Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure

Geometric phase around exceptional points

A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\pi$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\pi$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.Comment: 4 pages, 1 figure, with revisions in the introduction and conclusio

The Longitudinal Effects of STEM Identity and Gender on Flourishing and Achievement in College Physics

Background. Drawing on social identity theory and positive psychology, this study investigated women’s responses to the social environment of physics classrooms. It also investigated STEM identity and gender disparities on academic achievement and flourishing in an undergraduate introductory physics course for STEM majors. 160 undergraduate students enrolled in an introductory physics course were administered a baseline survey with self-report measures on course belonging, physics identification, flourishing, and demographics at the beginning of the course and a post-survey at the end of the academic term. Students also completed force concept inventories and physics course grades were obtained from the registrar. Results. Women reported less course belonging and less physics identification than men. Physics identification and grades evidenced a longitudinal bidirectional relationship for all students (regardless of gender) such that when controlling for baseline physics knowledge: (a) students with higher physics identification were more likely to earn higher grades; and (b) students with higher grades evidenced more physics identification at the end of the term. Men scored higher on the force concept inventory than women, although no gender disparities emerged for course grades. For women, higher physics (versus lower) identification was associated with more positive changes in flourishing over the course of the term. High-identifying men showed the opposite pattern: negative change in flourishing was more strongly associated with high identifiers than low identifiers. Conclusions. Overall, this study underlines gender disparities in physics both in terms of belonging and physics knowledge. It suggests that strong STEM identity may be associated with academic performance and flourishing in undergraduate physics courses at the end of the term, particularly for women. A number of avenues for future research are discussed

Information Geometry of Complex Hamiltonians and Exceptional Points

Information geometry provides a tool to systematically investigate the parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric

Breakdown of adiabatic transfer of light in waveguides in the presence of absorption

In atomic physics, adiabatic evolution is often used to achieve a robust and efficient population transfer. Many adiabatic schemes have also been implemented in optical waveguide structures. Recently there has been increasing interests in the influence of decay and absorption, and their engineering applications. Here it is shown that even a small decay can significantly influence the dynamical behaviour of a system, above and beyond a mere change of the overall norm. In particular, a small decay can lead to a breakdown of adiabatic transfer schemes, even when both the spectrum and the eigenfunctions are only sightly modified. This is demonstrated for the generalization of a STIRAP scheme that has recently been implemented in optical waveguide structures. Here the question how an additional absorption in either the initial or the target waveguide influences the transfer property of the scheme is addressed. It is found that the scheme breaks down for small values of the absorption at a relatively sharp threshold, which can be estimated by simple analytical arguments.Comment: 8 pages, 7 figures, revised and extende

On Nonlinear Dynamics of the Pendulum with Periodically Varying Length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Two types of transitions to chaos of the pendulum depending on problem parameters are investigated numerically.Comment: 8 pages, 8 figure