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

Exploring the formation of thermally detuned transverse patterns in a broad-area square VCSEL

The formation of thermally detuned transverse patterns of a broad-area square-aperture vertical-cavity surface-emitting laser (VCSEL) via cryogenic cooling is explored. It is found that the transverse wave vector gradually rotates from the horizontal or vertical direction to the diagonal direction of the square boundary as the transverse mode order increases. A model based on the quantum billiards with a finite potential well is developed to emulate the transition behavior of lasing modes. Combining the effective modal gain analysis with the response wave function of driven finite potential billiards, all experimental lasing patterns under different operation temperatures are well reconstructed.