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arXiv:2604.12741

Quantum Chaos in Phase Space

Martina Hentschel

quant-phnlin.CDphysics.opticsSubmitted Apr 14, 2026• Announced Apr 15, 2026
arXivHTMLPDF

Abstract

Mesoscopic devices, with system sizes in the range of several to several dozens wavelengths, represent paradigmatic model systems for the observation of quantum chaotic behaviour based on semiclassical concepts. Those electronic and photonic billiard cavities are small enough for interference effects not to be ignored. Nonetheless, the classical ray or particle tracing picture can often provide a substantial understanding of the dynamics of the system along the lines of classical-quantum, or ray-wave correspondence. This well-established principle turns out to be particularly useful when applied not only in real space, but by extending it to phase space such that both location and momentum information can contribute to a deeper and more comprehensive understanding of the dynamical behaviour.

Comments
Chapter for the Quantum Chaos volume in 'Comprehensive Quantum Mechanics', to be published by Elsevier (Main editor: R. Mann; volume editors: S. Gnutzmann and K. Życzkowski)

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Mesoscopic devices, with system sizes in the range of several to several dozens wavelengths, represent paradigmatic model systems for the observation of quantum chaotic behaviour based on semiclassical concepts. Those electronic and photonic billiard cavities are small enough for interference effects not to be ignored. Nonetheless, the classical ray or particle tracing picture can often provide a substantial understanding of the dynamics of the system along the lines of classical-quantum, or ray-wave correspondence. This well-established principle turns out to be particularly useful when applied not only in real space, but by extending it to phase space such that both location and momentum information can contribute to a deeper and more comprehensive understanding of the dynamical behaviour.
In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between phase-space structures, reflecting the distinction between chaotic, mixed or integrable classical dynamics, and the corresponding quantum properties. Well established techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ→0\hbar \to 0ℏ→0, where the classical limit is given by Hamiltonian mechanics of particles, constitute a now standard part of the toolkit of theoretical physics. During the last years, these ideas have been extended into the field theoretical domain of systems composed of NNN indistinguishable particles, aka quantum fields, displaying a different type of semiclassical limit ℏeff=1/N→0\hbar_{\rm eff}=1/N \to 0ℏeff​=1/N→0 and accounting for genuine many-body quantum interference. The foundational concept behind this idea of many-body interference, the many-body version of the van Vleck-Gutzwillers semiclassical propagator, is explained in detail. Based on this the corresponding semiclassical many-body theory is reviewed. It provides a unified framework for understanding a variety of quantum chaotic phenomena addressed, including random-matrix spectral correlations in many-body systems, the universal morphology of many-body eigenstates, interference effects kin to mesoscopic weak localization, and the key to the scrambling of many-body correlations characterized by out-of-time-order correlators.
Experimental progress in the miniaturization of electronic devices has made routinely available in the laboratory small electronic systems, on the micrometre or sub-micrometre scale, which at low temperature are sufficiently well isolated from their environment to be considered as fully coherent. Some of their most important properties are dominated by the interaction between electrons. Understanding their behaviour therefore requires a description of the interplay between interference effects and interactions. The goal of this review is to address this relatively broad issue and more specifically to address it from the perspective of the quantum chaos community. I will therefore present some of the concepts developed in the field of quantum chaos which may be applied to the study of many-body effects in mesoscopic and nanoscopic systems. Their implementation is illustrated in a few examples of experimental relevance such as persistent currents, mesoscopic fluctuations of Kondo properties or Coulomb blockade. I will furthermore try to bring out, from the various physical illustrations, some of the specific advantages on more general grounds of the quantum chaos based approach.
The intrinsic multivaluedness of interaction process, revealed in Part I of this series of papers, is interpreted as the origin of the true dynamical (in particular, quantum) chaos. The latter is causally deduced as unceasing series of transitions, dynamically probabilistic by their origin, between the equally real, but incompatible 'realisations' (modes of interaction) of a system. The obtained set of realisations form the causally derived, intrinsically complete "space of events" providing the crucial extension of the notion of probability and the method of its first-principle calculation. The fundamental dynamic uncertainty thus revealed is specified for Hamiltonian quantum systems and applied to quantum chaos description in periodically perturbed systems. The ordinary semiclassical transition in our quantum-mechanical results leads to exact reproduction of the main features of chaotic behaviour of the system known from classical mechanics, which permits one to "re-establish" the correspondence principle for chaotic systems (inevitably lost in any their conventional, single-valued description). The causal dynamical randomness in the extended quantum mechanics is not restricted, however, to semiclassical conditions and generically occurs also in essentially quantum regimes, even though partial "quantum suppression of chaos" does exist and is specified in our description, as well as other particular types of the quantum (truly) chaotic behaviour.
We report on the numerical simulation of the double-slit experiment, where the initial wave packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
Preface. 1. Genesis of Chaos and Breakdown of Quantization of Adiabatic Invariants. 2. Semiclassical Quantization of Chaos: Trace Formula. 3. Pseudo-Chaos Without Classical Counterpart in 1-Dimensional Quantum Transport. 4. Chaos and Quantum Transport in Open Magnetic Billiards: From Stadium to Sinai Billiards. 5. Chaotic Scattering on Hyperbolic Billiards: Success of Semiclassical Theory. 6. Nonadiabaticity-Induced Quantum Chaos. 7. Level Dynamics and Statistical Mechanics. 8. Towards Time-Discrete Quantum Mechanics. 9. Conclusions and Prospects. Index.
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.