The probability current statistics of two-dimensional open chaotic ballistic billiards is studied both analytically and numerically. Assuming that the real and imaginary parts of the scattering wave function are both random Gaussian fields, we find a universal distribution function for the probability current. In by-passing we recover previous analytic forms for wave function statistics. The expressions bridge the entire region from GOE to GUE type statistics. Our analytic expressions are verified numerically by explicit quantum-mechanical calculations of transport through a Bunimovich billiard. PACS 05.45.Mt,05.60.Gg,73.23.Ad

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We present numerical results for ballistic-electron quantum transport through weakly open integrable circle and chaotic stadium billiards. The geometry of the pair of conducting leads is chosen in accordance with recent experiments for semiconductor microstructures I Marcus et al , Phys. R. ev. Lett. 69, 506 (1992)].The conductance as a function of the Fermi wave number displays characteristic noisy fluctuations for both the integrable and the chaotic systems. We show that structures in the conductance autocorrelation function as a function of the Fermi wave number are related to short-length classical orbits. This correspondence permits incorporation of effects of phase decoherence due to incoherent scattering into the quantum calculation.During the last decade, remarkable advances in semiconductor fabrication technology have made it possible to produce quantum devices of submicrometer size. ' The size of such mesoscopic devices is large on the atomic scale but comparable to or smaller than the phase-coherence length l~d ue to inelastic electron-electron (and electron-phonon) scattering at low temperatures ((0.1 K). Both experimentally and theoretically, there has been considerable interest in the electronic and magnetic properties of such systems and as a result, a number of physical phenomena have been observed and predicted including universal conductance fiuctuations, weak localization, and Aharonov-Bohm conductance oscillations. One of the most interesting aspects is chaotic boundary scattering in systems formed in highmobility GaAs/Al"Ga, As heterostructures, where the system size is less than the elastic mean free path of electrons. In such systems, the two-dimensional motion of noninteracting electrons confined by a wall of arbitrary geometry is ballistic. In the ballistic regime, the shape of walls directly determines the motion of electrons colliding with them, and the classical "billiard" model successfully explains the experimental results of those systems. Nonlinear dynamics of the electrons play an important role in quantum transport through the microstructures (see, e. g., Ref. 6, and references therein).One of the prototypes of conservative chaotic systems is the "stadium billiard. " The study of its quantum mechanics has played an important role in the identification of quantum manifestations of classical chaos. ' Studies of opened billiards pertaining to transport through mesoscopic devices have only recently begun. ' Major progress has been made in relating quantum conductance fluctuations with classical (non)integrable dynamics' ' employing Miller's semiclassical formulation of the S matrix. Generic fingerprints of regular and chaotic dynamics were identified that are related to long classical orbits and are universal (i.e., system independent). Several quantum calculations (e.g., Refs. 6, 13, and 17) were performed for a symmetric geometry for which entrance and exit leads are on opposite sides of the device facing each other. Very recently, Marcus et al. ' reported experimental results of...