Results:
To investigate the relationship between entanglement dynamics and auditory perception, we analyzed the spectrograms and von Neumann entropy (SvN) evolution for different Hamiltonians and system sizes. In our analysis, we focus on two complementary waveshapes for sonification: triangular waveforms and ring modulation. The triangular waveform introduces a rich harmonic structure, often perceived as higher-pitched and spectrally rich. Among standard periodic waveforms such as square, sawtooth, and absolute sine, the triangle wave was chosen for its perceptual balance of harmonic content—providing sufficient upper harmonics for spectral richness without overwhelming the signal with high-frequency energy. This selection was based on a subjective evaluation of clarity and interpretability in the resulting sound.
In contrast, ring modulation—despite producing fewer harmonics—yields a more metallic and textured timbre. Its strong sensitivity to amplitude variation results in perceptible silences when the von Neumann entropy is at its maximum and louder output when the entropy is low. This creates a pronounced dynamic range, enhancing the auditory contrast between weakly and strongly entangled states. Taken together, the two waveshapes highlight different aspects of the quantum dynamics, providing complementary perceptual perspectives on structure, complexity, and temporal evolution.
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).
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.
Next steps:
This study demonstrates the value of sonification as both a scientific and artistic tool for exploring quantum dynamics. The auditory rendering of abstract quantities such as von Neumann entropy and Husimi Q distributions provides a new channel for engaging with quantum behavior—particularly entanglement growth and the distinction between integrable and chaotic systems. By translating mathematical structures into sound, the method complements conventional visualizations and facilitates a more intuitive, temporally resolved understanding of complex quantum phenomena.
• The artistic dimension of sonification not only enhances accessibility but also opens pathways to interdisciplinary collaboration. The generated audio material has potential for creative reuse in musical composition, sound design, and multimedia art. Providing access to sonified datasets through open platforms such as GitHub encourages broader engagement beyond the physics community.
• Future work should also explore more immersive spatialization techniques. In particular, deploying the sonification in ambisonic or dome-based audio systems would enable spatial encoding of the Bloch sphere or Husimi Q-function data, improving the perceptual mapping between quantum phase space and auditory space. Such high-dimensional rendering could enhance listeners’ ability to internalize features like angular spread, symmetry breaking, or localization.
• To rigorously evaluate the effectiveness of these methods, perceptual and cognitive studies should be conducted to assess whether sonification genuinely improves understanding of quantum behavior—especially among students or non-expert audiences. Such evaluations could combine quantitative metrics, such as learning gain, pattern recognition accuracy, or intuitive classification of chaotic versus regular systems, with qualitative insights from cognitive science. In particular, research highlights criteria that can make a sonification more effective: leveraging robust crossmodal correspondences between auditory and visual dimensions (Chen 2024), exploiting gestural similarities that link motor schemas to sound patterns (Mannone 2018), and aligning with the conditions of the supramodal-brain hypothesis, which Rosenblum and collaborators argue underpins cross-sensory integration (Rosenblum 2016).
• On the technical side, one challenge we encountered was the need to downsample high-resolution quantum simulation data for audio rendering. Although this compression minimally affected perceptual clarity in our experiments, future implementations could benefit from sampling that preserve all the features. Additionally, integrating interactive controls—such as real-time parameter modulation or spatial navigation—would make the sonification experience more exploratory and user-driven, though it may come at a significant computational cost.
• Finally, extending this framework to more complex systems presents a compelling opportunity. These systems often defy easy visual interpretation, making them prime candidates for perceptually grounded approaches like sonification. In particular, the proposed sonification approach enables the analysis of complex quantum circuits on digital quantum computers by sonifying each successive circuit layer. A natural next step would be to incorporate many‑body entanglement measures—such as entanglement depth and non‑k‑separability based on many-body Bell correlations (Płodzień 2024)—which capture many-body entanglement structure of a given quantum state. In particular, these many-body Bell correlations characterize entanglement structure of graph states (Płodzień 2024), thus combining with Husimi function analysis, would allow to provide audio-representation of a quantum states which are naturally represented as a mathematical graphs. Sonification of mixed quantum states, in particular finite temperature Gibbs state, is another exciting possibility. Using more complicating entanglement measures for sonification would reveal additional details about the state’s internal structure enhancing audio-visual content. Lastly, the proposed sonification method allows for audio-visual representation of many-body quantum Hamiltonian through sonification of its eigenstates, where the time domain is given by the eigenvalues order.
OAT Hamiltonian:
The OAT Hamiltonian produces smooth and periodic spectro-temporal patterns. For L = 2, the spectrograms show broad, slowly evolving harmonic structures, which correspond to the sinusoidal oscillations in the von Neumann entropy. For L = 8, they depict a richer yet still regular spectro-temporal texture, reflecting the more structured but smooth entropy dynamics. Sonically, this results in gradually evolving textures characterized by a clear sense of periodicity.
TACT Hamiltonian:
Compared to OAT, this produces faster entanglement generation and more rapid deformation of phase-space structures. As a result, the spectro-temporal patterns tend to evolve more quickly and develop richer harmonic complexity.
For L = 2, the spectrograms show structured but faster oscillatory features compared to the OAT case. For L = 8, the dynamics becomes significantly more complex. The spectrograms develop dense spectro-temporal structures at earlier times, reflecting rapid multipartite entanglement generation.
The transition from L = 2 to L = 8 highlights how larger collective spin systems enable richer spectral and sonic textures.
XXZ Heisenberg:
The XXZ Heisenberg Hamiltonian generates dynamics through local nearest-neighbor interactions rather than collective twisting. For Δ = 0.5, the system favors transverse spin fluctuations, leading to relatively efficient spreading of correlations and entanglement compared to Ising-like regimes.
For L = 2, the spectro-temporal patterns remain relatively simple and structured. The spectrograms show smooth oscillatory bands that evolve over time, similar to the OAT model. For L = 8, the behavior becomes significantly more complex. The spectrograms develop denser spectro-temporal textures as correlations propagate through the chain and entanglement spreads across multiple spins. The entropy typically grows more steadily before reaching a fluctuating plateau, reflecting many-body dephasing rather than strict periodic recurrence.
Overall, the sonification distinguished model-specific entanglement dynamics and facilitated intuitive perception of underlying quantum behaviors. Additional sonified examples—including audio, spectrograms, von Neumann entropy plots, and videos showing quantum state evolution on the Bloch sphere and in phase space—are available for a broader range of system sizes, parameters, and Hamiltonians on the project’s GitHub repository.
Final words:
This study establishes sonification as an intuitive method for interpreting and experiencing the dynamics of quantum entanglement. By translating abstract quantum features—such as bipartite entanglement entropy growth and Husimi function of many-qubit state—into auditory form, our approach enables both scientific insight and artistic engagement. The perceptual accessibility of sound offers a novel way to explore differences between regular and quantum chaotic dynamics, and to intuitively grasp entanglement structure in many-qubit dynamics.
Looking ahead, the sonification of quantum entanglement—particularly in many-body and time-dependent systems—offers a promising avenue for educational and scientific communication. By making abstract quantum behaviors perceptible, this approach could support interdisciplinary efforts in physics, auditory display, and cognition. Further development of quantum-specific sonification techniques will help refine their perceptual utility and clarify their role in both research and outreach.
Quantum Kicked Rotor:
As a reminder, the Quantum Kicked Rotor is a time-periodic system whose dynamics is governed by alternating nonlinear kicks and rotations. The degree of chaotic behavior is controlled by the parameter α, which determines the strength of the kicks. We present results for two values: α = 0.1 (regular dynamics) and α = 10 (strongly chaotic regime), for the initial state |θ=0, ϕ=0>, and system sizes L=2 and L=8.
For small kick strength (α = 0.1), the kicked rotor exhibits regular and smooth dynamics.
In the L = 2 case, the entropy evolution closely resembles that of the OAT Hamiltonian, showing periodic behavior. Nevertheless, the discrete nature of the kicks is clearly visible, and this structure is perceptibly reflected in the audio as rhythmic or pulsating modulations. The initial state modifies the evolution of both the entanglement entropy and the Husimi Q function, giving rise to slightly different dynamics. Even in a nearly integrable regime, this highlights how the choice of initial state influences both the entanglement scale and the perceived sonic features.
In the L = 8 case is characterized by gradual growth followed by slight fluctuations. A small dip in entropy is observed around kick number 170. This anomaly is particularly noticeable in the triangular waveform spectrogram, where it manifests as a brief reduction in spectral density and pitch activity.
In the strongly chaotic regime α = 10, the system exhibits dramatically different behavior. Regardless of initial condition or system size, the entropy becomes highly irregular and fluctuates in a seemingly random manner. This is reflected sonically in spectrograms, where the frequency components exhibit sudden shifts, producing a pitch pattern with intervallic skips—distinct from the more continuous profiles of the integrable OAT model and kicked rotor with small values of α.
Despite the shared chaotic structure, the system size still influences the entanglement profile. For L=2, entropy remains below 1 with large fluctuations, whereas for L=8, it rapidly rises and stabilizes near SvN=1, fluctuating around this plateau. This reflects faster entanglement generation and information scrambling in larger chaotic systems, where the effective Hilbert space is significantly larger.