Chaos-Assisted Tunneling in Microcavities

• Chaos-assisted tunneling (CAT) is a quantum phenomenon where energy transitions occur from stable, high-Q modes confined in regular islands to chaotic regions via dynamical tunneling.
• It facilitates directional emission in deformed microcavities by channeling light from rectangle-orbit modes into leaky regions governed by unstable manifolds.
• Experimental implementations using GaAs/AlGaAs microcavities and selective current injection validate CAT's role in producing controlled, high-brightness, directional optical outputs.

Chaos-assisted tunneling (CAT) is a phenomenon in quantum and wave systems with mixed regular–chaotic phase space, whereby quantum transitions occur between classically disconnected regular regions (“islands”) through intermediate chaotic states. In the context of optical microcavities, CAT underpins the coupling of light from high-quality factor (high-Q) modes localized on stable periodic ray orbits into chaotic ray dynamics, resulting in emission patterns (including highly directional emission) that cannot be explained by classical or integrable models. The key mechanism is that dynamical tunneling from the stable island into the surrounding chaotic sea enables energy or probability flow to leaky regions of phase space, which is fundamentally wave-mechanical in origin and provides a paradigmatic setting for observing and controlling quantum chaos effects in a solid-state experiment.

1. Chaotic Microcavity Geometry and Ray Dynamics

The system of interest is a semiconductor microcavity laser with a deliberately deformed disk geometry to induce predominantly chaotic internal ray dynamics. The boundary is specified in polar coordinates as

$$r(\phi) = R [1 + a \cos(2\phi) + b \cos(4\phi) + c \cos(6\phi)],$$

where $(a, b, c)$ are small deformation parameters (e.g., $a = 0.1$, $b = 0.01$, $c = 0.012$). The refractive index is $n = 3.3$ for GaAs/AlGaAs.

A phase-space (Poincaré surface of section; SOS) analysis in Birkhoff coordinates $(s, \sin\theta)$, with $s$ the normalized arc length along the boundary and $\theta$ the angle of incidence, reveals much of the phase space is chaotic. Crucially, regular islands persist, especially a dominant period-4 island corresponding to a rectangular periodic orbit fully confined by total internal reflection (TIR), defined by $\sin\theta > 1/n$.

2. Dynamical Tunneling and High-Q Modes

Optical modes (“rectangle-orbit modes”) efficiently localize on the stable period-4 island, forming high-Q resonances (with $Q \sim 10^5$–$10^6$) whose rays are confined by TIR. In the absence of nonintegrability, emission would be minimal.

Dynamical tunneling, in this context, permits wavefunctions concentrated on the regular island to leak—even though the Kolmogorov–Arnold–Moser (KAM) tori separate it from the chaotic sea classically. In Husimi phase-space projections, the rectangle-orbit mode’s wavefunction exhibits a distinct “tail” extending into the chaotic sea, not as a uniform background but funneled by the system's unstable manifolds—those associated with nearby unstable periodic points (period-3 in this case).

The presence of these tails is critical because, while the main mode profile remains trapped, the probability amplitude carried along unstable manifolds is routed toward phase space regions where TIR fails ($\sin\theta < 1/n$), thus serving as an open channel for light escape.

3. Directional Emission Mechanism: CAT and Unstable Manifolds

The emission mechanism proceeds in two interdependent stages:

1. Tunneling: Quantum light leaks from the regular island (rectangle-orbit) into the chaotic sea by dynamical tunneling (CAT), mediated by the inherent wave nature of light.
2. Chaotic Routing & Escape: Once in the chaotic region, light follows the unstable manifolds of nearby unstable periodic orbits toward boundary points $s \approx 0.04$ and $0.54$, where the incidence angle drops below the critical TIR threshold, allowing refractive escape.

These escape points, when mapped onto the emission pattern, concentrate far-field emission at $\phi \approx \pm90^\circ$ with a characteristic divergence angle ($\sim 30^\circ$), producing highly directional emission unexpected from the geometry of the rectangle orbit alone.

The observed interference oscillations in the far field arise from coherent emission from both escape points, with an angular period

$\Delta\theta = 360^\circ / (k d),$

where $d = 2 R (1 + a + b + c)$ is the effective width and $k$ the free-space wavenumber.

4. Experimental Implementation and Selective Excitation

Device fabrication employs a GaAs/AlGaAs quantum-well structure shaped into the deformed disk cavity. A critical aspect of the experimental setup is selective current injection using an electrode patterned strictly along the rectangle-orbit path, with current-injection margin parameters $M < 1$ implementing a spatial filter to favor rectangle-orbit modes and suppress competing modes (notably whispering-gallery modes).

Selective excitation is validated by:

• Far-field emission showing strong peaks at $\phi \approx \pm90^\circ$, consistent with theoretical predictions based on CAT and ray dynamics.
• Near-field boundary imaging via CCD revealing bright spots at predicted escape locations, corresponding to the unstable manifold intersection points.

The quantitative correspondence between measured emission directions, calculated mode patterns, and the structure of unstable manifolds directly verifies the hypothesis of chaos-assisted tunneling-derived emission.

5. Theoretical and Experimental Evidence for Chaos-Assisted Tunneling

Analytical, numerical, and experimental evidence jointly establish that:

• The primary emission channel is granted not by ordinary leakage, but by dynamical tunneling from the TIR-confined period-4 rectangle orbit to chaotic, leaky phase space.
• Theoretical analysis based on the Husimi function, the geometry of unstable manifolds, and computations of the SOS are quantitatively consistent with the measured emission directions and intensities.
• Experimental control via electrode geometry demonstrates that directional emission is not an artifact of boundary roughness or mode mixing, but a robust feature resulting from CAT.

The experimental system thus constitutes conclusive evidence for CAT in a solid-state optical system.

6. Implications for Photonic Device Engineering and Quantum Chaos

CAT in microcavities demonstrates that quantum chaos is not only observable but can be exploited to engineer photonic devices with emission properties unattainable by conventional design. Directional emission patterns, mode lifetimes, and field distributions can all be shaped by leveraging dynamical tunneling through controlled modifications of the phase-space structure.

This also provides a classical–quantum correspondence roadmap: while the classical ray picture predicts certain invariances and forbidden transitions, the inclusion of wave effects and phase-space mixing fundamentally changes the allowed emission routes and efficiency.

The characterization of directional emission in microcavity lasers via CAT opens avenues for compact, high-brightness light sources with tailored emission, as well as for wave-dynamical investigations in quantum chaos.

Mechanism	Classical Scenario	CAT–Driven Wave Scenario
Emission pathway	Only at TIR-leaky regions, predictable	Assisted by tunneling via chaos, allowed
Emission direction	Dictated by periodic orbits	Determined by unstable manifolds, often unexpected
Mode Q factor	Extremely high for TIR-confined modes	Degraded by CAT, but still high

7. Summary and Broader Context

Chaos-assisted tunneling provides a robust mechanism linking wave chaos theory with practical photonic device physics. In the deformed disk microcavity system, CAT mediates the transfer of optical energy from well-confined high-Q modes to chaotic rays, resulting in directional, refractingly-coupled output. The detailed agreement between experiment and theory demonstrates that precise engineering of mixed phase-space structures enables control over tunneling-induced emission. These results extend the understanding of dynamical tunneling well beyond atomic physics, validating its significance for mesoscopic, photonic, and quantum technological platforms (Shinohara et al., 2010).