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

Signatures of the correlation hole in total and partial cross sections

In a complex scattering system with few open channels, say a quantum dot with leads, the correlation properties of the poles of the scattering matrix are most directly related to the internal dynamics of the system. We may ask how to extract these properties from an analysis of cross sections. In general this is very difficult, if we leave the domain of isolated resonances. We propose to consider the cross correlation function of two different elastic or total cross sections. For these we can show numerically and to some extent also analytically a significant dependence on the correlations between the scattering poles. The difference between uncorrelated and strongly correlated poles is clearly visible, even for strongly overlapping resonances.Comment: 25 pages, 13 Postscript figures, typos corrected and references adde

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

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

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

Decay of Classical Chaotic Systems - the Case of the Bunimovich Stadium

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.Comment: 16 pages, RevTex, 3 figures are available upon request from [email protected], to be published in Phys.Rev.

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

Using the maternal-fetal genotype incompatibility test to assess non-inherited maternal HLA-DRB1 antigen coding alleles as rheumatoid arthritis risk factors

Non-inherited maternal antigens encoded by specific HLA-DRB1 alleles (NIMA) have been implicated as a rheumatoid arthritis (RA) risk factor. Using genotype data from North American Rheumatoid Arthritis Consortium study participants and the maternal-fetal genotype incompatibility (MFG) test, we find evidence for offspring allelic effects but no evidence for NIMA as a RA risk factor. We discuss possible reasons why our result conflicts with several previous studies (including one of our own) that used RA patients from northern Europe

Classical and quantum decay of one dimensional finite wells with oscillating walls

To study the time decay laws (tdl) of quasibounded hamiltonian systems we have considered two finite potential wells with oscillating walls filled by non interacting particles. We show that the tdl can be qualitatively different for different movement of the oscillating wall at classical level according to the characteristic of trapped periodic orbits. However, the quantum dynamics do not show such differences.Comment: RevTeX, 15 pages, 14 PostScript figures, submitted to Phys. Rev.