From Quantum Chaos to Classical Chaos via Gain-Induced Measurement Dynamics in a Photon Gas

TL;DR

Problem addressed: how classical chaos with sensitivity to initial conditions emerges from quantum dynamics in chaotic systems, where unitary evolution lacks exponential diverging trajectories. Approach: identify gain-induced measurement dynamics in a chaotic photon gas confined to a gravitational wedge, using non-resonant pumping to trigger mode competition and resonant pumping as a control. Key findings: nonlinearly amplified gain selects a single chaotic eigenmode, producing Born-rule-like statistics and an irreversible outcome; this yields sensitivity to initial conditions and an operational mechanism for quantum-classical transition. They formalize a generalized Born-rule-like selection P_i ≈ G_i|a_i(0)|^2 / Σ_j G_j|a_j(0)|^2 with growth factors G_i ≈ exp(3(g_i-κ_i)t^*) and support it with numerical tests and a stability-map framework. Significance: shows that essential aspects of quantum measurement can emerge from intrinsic gain dynamics, providing a concrete physical mechanism for the quantum-classical boundary in chaotic systems and informing interpretations of quantum measurement.

Abstract

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.

Figures (10)

• Figure 1: Gravitational wedge for two-dimensional light. (a) Schematic of the experimental setup. The microcavity consists of two planar mirrors, one of which is nanostructured to create an effective wedge potential for the cavity photons. The cavity can be driven resonantly (at 655 nm) or optically pumped via the optical medium (at 532 nm). A spatial light modulator (SLM) and a motorized mirror allow for adjusting the position of the laser spots along the cavity plane. Part of the cavity light is transmitted through one of the mirrors and imaged onto a camera, giving access to the system's state. One of the cavity mirrors is tilted along the y-direction to create an effective gravitational potential. (b) Two excitation schemes are used in the experiment. For optical pumping (top), the optical medium is excited non-resonantly to induce a lasing process. This process amplifies modes depending on their overlap with the gain region (i.e., the pump spot). In the case of resonant driving of the cavity (bottom), modes are excited that have both spectral and spatial overlap with the driving field. (c) Microscope image of the nanostructured mirror surface used to create the wedge potential for the photons. The wedge is characterized by an opening angle $\theta$. (d) Numerically calculated trajectories of classical particles in a gravitational wedge. Depending on $\theta$, the trajectories are regular ($\theta=45{^\circ}$) or chaotic ($\theta=55{^\circ}$). In other cases ($\theta=35{^\circ}$), both types of motion coexist ('mixed' regime), and the behavior then depends on the initial conditions.
• Figure 2: Regular and chaotic mode patterns. The images show experimentally obtained mode patterns for a wedge opening angle of $\theta=55{^\circ}$ (top row) and $\theta=45{^\circ}$ (bottom row). The cavity is excited by optical pumping, with the pump spot position indicated by the red circle. Unlike regular patterns, chaotic patterns are ergodic, exploring and effectively filling the entire (phase) space that is accessible at a given energy.
• Figure 3: Stability map in a $\theta=35{^\circ}$ gravitational wedge. The top row shows a map of the entropy derived from the particle density distribution measured at different pump spot positions. High entropy values correspond to chaotic eigenstates, while low values indicate regular ones. The middle and bottom rows show the density distributions and camera images for specific pump spots, indicated by distinct symbols, each corresponding to either regular or chaotic motion. For chaotic probability density functions, Porter--Thomas fits are shown.
• Figure 4: Sensitivity to pump spot position inside a chaotic $\theta=55{^\circ}$ wedge billiard. (a) To test for sensitivity to initial conditions, we scan the excitation laser along an equipotential line using a fixed step size and compare neighboring particle density patterns by calculating their Pearson correlation coefficient (see Methods). Excitation occurs either through resonant driving or non-resonant optical pumping. (b-d) Correlation histograms derived from comparisons of neighboring density patterns for three different step sizes: 11 µ m, 3.7 µ m, and 1.2 µ m. The histograms are normalized to relative occurrence. With decreasing step size, correlations increase under resonant driving. For non-resonant optical pumping, low-correlation events remain frequent, indicating jumps between uncorrelated eigenmodes.
• Figure S1: Stochastic trajectories for two modes following a numerical simulation of Eq. \ref{['eq:laser']}.