to the presence of the cavities, ensuring causality and orthogonality.
• Hamiltonian Formulation:
  • The total Hamiltonian is split into System ($H_S$), Bath ($H_B$), and System-Bath coupling ($H_{SB}$).
  • $H_S$ includes direct TLS-QNM coupling and intra-cavity QNM interactions.
  • $H_{SB}$ describes the coupling of both TLSs and QNMs to the bath, which mediates the retarded interactions between separated systems.
• Derivation of Correlation Functions:
  • The core of the methodology involves deriving system-bath correlation functions ($C_{xy}(t, t')$). These functions fully describe the memory effects and time delays.
  • The authors solve the Heisenberg equations of motion for the non-bosonic bath operators, treating the interaction as a scattering process. This leads to a recursive expansion involving inter-cavity scattering terms ($K^N_{i\mu j\nu}$) of order $N$.
  • They introduce retarded overlap matrices ($S_{i\mu \leftarrow j\eta}$) to handle the spatial integrals efficiently, separating fast-rotating phase factors (time delays) from slow-varying envelope functions.
• Numerical Implementation:
  • The theory is applied to a model system of two spatially separated metal dimers (acting as QNM cavities) with TLSs in the gaps.
  • Numerical calculations use Maxwell's equations with a Drude model for metal dispersion and losses, solved via COMSOL Multiphysics (Finite Element Method).
  • The Green's function is expanded using dominant QNMs to calculate coupling elements and correlation functions.

3. Key Contributions

1. General Theory for Retarded Interactions: The paper provides the first rigorous derivation of system-bath correlation functions for coupled QNM cavities that explicitly include finite-time delays (retardation) mediated by a non-bosonic photonic bath.
2. Definition of "Area of Direct Influence": The authors define a metric $P_i(\mathbf{R}_a)$ that quantifies the spatial region around a cavity where direct, non-retarded coupling between the QNM and a TLS is significant. Outside this area, coupling is exponentially suppressed unless mediated by time-delayed bath photons.
3. Non-Bosonic Bath Dynamics: The work explicitly treats the bath operators as non-bosonic, correcting previous approximations that assumed simple bosonic time evolution. This is shown to be critical for accurate multi-cavity dynamics, although the bosonic approximation holds for single-cavity cases.
4. Efficient Numerical Formulas: The derivation of retarded overlap matrices allows for the calculation of coupling parameters using standard numerical solvers for Maxwell's equations, avoiding the need for full-space time-domain simulations of the quantum dynamics.
5. Unified Framework: The theory unifies three types of interactions into a single formalism:
   • QNM-QNM: Energy transfer between cavities.
   • QNM-TLS: Interaction between a cavity mode and an emitter.
   • TLS-TLS: Direct and mediated interaction between emitters.

4. Results

• Correlation Functions: The derived correlation functions (Eqs. 37, 48, 50) show that interactions are dominated by $\delta$-like peaks at time delays corresponding to the photon travel time ($\tau_{ij} = n_B R_{ij}/c$).
• Metal Dimer Example:
  • For two coupled gold nanorod dimers separated by ~2 $\mu$m, the QNM-QNM correlation function exhibits peaks at $t - t' = \pm \tau$, confirming the time-delayed nature of the interaction.
  • Direct vs. Bath-Mediated Coupling: For a TLS inside a cavity, the direct coupling to the local QNM dominates (by orders of magnitude) over the bath-mediated coupling. However, for a TLS in a distant cavity, the bath-mediated coupling is the only significant interaction mechanism.
  • TLS-TLS Interaction: The bath-mediated coupling between separated TLSs includes both instantaneous components (due to the non-zero tail of QNMs outside the cavity) and retarded components. The instantaneous part is weak but non-zero, while the retarded part carries the coherent energy exchange.
• Validity of Approximations: The study confirms that for well-separated cavities, the correlation functions can be truncated at the first order of scattering ($N \le 1$), significantly simplifying calculations while maintaining high accuracy.

5. Significance

• Quantum Networks: This work provides the theoretical backbone for designing and simulating quantum networks where quantum states are transmitted between distant nodes via propagating photons. It moves beyond Markovian approximations to capture non-Markovian memory effects essential for long-distance entanglement distribution.
• Rigorous Modeling of Lossy Systems: By using quantized QNMs, the theory avoids the need for phenomenological loss parameters. All decay rates and coupling strengths are derived from first principles (Maxwell's equations) and are numerically calculable for arbitrary 3D geometries.
• Scalability: The separation of the problem into "system" (QNMs) and "bath" (propagating photons) with pre-calculated correlation functions allows these methods to be plugged into existing open quantum system solvers (e.g., TCL, HEOM, Tensor Networks). This enables the simulation of complex, large-scale quantum photonic devices.
• Future Applications: The framework is a stepping stone toward simulating structured environments (e.g., waveguide-coupled cavities) and on-chip quantum information technologies, offering a path to rigorous, parameter-free simulation of quantum devices.