Microwave billiards with broken time reversal invariance

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Abstract
We consider a microwave resonator with three single-channel waveguides attached. One of these serves to couple waves into and out of the resonator; the remaining two are connected to form a one-way handle so as to break time reversal invariance. The poles of the input - output scattering coefficient of such a resonator are shown to be the eigenvalues of a non-Hermitian effective `Hamiltonian' , the anti-Hermitian part of which has rank 1 and is responsible for the breaking of time reversal invariance. All of the spectral statistics recently observed for such a microwave billiard are reproduced quantitatively by taking H and as random matrices. In particular, the distribution of nearest-neighbour spacings of the resonances is close to that of the GUE when H belongs to the GOE corresponding to a Sinai shape of the resonator; linear level repulsion results when H belongs to the Poissonian ensemble as it corresponds to a rectangular resonator.