This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as =u+iv. With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T. The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.

We consider a transmission through the potential relief created by a split gate constriction (quantum point contact). Simultaneously, dc and ac voltages Vup (t) = V0 + V1 cos ?t and Vdw (t) = V0 + V1 cos (?t+?) are applied to the gates. We show numerically that the in-phase ac voltages (?=0) smear the conductance steps of the stationary conductance, while the antiphase ac voltages (?=p) only shift the conductance steps. Moreover, computation of currents in probing wires connected cross to the time-periodic quantum point contact reveals a net current for ? 0,p. This implies that the Schrödinger equation described by the electron transport under the effect of the time-periodic long electrodes is equivalent to the transmission in the crossed effective magnetic and electric fields, where the in-plane magnetic field b~? is directed along the transport axis and the electric field e~? is directed perpendicular to the plane of electron transport. Then the vector e×b gives rise to the galvanomagnetic current directed cross to the electron transport. © 2007 The American Physical Society.

Similar to the Berry conjecture of quantum chaos, an elastic analogue which incorporates longitudinal and transverse elastic displacements with corresponding wave vectors is considered. The correlation functions are derived for the amplitudes and intensities of elastic displacements. A comparison to the numerics in a quarter-Bunimovich stadium demonstrates excellent agreement. © 2007 Pleiades Publishing, Ltd.

The phase correlation function for the complex random Gaussian field (x)=(x)exp[i(x)] is derived. It is compared to the numerical scattering wave function in the open Sinai billiard. © Europhysics Letters Association.

The S matrix theory with use of the effective Hamiltonian is sketched and applied to the description of the transmission through double quantum dots. The effective Hamilton operator is non-hermitian, its eigenvalues are complex, the eigenfunctions are bi-orthogonal. In this theory, singularities occur at points where two (or more) eigenvalues of the effective Hamiltonian coalesce. These points are physically meaningful: they separate the scenario of avoided level crossings from that without any crossings in the complex plane. They are branch points in the complex plane. Their geometrical features are different from those of the diabolic points. © 2007 Springer Science+Business Media, LLC.

We consider numerically the L-, T-, and X-shaped elastic waveguides with the Dirichlet boundary conditions for in-plane deformations (displacements) which obey the vectorial Navier-Cauchy equation. In the X-shaped waveguide we show the existence of a doubly degenerate bound state with frequency below the first symmetrical cutoff frequency, which belongs to the two-dimensional irreducible representation E of symmetry group C-4v. Moreover the next bound state is below the next antisymmetric cutoff frequency. This bound state belongs to the irreducible representation A(2). The T-shaped waveguide has only one bound state while the L-shaped one has no bound states.

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions ?, and define the value r? = (?/?) ?/? that characterizes the phase rigidity of the eigenfunctions ?. In the scenario with avoided level crossings, r? varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of r? is an internal property of an open quantum system. In the literature, the phase rigidity ? of the scattering wave function ?CE is considered. Since ?CE can be represented in the interior of the system by the ?, the phase rigidity ? of the ?CE is related to the r? and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity ? to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant at energies that are determined by the real part of the eigenvalues of the effective Hamiltonian. We illustrate the relation between phase rigidity ? and transmission numerically for small open cavities. © 2006 The American Physical Society.

We study the spectrum of an open double quantum dot as a function of different system parameters in order to receive information on the geometric phases of branch points in the complex plane (BPCP). We relate them to the geometrical phases of the diabolic points (DPs) of the corresponding closed system. The double dot consists of two single dots and a wire connecting them. The two dots and the wire are represented by only a single state each. The spectroscopic values follow from the eigenvalues and eigenfunctions of the Hamiltonian describing the double dot system. They are real when the system is closed, and complex when the system is opened by attaching leads to it. The discrete states as well as the narrow resonance states avoid crossing. The DPs are points within the avoided level crossing scenario of discrete states. At the BPCP, width bifurcation occurs. Here, different Riemann sheets evolve and the levels do not cross anymore. The BPCP are physically meaningful. The DPs are unfolded into two BPCP with different chirality when the system is opened. The geometric phase that arises by encircling the DP in the real plane, is different from the phase that appears by encircling the BPCP. This is found to be true even for a weakly opened system and the two BPCP into which the DP is unfolded. ©2005 The American Physical Society.

We consider two electric RLC resonance networks that are equivalent to quantum billiards. In a network of inductors grounded by capacitors, the eigenvalues of the quantum billiard correspond to the squared resonant frequencies. In a network of capacitors grounded by inductors, the eigenvalues of the billiard are given by the inverse of the squared resonant frequencies. In both cases, the local voltages play the role of the wave function of the quantum billiard. However, unlike for quantum billiards, there is a heat power because of the resistance of the inductors. In the equivalent chaotic billiards, we derive a distribution of the heat power which describes well the numerical statistics. © 2005 The American Physical Society.

We consider a system that consists of two single-quantum billiards (QBs) coupled by a waveguide and study the transmission through this system as a function of length and width of the waveguide. To interpret the numerical results for the transmission, we explore a simple model with a small number of states which allows us to consider the problem analytically. The transmission is described in the S-matrix formalism by using the non-Hermitian effective Hamilton operator for the open system. The coupling of the single QBs to the internal waveguide characterizes the 'internal' coupling strength u of the states of the system while that of the system as a whole to the attached leads determines the 'external' coupling strength v of the resonance states via the continuum (waves in the leads). The transmission is resonant for all values of v/u in relation to the effective Hamiltonian. It depends strongly on the ratio v/u via the eigenvalues and eigenfunctions of the effective Hamiltonian. The results obtained are compared qualitatively with those from simulation calculations for larger systems. Most interesting is the existence of resonance states with vanishing widths that may appear at all values of v/u. They cause zeros in the transmission through the double QB due to trapping of the particle in the waveguide. © 2005 IOP Publishing Ltd.