We report on a microwave cavity experiment where exceptional points (EPs), which are square root singularities of the eigenvalues as function of a complex interaction parameter, are encircled in the laboratory. The real and imaginary parts of an eigenvalue are given by the frequency and width of a resonance and the eigenvectors by the field distributions. Repulsion of eigenvalues--always associated with EPs--implies frequency anticrossing (crossing) whenever width crossing (anticrossing) is present. The eigenvalues and eigenvectors are interchanged while encircling an EP, but one of the eigenvectors undergoes a sign change which can be discerned in the field patterns.

A microwave experiment has been realized to measure the phase difference of the oscillating electric field at two points inside the cavity. The technique has been applied to a dissipative resonator which exhibits a singularity -called exceptional point -in its eigenvalue and eigenvector spectrum. At the singularity, two modes coalesce with a phase difference of π/2 . We conclude that the state excited at the singularity has a definitiv chirality.PACS numbers: 05.45. Mt, 41.20.Jb, 03.65.Vf, Recently a surprising phenomenon occurring in systems described by non-hermitian Hamiltonians has received considerable attention: the coalescence of two eigenmodes. If the system depends on some interaction parameter λ , the value λ EP at which the coalescence occurs is called an exceptional point (EP) [1]. At an EP, the eigenvalues and eigenvectors show branch point singularities [1,2,3,4,5] [3,13]. So far EPs have been observed in decaying systems described by a complex symmetric effective Hamiltonian [14]. While the theoretical and experimental articles cited above discuss the properties of systems in the vicinity of an EP, a recent theoretical work [15] investigates the complex symmetric Hamiltonian of a two-level system at the EP. The eigenfunction at the EP turns out to be( 1) for any choice of the basis states |1 and |2 . This is a chiral state: in quantum, acoustical and electromagnetic systems the two orthogonal basis states oscillate in time; if they are superimposed according to Eq. (1) -where they follow each other with a time lag of a quarter period -the result is rotating either clockwise or counter-clockwise. This is in analogy with the

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We calculate analytically the geometric phases that the eigenvectors of a parametric dissipative two-state system described by a complex symmetric Hamiltonian pick up when an exceptional point (EP) is encircled. An EP is a parameter setting where the two eigenvalues and the corresponding eigenvectors of the Hamiltonian coalesce. We show that it can be encircled on a path along which the eigenvectors remain approximately real and discuss a microwave cavity experiment, where such an encircling of an EP was realized. Since the wavefunctions remain approximately real, they could be reconstructed from the nodal lines of the recorded spatial intensity distributions of the electric fields inside the resonator. We measured the geometric phases that occur when an EP is encircled four times and thus confirmed that for our system an EP is a branch point of fourth order.

We report on first experimental signatures for chaos-assisted tunneling in a two-dimensional annular billiard. Measurements of microwave spectra from a superconducting cavity with high frequency resolution are combined with electromagnetic field distributions experimentally determined from a normal conducting twin cavity with high spatial resolution to resolve eigenmodes with properly identified quantum numbers. Distributions of quasidoublet splittings serve as basic observables for the tunneling between whispering gallery-type modes localized to congruent, but distinct tori which are coupled weakly to irregular eigenstates associated with the chaotic region in phase space.

The escape of an ensemble of particles from the Bunimovich stadium via a small hole has been studied numerically. The decay probability starts out exponentially but has an algebraic tail. The weight of the algebraic decay tends to zero for vanishing hole size. This behaviour is explained by the slow transport of the particles close to the marginally stable bouncing ball orbits. It is contrasted with the decay function of the corresponding quantum system.

We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of nongeneric modes and from unstable periodic orbits of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.

Two superconducting microwave billiards have been electromagnetically coupled in a variable way. The spectrum of the entire system has been measured and the spectral statistics analyzed as a function of the coupling strength. It is shown that the results can be understood in terms of a random matrix model of quantum mechanical symmetry breaking -- as e.g. the violation of parity or isospin in nuclear physics.Comment: 4 pages, 5 figure