MICROWAVE BILLIARDS WITH BROKEN TIME-REVERSAL INVARIANCE

We consider a microwave resonator with three single-channel waveguides attached. One of these serves to couple waves into and out of the res onator; the remaining two are connected to form a one-way handle so as to break time reversal invariance. The poles of the input-output scat tering coefficient of such a resonator are shown to be the eigenvalues of a non-Hermitian effective 'Hamiltonian' H-eff = H - i Gamma, the a nti-Hermitian part Gamma of which has rank 1 and is responsible for th e breaking of time reversal invariance. All of the spectral statistics recently observed for such a microwave billiard are reproduced quanti tatively by taking H and Gamma 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 resonato r.