# Quantum chaos and electron transport properties in a quantum waveguide

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We numerically investigate electron transport properties in an electron waveguide which can be constructed in 2DEG of the heterostructure of GaAs and AlGaAs. We apply R-matrix theory to solve a Schrödinger equation and construct a S-matrix, and we then calculate conductance of an electron waveguide. We study single impurity scattering in a waveguide. A [delta]-function model as a single impurity is very attractive, but it has been known that [delta]-function potential does not give a convergent result in two or higher space dimensions. However, we find that it can be used as a single impurity in a waveguide with the truncation of the number of modes. We also compute conductance for a finite size impurity by using R-matrix theory. We propose an appropriate criteria for determining the cut-off mode for a [delta]-function impurity that reproduces the conductance of a waveguide when a finite impurity presents. We find quantum scattering echoes in a ripple waveguide. A ripple waveguide (or cavity) is widely used for quantum chaos studies because it is easy to control a particle's dynamics. Moreover we can obtain an exact expression of Hamiltonian matrix with for the waveguide using a simple coordinate transformation. Having an exact Hamiltonian matrix reduces computation time significantly. It saves a lot of computational needs. We identify three families of resonance which correspond to three different classical phase space structures. Quasi bound states of one of those resonances reside on a hetero-clinic tangle formed by unstable manifolds and stable manifolds in the phase space of a corresponding classical system. Resonances due to these states appear in the conductance in a nearly periodic manner as a function of energy. Period from energy frequency gives a good agreement with a prediction of the classical theory. We also demonstrate wavepacket dynamics in a ripple waveguide. We find quantum echoes in the transmitted probability of a wavepacket. The period of echoes also agrees with the classical predictions. We also compute the electron transmission probability through a multi-ripple electron waveguide. We find an effect analogous to the Dicke effect in the multi-ripple electron waveguide. We show that one of the S-matrix poles, that of the super-radiant resonance state, withdraws further from the real axis as each ripple is added. The lifetime of the super-radiant state, for N quantum dots, decreases as [1/N] . This behavior of the lifetime of the super-radiant state is a signature of the Dicke effect.