# Quantum-inspired reverberation using Feedback Delay Network

Juliette Tudoce, Antoine Reserbat-Plantey, Maciej Lewenstein, Reiko Yamada

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Juliette Tudoce, Antoine Reserbat-Plantey, Maciej Lewenstein, Reiko Yamada. Quantum-inspired reverberation using Feedback Delay Network. Proceedings of meetings on acoustics, 2025, 56 (1), pp.015007. 10.1121/2.0002128. hal-05395160

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# Quantum-inspired reverberation using Feedback Delay Network

Juliette Tudoce; Antoine Reserbat-Plantey ID ; Maciej Lewenstein ID ; Reiko Yamada

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Proc. Mtgs. Acoust. 56, 015007 (2025)

https://doi.org/10.1121/2.0002128

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Quantum-inspired reverberation using Feedback Delay Network

**Juliette Tudoce**

Department of Quantum Optics Theory, ICFO-Institut de Sciences Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona, 08860, SPAIN; juliettetudoce@me.com

**Antoine Reserbat-Plantey**

Universite Cote d'Azur, CNRS, CRHEA Valbonne, Sophia-Antipolis, FRANCE; antoine.reserbat@cnrs.fr

**Maciej Lewenstein**

Department of Quantum Optics Theory, ICFO-Institut de Sciences Fotoniques, ICREA, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona, 08860, SPAIN; maciej.lewenstein@icfo.eu;

**Reiko Yamada**

Department of Quantum Optics Theory, ICFO-Institut de Sciences Fotoniques, ESMUC, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona, 08860, SPAIN; Reiko.Yamada@icfo.eu

This work presents a quantum-inspired modification to the classical Feedback Delay Network (FDN) structure for artificial reverberation. Drawing on analogies between wave behavior in acoustic and optical cavities, we incorporate distance-dependent coupling functions modeled after dipolar interactions from cavity quantum electrodynamics (CQED). Nodes are distributed in a 2D spatial domain, and coupling strength decays according to physical power laws: dipole–dipole $1/r^3$, van der Waals $1/r^6$, and Casimir–Polder $1/r^7$. These interaction laws are embedded into the FDN as a gain matrix, modulating the signal propagation through the network. We simulate impulse responses under each model and analyze their decay characteristics, then convolve the responses with audio signals to evaluate perceptual effects. Results show that different interaction models produce noticeably varied reverberation behaviors, highlighting the influence of spatial coupling structure on acoustic decay. This framework opens new directions for exploring physically inspired reverber models with potential applications in spatial audio and sound design.
# 1. INTRODUCTION

Feedback Delay Networks (FDNs) are recursive structures for artificial reverberation, composed of multiple delay lines whose outputs are linearly combined and fed back into the system via a coupling matrix.¹,² First proposed by Gerzon in the early 1970s,³ FDNs generalize earlier models such as Schroeder's parallel comb filters,⁴ enabling arbitrary signal routing between delays. In typical implementations, delay lengths are chosen to approximate acoustic propagation paths, while the feedback matrix governs how energy is distributed across the network. Orthogonal matrices are often used to ensure stability and uniform energy spread, but later approaches have incorporated perceptual or geometric constraints to better reflect spatial structure.⁵⁻⁷ Nevertheless, the interaction strength between nodes is typically determined by the chosen feedback structure and does not explicitly reflect distance-based attenuation laws.

Meanwhile, cavity quantum electrodynamics (CQED) provides a framework for describing how dipolar emitters interact through vacuum field fluctuations, often mediated by virtual photon exchange in confined or structured electromagnetic environments.⁸ These interactions depend strongly on distance and geometry, following characteristic power laws that describe how coupling strength decays with separation. Such models are central to phenomena like dipole-dipole interactions, van der Waals forces, and Casimir-Polder effects.⁹

Motivated by the analogy between wave propagation in optical cavities and acoustic reverberation in enclosed spaces, we propose a modification to the standard FDN framework that incorporates distance-dependent gain terms modeled after dipolar interactions. This allows us to explore how reverberation behavior changes when energy exchange within the network is spatially modulated. Specifically, we introduce gain matrices that decay with distance according to power laws and compute impulse responses for different interaction models to analyze their temporal behavior. These impulse responses are then convolved with audio signals to demonstrate perceptual differences in reverberation.

The structure of the paper is as follows. Section 2 introduces the dipolar interaction models drawn from quantum optics that motivate our approach. Section 3 describes the construction of the modified FDN and the implementation of distance-dependent gain matrices. Section 4 presents the simulation outcomes and analyzes the resulting audio characteristics. Section 5 explores the broader implications of the model and potential directions for future work, and Section 6 summarizes the main findings.

# 2. PRELIMINARIES: DIPOLAR INTERACTION MODELS

Dipolar interactions arise from different physical mechanisms depending on the regime—ranging from resonant coupling via real photon exchange to dispersion forces mediated by vacuum fluctuations. In all cases, the interaction strength decays with distance according to characteristic power laws. These models, while originally defined for microscopic quantum systems, are used here as conceptual tools to structure energy exchange in an audio simulation framework.

## A. DIPOLE-DIPOLE INTERACTION

The dipole-dipole interaction potential $V_{dd}$ describes the mutual coupling between two electric or magnetic dipoles. This interaction can arise between atoms, molecules, or between an emitter and its image dipole near a reflective boundary. In quantum optics and CQED, it governs resonant energy exchange through near-field radiation or photon-mediated coupling.¹⁰

The total Hamiltonian for two interacting dipoles can be written as:

$$H = H_0 + V_{dd}, \qquad (1)$$
where $H_0$ is the unperturbed Hamiltonian of the two systems and $V_{dd}$ captures the dipolar interaction between them. In the near-field regime—where the inter-dipole distance $r$ is much smaller than the characteristic wavelength $\lambda$—the potential scales as:

$$V_{dd}(r) \propto \frac{1}{r^3}. \qquad (2)$$

The full form of the potential is anisotropic and depends on the relative orientations of the dipoles, but for our purposes, we adopt an isotropic scalar approximation to focus solely on distance dependence. Interestingly, dipoles can also be acoustic, as shown in a recent experiment involving a vibrating gong,¹¹ where the frequency and decay time of the sound depend on the distance to a nearby reflector (e.g., a concrete wall). This setup serves as an acoustic analogue of the Dreхage experiment, in which the spontaneous emission rate of an electric dipole varies with its distance from a mirror—an effect that illustrates how dipolar interactions are shaped by the surrounding environment, altering radiative linewidths and spectral properties.

## B. VAN DER WAALS INTERACTION

The van der Waals (vdW) interaction arises from quantum fluctuations of the electromagnetic field and is treated as a second-order correction to the dipole-dipole interaction. Such interaction leads to a force between neutral atoms or molecules in their ground state.⁹ The vdW forces are also responsible for adhesion between layers in layered materials such as graphite, for instance.

The energy shift of the ground state due to this interaction is given by:

$$\Delta E^{(2)} = \sum_{n \neq 0} \frac{|\langle n | V_{dd} | 0 \rangle|^2}{E_0 - E_n}, \qquad (3)$$

where $|0\rangle$ is the ground state of the unperturbed Hamiltonian $H_0$, and $|n\rangle$ are its excited eigenstates with energies $E_0$ and $E_n$, respectively

In the non-retarded regime ($r \ll \lambda$), the interaction scales as:

$$V_{vdW}(r) \propto \frac{1}{r^6}. \qquad (4)$$

## C. CASIMIR-POLDER INTERACTION

The Casimir-Polder (CP) interaction extends van der Waals theory by including retardation effects that arise from the finite speed of light. It describes long-range interactions between a neutral polarizable atom and a macroscopic boundary, mediated by vacuum field fluctuations.¹²

In the far-field regime ($r \gg \lambda$), where retardation dominates, the potential decays more steeply as:

$$V_{CP}(r) \propto \frac{1}{r^7}. \qquad (5)$$

One example of how such forces can be operationalized is in the context of quantum sensing, where Casimir-Polder interactions have been proposed to detect the motion of suspended graphene membranes through vacuum-mediated coupling.¹³ This illustrates how distance-dependent interactions, though originally quantum in nature, can be repurposed or reimagined in engineered systems—including, as explored here, in acoustic analogues. Casimir forces have been extensively studied in micro-mechanical systems,¹⁴ demonstrating their relevance for nanomechanics and, by extension, for nano-acoustics. When two mirrors face each other in vacuum, quantum vacuum fluctuations exert radiation pressure, resulting in a net attractive force—the Casimir force—due to higher external than internal equivalent “radiation pressure”. This force scales as $F \propto d^{-4}$ for planar mirrors. The distance dependence is highly sensitive to the system’s dimensionality. In this proposal, we focus on interactions between two 0D objects, which scale as $F \propto r^{-7}$.
### 3. MODELING FRAMEWORK

To simulate reverberation with quantum-inspired distance dependency, we model a set of $N$ nodes placed randomly in a 2D spatial domain. Each node represents a virtual acoustic emitter or receiver, analogous to dipolar particles in a quantum optical system. These nodes are connected via delay lines and form a Feedback Delay Network (FDN), where energy propagates through both structural feedback and spatially modulated interaction strength.

The key innovation lies in modifying the feedback mechanism using a distance-dependent gain matrix $G$, whose elements $G_{ij}$ model the strength of coupling between node $i$ and node $j$ as a power-law function of their separation:

$$G_{ij} = \frac{1}{d_{ij}^n}, \qquad (6)$$

where $d_{ij}$ is the Euclidean distance between nodes $i$ and $j$, and $n \in \{3, 6, 7\}$ corresponds to the selected quantum-inspired decay model: dipole-dipole, van der Waals, or Casimir-Polder, respectively.

This gain matrix is combined with a traditional feedback matrix, specifically a Householder matrix $H$, which ensures energy conservation and orthogonality within the network:

$$H = I - 2 \frac{vv^T}{v^T v}, \qquad (7)$$

where $v$ is a non-zero vector and $I$ is the identity matrix. The update rule for the system state $x(t)$ at discrete time $t$ becomes:

$$x(t + 1) = H \cdot (G \odot x(t - \tau)), \qquad (8)$$

where $\tau$ represents the delay time vector and $\odot$ denotes element-wise multiplication. This formulation differs from classical FDNs by embedding the gain modulation directly into the feedback loop, allowing spatial decay characteristics to shape the impulse response dynamically.

To generate impulse responses (IRs), we inject a unit Dirac impulse into node 0 at $t = 0$. The signal then propagates through the network according to the modified feedback dynamics. At each timestep, we track the amplitude at each node, resulting in a multichannel IR profile—one per node—for each interaction model. The delay length between two nodes, $\tau_{ij}$, is determined by the physical distance $d_{ij}$ between them. To express this delay in samples, we scale the distance by an effective propagation speed $c$ (analogous to the speed of sound), and multiply by the sampling frequency $f_s$. The result is then rounded to the nearest integer:

$$\tau_{ij} = \text{round} \left( \frac{d_{ij}}{c} \cdot f_s \right). \qquad (9)$$

This mapping preserves spatial relationships in the temporal domain and ensures that energy travels through the network with delays that reflect physical distances.

The simulation is implemented in Python, using numpy-based matrix operations. Node positions are randomly initialized within a bounded 2D domain of 10x10 meters, and distances are precomputed to form the gain matrix. The system is iterated over a fixed number of timesteps to ensure the impulse responses decay fully.

### 4. IMPULSE RESPONSE AND AUDIO ANALYSIS

We analyze the reverberation characteristics by observing the impulse responses at node 0 under three decay laws. Figure 1 summarizes the observed behaviors:

* For $n = 3$, the interaction is strong and global, causing energy to disperse rapidly, potentially leading to more destructive interference and shorter reverberation tails.
* For $n = 6$ and $n = 7$, interactions are more localized. Energy remains confined longer, producing longer decay tails due to reduced interference.

<figure><img src="image_6.png" /><figcaption>Figure 1: Impulse responses for node 0 under different power laws.</figcaption></figure>

To evaluate how the impulse responses generated by each interaction model affect audio perception, we convolve them with a reference audio signal. This process is repeated for each node in the network and for each value of the power-law exponent $n$. For clarity, we focus again on the output at Node 0, though we note that, under this configuration, perceptual differences are primarily driven by the interaction law rather than the specific node.

Figure 2 compares the waveform of the original (dry) signal with its reverberated versions. The reverberated signals consistently exhibit broader envelopes, indicating the presence of reflections and temporal smearing introduced by the network. As expected, the decay characteristics vary significantly with the value of $n$: for smaller $n$ (e.g., $n = 3$), energy disperses rapidly through the network, resulting in a short, dense reverberation reminiscent of a small or medium-sized room. For higher values of $n$ (e.g., $n = 6$ or $n = 7$), the energy is more confined and circulates longer within local clusters of nodes, producing longer reverberation tails and echo-like effects that extend several seconds beyond the original signal.

An interesting spectral effect emerges: certain frequency bands are attenuated more than others, leading to the formation of formants—peaks and dips in the spectral envelope—caused by destructive interference patterns introduced by the impulse response. This results in time-localized regions where the envelope of the convolved signal is lower than that of the dry input. These interactions, although abstracted from quantum electrodynamics, introduce perceptually meaningful colorations in the audio signal, differentiating the reverberation texture beyond simple decay time differences.

Additionally, the resulting audio has a distinct metallic character, especially pronounced for higher values of $n$, which may be reminiscent of early comb-filter-based reverberators. This effect becomes even more noticeable when the input audio is not entirely clean, suggesting sensitivity to high-frequency content or transients. We also observe a consistent shift in amplitude between the dry and convolved signals, the origin of which remains unclear and may be linked to the energy redistribution properties of the gain-modulated feedback network.

## 5. DISCUSSION AND APPLICATIONS

The proposed framework reveals how altering network topology via physical-inspired decay functions can affect reverberation characteristics. Although currently limited to 2D and static geometries, the model offers avenues for enhancement:

* **3D Extensions:** Modeling in three-dimensional space could introduce more realistic spatiality.
<figure><img src="image_7.png" /><figcaption>Figure 2: Convolution waveforms at node 0 using impulse responses generated from different power-law interaction models: (a) $1/d^3$ (dipole-dipole), (b) $1/d^6$ (van der Waals), and (c) $1/d^7$ (Casimir-Polder). Each plot shows the original dry signal (black) and the reverberated signal (blue).</figcaption></figure>

* **Directional Propagation:** Integrating angular dependencies, as in real dipolar fields, may enrich acoustic texture.

* **Real-Time Processing:** Implementing efficient convolution engines may allow real-time application in music and virtual acoustics.

* **Artistic Exploration:** The robotic, stylized sound artifacts produced by the model may find use in creative sound design.

## 6. CONCLUSION

We have presented a quantum-inspired reverberation model based on distance-modulated Feedback Delay Networks. By using analogs of dipolar decay laws, we obtained impulse responses with distinct temporal and spectral properties. While based in an acoustic simulation framework, this approach takes inspiration from cavity quantum electrodynamics and offers a new perspective for reverber design. Future work could extend the model to support more complex spatial configurations, directional propagation, and real-time audio applications.
ACKNOWLEDGMENTS

We gratefully acknowledge the funding and support provided by ICFO for our research, as well as the
Barcelona Institute of Science and Technology (BIST).

A. R-P acknowledges funding from French ANR JCJC "NEAR 2D".

ICFO-QOT group acknowledges support from: European Research Council AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project funded by MCIN/AEI/10.13039/501100011033 and by the "European Union NextGenerationEU/PRTR" (PRTR-C17.I1), FPI); QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European Union's Horizon 2020 program under Grant Agreement No 101017733; Ministry for Digital Transformation and of Civil Service of the Spanish Government through the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda; Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund FEDER and CERCA program; Barcelona Supercomputing Center MareNostrum (FI-2023-3-0024); Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union, European Commission, European Climate, Infrastructure and Environment Executive Agency (CINEA), or any other granting authority. Neither the European Union nor any granting authority can be held responsible for them (HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU Horizon 2020 FET-OPEN OPTOlogic, Grant No 899794, QU-ATTO, 101168628), EU Horizon Europe Program (This project has received funding from the European Union's Horizon Europe research and innovation program under grant agreement No 101080086 NeQSTGrant Agreement 101080086 — NeQST); ICFO Internal "QuantumGaudi" project

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