Reaction matrix for Dirichlet billiards with attached waveguides

The reaction matrix of a cavity with attached waveguides connects scattering properties to properties of a corresponding closed billiard for which the waveguides are cut off by straight walls. On the one hand this matrix is directly related to the S-matrix, on the other hand it can be expressed by a spectral sum over all eigenfunctions of the closed system. However, in the physically relevant situation where these eigenfunctions vanish on the impenetrable boundaries of the closed billiard, the spectral sum for the reaction matrix, as it was used before, fails to converge and does not reliably reproduce the scattering properties. We derive here a convergent representation of the reaction matrix in terms of eigenmodes satisfying Dirichlet boundary conditions and demonstrate its validity in the rectangular and the Sinai billiards. Recently, there has been some interest in the application of the reaction-matrix theory of Wigner and Eisenbud [1] and the projection-operator formalism of Feshbach [2], originally developed for the description of nuclear collisions, to chaotic cavities with attached scattering channels [3, 4, 5, 6, 7]. Such models are frequently used as paradigms of chaotic scattering [8, 9, 10, 11] and found important experimental realizations by electron transport through open quantum dots [12, 13], lasing optical micro-cavities [14, 15, 16] and scattering of microwaves in resonators with attached waveguides [17, 6]. It is known that the quantum scattering in the open system shows signatures of the classical dynamics in the closed system. For example, conductance fluctuations of open quantum dots are different for systems whose closed counterparts have integrable, fully chaotic or mixed phase space [12, 13]. In fact many statistical results for quantum chaotic scattering rely on this connection as for their derivation an ad-hoc formulation of the scattering problem in terms of an effective non-Hermitian Hamiltonian is used [2, 21], which describes the dynamics inside a closed system and an additional coupling of the corresponding eigenstates to some scattering channels. Classical chaos enters via the random-matrix assumption for the Hermitian part of this Hamiltonian [18, 19, 20]. However, while the effective Hamiltonian appears naturally within the formalism from ‡ [email protected] Reaction matrix for Dirichlet billiards 2 nuclear physics, it is not a priori clear when this formalism applies to some given billiard system and how the parameters of the two different models are related. Therefore it is important and interesting to develop a thorough understanding of the connection between the scattering properties of billiards, in particular the S-matrix, and the properties of the corresponding closed system, i. e. spectrum and eigenfunctions. For example, today it is well known that the spectrum can be found from a secular equation involving the S-matrix [8, 9, 10]. For scattering from the outside of convex billiards in R2 this scattering approach to quantization allows even for a mathematically rigorous formulation [22]. Unfortunately, the opposite direction is less profoundly understood. Here, the unitary S-matrix is related to a Hermitian reaction matrix, and this can in turn be expressed as a sum over the internal spectrum with coefficients reflecting the behavior of the internal wavefunction at the boundary separating billiard and waveguide. From the physical point of view, and in particular for the aforementioned applications [12, 13, 14, 15, 16, 17], the most natural situation is a wavefunction which vanishes outside the billiard and on the boundary (Dirichlet b. c.): Electrons in a quantum dot are depleted from the boundary by a high negative gate voltage, the radiation field is restricted to the optical cavity by total internal reflection or additional mirrors, and in microwave resonators the metallic walls do not admit the electrical field. However, for this of all choices of boundary conditions no consistent representation of the reaction matrix as a spectral sum is known. We derived a formal expression [3, 4] but found both numerically and from semiclassical estimates that it fails to converge. Recently, this conclusion was confirmed with different methods [7]. It is certainly possible to circumvent this problem: One option is to ignore the divergence and to restrict the spectral sum by hand to some set of levels. Naturally this must fail in a generic situation, but it can give reasonable results in some special cases, e. g. when the spectrum has a doublet or band structure [7]. A second option relies on the fact that in principle all boundary conditions providing a self-adjoint Hamiltonian are admissible for defining the closed system, see [7] for a nice explanation of this point. In particular Neumann b. c. with finite wave function but zero derivative at the interface were considered previously [3, 4, 5, 6, 7]. In this case the spectral sum converges, and the resulting reaction matrix successfully reproduces numerical or experimental scattering data. Nevertheless it would be very unsatisfying, could the formulation of a proper reaction-matrix theory for billiards not be based on the physically relevant boundary conditions. To remedy this situation we establish in the present paper a convergent expansion of the reaction matrix for Dirichlet billiards with attached waveguides. Our main result is Eq. (15) below, where the coupling constants entering this series are given. Before we get to this equation, we recall some results from previous work. Following Eq. (15) we give a derivation of this formula, discuss some interesting aspects related to it and show with two examples how it works. We consider the same ”frying-pan” setup as in Refs. [9, 4], i. e. a cavity with an Reaction matrix for Dirichlet billiards 3