Scattering Experiments with Microwave Billiards at an Exceptional Point under Broken Time Reversal Invariance

Scattering experiments with microwave cavities were performed and the effects of broken time-reversal invariance (TRI), induced by means of a magnetized ferrite placed inside the cavity, on an isolated doublet of nearly degenerate resonances were investigated. All elements of the effective Hamiltonian of this two-level system were extracted. As a function of two experimental parameters, the doublet and also the associated eigenvectors could be tuned to coalesce at a so-called exceptional point (EP). The behavior of the eigenvalues and eigenvectors when encircling the EP in parameter space was studied, including the geometric amplitude that builds up in the case of broken TRI. A one-dimensional subspace of parameters was found where the differences of the eigenvalues are either real or purely imaginary. There, the Hamiltonians were found PT-invariant under the combined operation of parity (P) and time reversal (T) in a generalized sense. The EP is the point of transition between both regions. There a spontaneous breaking of PT occurs

Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation

We report on the experimental study of an exceptional point (EP) in a dissipative microwave billiard with induced time-reversal invariance (T) violation. The associated two-state Hamiltonian is non-Hermitian and non-symmetric. It is determined experimentally on a narrow grid in a parameter plane around the EP. At the EP the size of T violation is given by the relative phase of the eigenvector components. The eigenvectors are adiabatically transported around the EP, whereupon they gather geometric phases and in addition geometric amplitudes different from unity

Induced Time-Reversal Symmetry Breaking Observed in Microwave Billiards

Using reciprocity, we investigate the breaking of time-reversal (T) symmetry due to a ferrite embedded in a flat microwave billiard. Transmission spectra of isolated single resonances are not sensitive to T-violation whereas those of pairs of nearly degenerate resonances do depend on the direction of time. For their theoretical description a scattering matrix model from nuclear physics is used. The T-violating matrix elements of the effective Hamiltonian for the microwave billiard with the embedded ferrite are determined experimentally as functions of the magnetization of the ferrite.Comment: 4 pages, 4 figure

Chaotic Scattering in the Regime of Weakly Overlapping Resonances

We measure the transmission and reflection amplitudes of microwaves in a resonator coupled to two antennas at room temperature in the regime of weakly overlapping resonances and in a frequency range of 3 to 16 GHz. Below 10.1 GHz the resonator simulates a chaotic quantum system. The distribution of the elements of the scattering matrix S is not Gaussian. The Fourier coefficients of S are used for a best fit of the autocorrelation function if S to a theoretical expression based on random--matrix theory. We find very good agreement below but not above 10.1 GHz

Quantum Chaotic Scattering in Microwave Resonators

In a frequency range where a microwave resonator simulates a chaotic quantum billiard, we have measured moduli and phases of reflection and transmission amplitudes in the regimes of both isolated and of weakly overlapping resonances and for resonators with and without time-reversal invariance. Statistical measures for S-matrix fluctuations were determined from the data and compared with extant and/or newly derived theoretical results obtained from the random-matrix approach to quantum chaotic scattering. The latter contained a small number of fit parameters. The large data sets taken made it possible to test the theoretical expressions with unprecedented accuracy. The theory is confirmed by both, a goodness-of-fit-test and the agreement of predicted values for those statistical measures that were not used for the fits, with the data

PT symmetry and spontaneous symmetry breaking in a microwave billiard

We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP

Towards an analytical framework of science communication models

This chapter reviews the discussion in science communication circles of models for public communication of science and technology (PCST). It questions the claim that there has been a large-scale shift from a ‘deficit model’ of communication to a ‘dialogue model’, and it demonstrates the survival of the deficit model along with the ambiguities of that model. Similar discussions in related fields of communication, including the critique of dialogue, are briefly sketched. Outlining the complex circumstances governing approaches to PCST, the author argues that communications models often perceived to be opposed can, in fact, coexist when the choices are made explicit. To aid this process, the author proposes an analytical framework of communication models based on deficit, dialogue and participation, including variations on each

Encircling an Exceptional Point

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.Comment: RevTex 4.0, four eps-figures (low resolution

Observation of a Chiral State in a Microwave Cavity

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 $\pi/2 .$ We conclude that the state excited at the singularity has a definitiv chirality.Comment: RevTex 4, 5 figure

Analyzing symmetry breaking within a chaotic quantum system via Bayesian inference

Bayesian inference is applied to the level fluctuations of two coupled microwave billiards in order to extract the coupling strength. The coupled resonators provide a model of a chaotic quantum system containing two coupled symmetry classes of levels. The number variance is used to quantify the level fluctuations as a function of the coupling and to construct the conditional probability distribution of the data. The prior distribution of the coupling parameter is obtained from an invariance argument on the entropy of the posterior distribution.Comment: Example from chaotic dynamics. 8 pages, 7 figures. Submitted to PR