Wave functions with localizations on classical periodic orbits in weakly perturbed quantum billiards

Quantum states resembling classical periodic trajectories in mesoscopic elliptic billiards

A quantum wave function with localization on classical periodic orbits in a mesoscopic elliptic billiard has been achieved by appropriately superposing nearly degenerate eigenstates expressed as products of Mathieu functions. We analyze and discuss the rotational and librational regimes of motion in the elliptic billiard. Simplified line equations corresponding to the classical trajectories can be extracted from the quantum state as an integral equation involving angular Mathieu functions. The phase factors appearing in the integrals are connected to the classical initial positions and velocity components. We analyze the probability current density, phase maps, and vortex distributions of the periodic orbit quantum states for both rotational and librational motions; furthermore, they may represent traveling and standing trajectories inside the elliptic billiard.

Polygon Coherent Modes in a Weakly Perturbed Whispering Gallery Microresonator for Efficient Second Harmonic, Optomechanical, and Frequency Comb Generations

We observe high optical quality factor (Q) polygonal and star coherent optical modes in a lithium niobate microdisk. In contrast to the previous polygon modes achieved by deformed microcavities at lower mechanical and optical Q, we adopt weak perturbation from a tapered fiber for the polygon mode formation. The resulting high intracavity optical power of the polygon modes triggers second harmonic generation at high efficiency. With the combined advantages of a high mechanical Q cavity, we observe optomechanical oscillation in polygon modes for the first time. Finally, we observe frequency microcomb generation from the polygon modes with an ultrastable taper-on-disk coupling mechanism.

Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional integrable billiards

The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.

Strong quantum scarring by local impurities

We discover and characterise strong quantum scars, or quantum eigenstates resembling classical periodic orbits, in two-dimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum near-degeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremise the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wave-packet recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.

Fermi resonance in optical microcavities

Fermi resonance is a phenomenon of quantum mechanical superposition, which most often occurs between normal and overtone modes in molecular systems that are nearly coincident in energy. We find that scarred resonances in deformed dielectric microcavities are the very phenomenon of Fermi resonance, that is, a pair of quasinormal modes interact with each other due to coupling and a pair of resonances are generated through an avoided resonance crossing. Then the quantum number difference of a pair of quasinormal modes, which is a consequence of quantum mechanical superposition, equals periodic orbits, whereby the resonances are localized on the periodic orbits. We derive the relation between the quantum number difference and the periodic orbits and confirm it in an elliptic, a rectangular, and a stadium-shaped dielectric microcavity.

Exploring the origin of the directional emission from a microcavity with a large-aperture surface-emitting laser

The origin of the direction emission from a microcavity is investigated with a large-aperture surface-emitting laser with an equilateral-triangular shape. Experimental results reveal that a wide-spread eigenmode and a localized superscar mode can display quite similar far-field directional emissions. This experimental finding is theoretically confirmed with the quantum-billiard wave function and the approximation of the paraxial propagation.

Resonances near the classical separatrix of a weakly deformed circular microcavity

We study localized resonance patterns of a weakly deformed circular dielectric microcavity. The dominant pattern among them is detected when the classical motion of the light is on the separatrix of the effective potential. In this case, the topological shape of the pattern is automatically determined by the refractive index of the microcavity n. Also, the pattern can be interpreted as a linear combination of nearly degenerate eigenstates in the circular dielectric microcavity. Applying this analysis to a spiral-shaped microcavity, we generate quasiscarred patterns for n=2 and n=3.