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Signatures of Classical Chaos in a Non-Hermitian Quantum System

Violetta Sharoglazova, Marius Puplauskis, Lotte Hof, Jan Klaers

Published Jan 1, 2024• 0 citations
OpenAlex

Abstract

Quantum mechanics is considered the most fundamental theory for describing nature. Classical mechanics is thought to arise from quantum mechanics in the limit of high quantum numbers. Unlike quantum mechanics, classical mechanics allows for chaotic motion, characterized by sensitivity to initial conditions, a feature ambiguous for translation to quantum systems. Here, we investigate the onset of classical chaos in a non-Hermitian quantum system. We study photons in a microcavity, treated as two-dimensional quantum particles moving in the plane of the resonator. By nanostructuring the resonator's mirrors, we can confine the photons in arbitrary potential landscapes, particularly billiard-like potentials. Non-resonant optical pumping inside of these potentials creates gain in the optical medium. We anticipate that introducing latter will amplify modes with maximum overlap with the gain region, introducing sensitivity to initial conditions, specifically to the pumping spot position. This investigation is expected to shed light on the quantum-to-classical transition in a chaotic system.

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Quantum mechanics is considered the most fundamental theory for describing nature. Classical mechanics is thought to arise from quantum mechanics in the limit of high quantum numbers. Unlike quantum mechanics, classical mechanics allows for chaotic motion, characterized by sensitivity to initial conditions, a feature ambiguous for translation to quantum systems. Here, we investigate the onset of classical chaos in a non-Hermitian quantum system. We study photons in a microcavity, treated as two-dimensional quantum particles moving in the plane of the resonator. By nanostructuring the resonator's mirrors, we can confine the photons in arbitrary potential landscapes, particularly billiard-like potentials. Non-resonant optical pumping inside of these potentials creates gain in the optical medium. We anticipate that introducing latter will amplify modes with maximum overlap with the gain region, introducing sensitivity to initial conditions, specifically to the pumping spot position. This investigation is expected to shed light on the quantum-to-classical transition in a chaotic system.
Understanding how classical and quantum descriptions of nature connect remains one of the central challenges in physics. Classical mechanics provides a deterministic picture for macroscopic systems, while quantum mechanics is governed by probabilistic behavior at microscopic scales. In the intermediate regime — where quantum effects clash with classical intuition, such as tunneling through forbidden regions or the appearance of chaos-like features in quantum systems — new experimental insight is essential. This thesis investigates two questions: How fast can a particle move within a classically forbidden region? and In what ways can classical chaos leave signatures in quantum systems? Both problems are explored experimentally using an optical microcavity filled with dye, in which confined photons behave as massive particles. This platform provides offers high-resolution spatial and temporal access to quantities that are difficult to measure in traditional quantum systems. In the first part of this thesis, we examine the kinematics of classically forbidden motion, where the particle energy is lower than the potential energy of a barrier or a potential step. While classical mechanics forbids such motion, quantum mechanics allows particles to explore these regions due to an effectively negative local kinetic energy, raising the long-standing question of tunneling time: How long does it take to traverse a barrier? We address this by engineering a system in which the speed associated with forbidden motion can be derived and directly measured. Different theoretical approaches to tunneling time are compared, and we discuss how our results relate to and challenge existing models, with particular attention to Bohmian mechanics. The second half of this thesis focuses on the emergence of classical chaos signatures in quantum systems. While the correspondence principle suggests that quantum behavior should resemble classical physics in certain limits, quantum systems are fundamentally linear and therefore cannot exhibit chaos in the classical sense. Nonetheless, under specific conditions – such as non-resonant optical pumping leading to lasing – we observe sensitivity to initial conditions in a billiard-shaped potential. By combining novel experimental techniques with foundational theoretical questions, this thesis contributes to the ongoing dialogue between classical and quantum physics.
How classical chaos emerges from quantum mechanics remains a central open question, as the unitary evolution of isolated quantum systems forbids exponential sensitivity to initial conditions. A key insight is that this quantum-classical link is provided by measurement processes. In this work, we identify gain competition in a chaotic photon gas as an operational quantum measurement that selects single motional modes from an initial superposition through stochastic, nonlinear amplification. We show that this mechanism naturally gives rise to classical chaotic behavior, most notably sensitivity to initial conditions. Our results provide a concrete physical mechanism for the quantum-classical transition in a chaotic system and demonstrate that essential aspects of quantum measurement-state projection, Born-rule-like selection, and irreversibility-can naturally emerge from intrinsic gain dynamics.
This chapter provides an overview of chaotic billiard lasers as a prominent branch of quantum chaos. These lasers offer an ideal experimental platform for demonstrating the principles of quantum chaos within a physical system. We begin by introducing the fundamental principles of chaotic ray dynamics in optical microcavities, where the transition from regular to fully chaotic dynamics fundamentally alters the underlying wavefunctions and lasing properties. A central focus is placed on "chaos-assisted light emission," which serves as a practical manifestation of "chaos-assisted tunneling" -- a hallmark phenomenon in the study of quantum chaos. We discuss both theoretical frameworks and experimental validations, demonstrating how chaotic orbits facilitate the coupling between evanescently localized modes and far-field emission. Furthermore, exploring how the presence of a gain medium influences established results from quantum chaos research remains a fundamental and intriguing problem in physics. To address this, we establish a rigorous and comprehensive derivation of the Maxwell-Bloch equations for two-dimensional microcavity lasers, specifically examining their application to fully chaotic, stadium-shaped billiard lasers. By bridging the gap between nonlinear lasing processes and chaotic wavefunctions, this chapter highlights the unique potential of chaotic billiards for controlling light-matter interactions and shaping the next generation of unconventional coherent light sources.
Quantum mechanics is the most fundamental theory describing nature, and according to the correspondence principle, classical mechanics should emerge from it. However, in areas like chaotic motion, the connection between classical and quantum descriptions is not straightforward. A key challenge is understanding how classical chaotic behaviour, especially sensitivity to initial conditions, arises from quantum mechanics. Measurement processes are expected to drive such transitions, yet they raise many unresolved questions, known collectively as the measurement problem—including whether wavefunction collapse is a real physical process. Here, we experimentally study the quantum-to-classical transition in a two-dimensional billiard system. Position-dependent gain in this system acts similarly to a measurement, reducing quantum superpositions to eigenstates. By varying the gain position, we observe sensitivity to initial conditions via mode jumps. Our experiment offers a concrete example of how measurement-like processes can yield classical chaotic behaviour in a quantum system. Unlike idealized axiomatic measurements, our process is a tangible physical phenomenon, offering new insight into the nature of quantum measurements.
How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.
Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom-field system as a quantum-classical hybrid with dynamically coupled quantum and classical egrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. It manifest itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with a prominent fractal-like behavior and in the quantum atom-filed degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom-field resonance correlate strongly. This result provides a quantum-classical correspondence in a closed physical system.