⚛️ quantum physics

QuantumBilliards.jl: An extensible, user friendly, high-performance Julia library for the large scale computation of spectra and eigenstates of quantum billiards and related problems

This paper introduces QuantumBilliards.jl, an open-source, high-performance Julia library designed to efficiently compute spectra and eigenstates of two-dimensional quantum billiards by offering a unified framework for various advanced numerical solvers, geometry construction, and phase-space analysis.

Original authors: Matic Orel, Črt Lozej

Published 2026-10-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matic Orel, Črt Lozej

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

Imagine a room with perfectly smooth, hard walls where a tiny, invisible particle bounces around forever. In the classical world, this is a simple game of pool: the particle travels in straight lines, hits a wall, and bounces off at the same angle it arrived. If you know where it started and how fast it was going, you can predict its entire path. But in the quantum world, that same particle behaves like a wave. It doesn't just bounce; it spreads out, interferes with itself, and creates a complex pattern of high and low points across the entire room. This is the realm of quantum billiards, a playground for physicists trying to understand how the chaotic, unpredictable motion of the classical world transforms into the rigid, mathematical rules of quantum mechanics. Scientists have long used these "billiard tables" to test theories about chaos, randomness, and the fundamental nature of energy, but calculating the exact patterns these waves make has been a massive computational challenge, especially as the waves get smaller and more numerous.

To solve this, a team of researchers has created a new, open-source software tool called QuantumBilliards.jl. Think of this as a powerful, all-in-one digital laboratory designed specifically to simulate these quantum rooms. Before this tool existed, researchers often had to piece together different, incompatible programs to study the same problem, or they were limited to simple shapes like circles and squares. The new library unifies a wide variety of advanced mathematical techniques into a single, user-friendly system. It allows scientists to design complex, irregular shapes—some with sharp corners, some with holes, and some that are perfectly symmetrical—and then calculate the exact energy levels and wave patterns inside them with unprecedented speed and accuracy. The software is built on the Julia programming language, which is known for being both fast enough for heavy scientific work and easy enough for researchers to use without getting bogged down in code.

The core achievement of this work is the ability to find thousands of these energy patterns at once, rather than hunting for them one by one. In the past, finding a single pattern required a slow, repetitive process of guessing a value, checking the result, and guessing again. The new software uses several clever shortcuts, or "accelerated solvers," to leap directly to the answers. One method looks at how the patterns change when the room is slightly stretched, allowing the computer to predict a whole cluster of solutions from a single calculation. Another method treats the problem as a search for specific points in a complex mathematical landscape, using a circular path to find all the answers hidden inside that loop at the same time. By combining these different approaches, the researchers can verify their results against each other, ensuring that the numbers they find are correct and not just artifacts of a specific calculation method.

The software is not just about finding numbers; it also reconstructs the full picture of what the quantum wave looks like inside the room. It can show the "boundary function," which describes how the wave behaves right at the walls, and the "wavefunction," which maps the wave's intensity throughout the entire space. Perhaps most importantly, it can translate these complex wave patterns into a phase-space representation, a type of map that shows how the quantum state relates to the classical bouncing motion. This allows researchers to see "scars," which are regions where the quantum wave seems to cling to the path of a classical trajectory, even though the rest of the wave is chaotic. The team demonstrated the power of their tool by running simulations on a variety of challenging shapes, including a mushroom-shaped table with a stable island in the middle of a chaotic sea, and a star-shaped table with five-fold symmetry. In one large-scale test, the software successfully computed over 100,000 distinct energy states for a single complex shape, a task that would have been prohibitively slow with older methods.

What makes this tool particularly valuable is its flexibility and its ability to handle the messy realities of real-world physics. Many physical systems, from microwave cavities to semiconductor devices, behave like these quantum billiards, but they often have corners, multiple disconnected parts, or specific symmetries that complicate the math. The new library handles all of these cases seamlessly, automatically adjusting its calculations for sharp corners or multiple separate boundaries. It also respects the symmetries of the shape, breaking the problem down into smaller, manageable pieces based on how the wave behaves when reflected or rotated. This means researchers can study the most intricate and chaotic systems without having to rewrite their code for every new shape they want to test. By providing a common framework where different mathematical methods can be applied to the same problem, the software allows the scientific community to cross-check their findings and build a more reliable understanding of quantum chaos.

The impact of this work extends beyond just faster calculations; it opens the door to exploring new frontiers in quantum physics. The researchers have designed the software to be easily expanded, meaning that future updates can include new types of billiards, such as those influenced by magnetic fields or moving in curved spaces. They have already laid the groundwork for studying time-dependent systems and other related problems. By making this powerful tool freely available to everyone, the authors hope to accelerate the pace of discovery in the field of quantum chaos, allowing more scientists to tackle the deep questions about how order emerges from chaos and how the quantum world connects to our everyday experience. The result is a robust, high-performance engine for exploring the hidden patterns of the quantum universe, turning a decades-old computational bottleneck into a streamlined path for discovery.

Technical Summary of QuantumBilliards.jl: An extensible, user friendly, high-performance Julia library for the large scale computation of spectra and eigenstates of quantum billiards and related problems

Problem Statement
The paper addresses the numerical solution of the two-dimensional interior Dirichlet Helmholtz eigenproblem, a paradigmatic model in quantum chaos. The objective is to determine discrete eigenwavenumbers $k_n$ and corresponding eigenfunctions $\psi_n$ for a particle confined within a bounded domain $B$ with Dirichlet boundary conditions ( $\psi|_{\partial B} = 0$ ). While classical billiard dynamics are well-studied, the quantum generalization requires solving the stationary Schrödinger equation (equivalent to the Helmholtz equation) in the semiclassical regime ( $k \to \infty$ ). This regime presents significant computational challenges: the number of required collocation points grows linearly with $k$ , geometric singularities (corners) demand specialized treatment, and standard "sweep" methods (scanning wavenumbers to find local minima of tension norms) become computationally prohibitive for large-scale studies requiring dense spectra and accurate eigenstates.

Methodology
The authors present QuantumBilliards.jl, an open-source Julia library that unifies two primary numerical formulations:

Basis-Expansion Methods: The wavefunction is approximated by a linear combination of basis functions that satisfy the Helmholtz equation in free space (e.g., real plane waves, Fourier-Bessel functions, or corner-adapted functions). The boundary conditions are enforced numerically by minimizing a "tension" norm on the boundary.
Boundary-Integral Methods (BIM): The problem is reduced to a nonlinear operator equation on the one-dimensional boundary using free-space Green's functions. This leads to a Fredholm matrix eigenproblem $A(k)\mu = 0$ , where $\mu$ is the layer density. The library implements both Double-Layer Potential (DLP) and Combined-Field Integral Equation (CFIE) formulations, the latter being more robust against spurious resonances.

To overcome the inefficiency of direct wavenumber sweeping, the library integrates four accelerated eigensolvers that extract multiple eigenvalues within a spectral window simultaneously:

Vergini–Saraceno (VS) Scaling Method: A basis-based method that uses a Taylor expansion of a weighted tension norm around a reference wavenumber to solve a generalized eigenvalue problem, extracting a cluster of nearby states.
Expanded Boundary Integral Method (EBIM): Applies the same local spectral expansion idea directly to the nonlinear boundary-integral equation, expanding the Fredholm matrix to second order.
Beyn's Contour-Integral Method: Treats the problem as a nonlinear eigenvalue problem, extracting all eigenvalues enclosed by a contour in the complex $k$ -plane using moment matrices derived from contour integration.
Compact Rational Krylov (CORK) Method: Approximates the frequency dependence of the boundary operator using Chebyshev polynomials within a window and solves the resulting polynomial eigenvalue problem via a compact linearization that reduces storage costs.

The library employs Kress quadrature to handle logarithmic singularities in the integral kernels and supports graded parametrizations for piecewise-smooth boundaries with corners. It also exploits discrete symmetries to reduce computational cost and handles multiply connected domains via composite solvers.

Key Contributions

Unified Framework: The primary contribution is a single, modular Julia framework that supports both basis and boundary-integral formulations, allowing different numerical approaches to be applied to the same geometry and cross-validated.
Extensibility: Leveraging Julia's multiple dispatch, the code architecture separates the API from specific solver implementations. This allows for the easy addition of new basis sets, solvers, or geometries (e.g., magnetic billiards, hyperbolic metrics) without rewriting core infrastructure.
Comprehensive Post-Processing: The library provides tools to reconstruct physical quantities from the raw numerical output, including:
- Normalized boundary functions (outward normal derivatives).
- Interior wavefunctions on arbitrary grids.
- Poincaré–Husimi (PH) phase-space representations, essential for analyzing quantum scars and state localization.
High-Performance Implementation: The library is designed for large-scale calculations, capable of computing thousands of eigenstates in the semiclassical regime. It includes optimized backends for evaluating special functions (Hankel functions) and handling dense matrix operations.

Results and Benchmarks
The paper demonstrates the library's capabilities through several examples:

Geometries: Successful computation of spectra and eigenstates for smooth domains (stadium, limaçon), polygonal domains (triangles, Veech polygons), and domains with divided phase spaces (mushroom, annular, Robnik billiards).
Accuracy: The library correctly reproduces known spectral statistics (e.g., GOE for chaotic systems, intermediate statistics for Veech triangles) and visualizes phenomena such as quantum scarring and eigenstate localization.
Performance: Benchmarks on a 64-core AMD processor show that the accelerated solvers can compute over 100,000 eigenvalues in the range $500 \le k \le 1400$ in approximately 48 to 69 minutes (corresponding to runtimes of 2875s, 3338s, and 4125s for the Vergini–Saraceno, Beyn, and CORK solvers, respectively). The Vergini–Saraceno solver achieved the highest throughput (~~37 eigenvalues/s), followed by Beyn (~~32/s) and CORK (~26/s). The authors note that while direct sweeps are useful for validation, the accelerated methods are necessary for production-scale spectral calculations.
Completeness: A Weyl-law completeness test confirms that the computed spectra are complete, with no systematic drift in the spectral staircase compared to the theoretical prediction.

Significance
The paper positions QuantumBilliards.jl as a critical tool for the study of quantum chaos, addressing the increasing demand for large ensembles of highly excited eigenstates and accurate phase-space representations. By providing a common environment where different high-frequency quantization methods can be applied to the same problem, the library facilitates rigorous cross-validation of numerical results. The authors emphasize that the library is not limited to a single numerical formulation but serves as a "ground layer" for future extensions to related problems, such as dielectric micro-cavities and time-dependent billiards. It complements existing scientific computing ecosystems (e.g., DynamicalSystems.jl, SpectralStatistics.jl) and aims to lower the barrier to entry for high-performance quantum billiard simulations.

Technical Summary of QuantumBilliards.jl: An extensible, user friendly, high-performance Julia library for the large scale computation of spectra and eigenstates of quantum billiards and related problems

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