arXiv subjects

O. N. Kirillov

Publications and source records attributed to O. N. Kirillov.

5 recordsLinked to original sources

arXiv2010

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.

nlin.CD↗

arXiv2009

Singularities on the boundary of the stability domain near 1:1 resonance

We study the linear differential equation x' = Lx in 1:1 resonance. That is, x in R^4 and L is a 4 by 4 matrix with a semi-simple double pair of imaginary eigenvalues (ib,-ib,ib,-ib). We wish to find all perturbations of this linear system such that the perturbed system is stable. Since linear differential equations are in one to one correspondence with linear maps we translate this problem to gl(4,R). In this setting our aim is to determine the stability domain and the singularities of its boundary. The dimension of gl(4,R) is 16, therefore we first reduce the dimension as far as possible. Here we use a versal unfolding of L ie a transverse section of the orbit of L under the adjoint action of Gl(4,R). Repeating a similar procedure in the versal unfolding we are able to reduce the dimension to 4. A 3-sphere in this 4-dimensional space contains all information about the neighborhood of L in gl(4,R). In this 3-sphere the boundary of the stability domain is a surface with singularities: transverse self-intersections, Whitney umbrellas and an intersection of self-intersections where the surface has a self-tangency.

math.DS↗

arXiv2005

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 $π$ 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 $π$ 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.

quant-ph↗

arXiv2005

Coupling of eigenvalues of complex matrices at diabolic and exceptional points

The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

math-ph↗

arXiv2004

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation

The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied.

math-ph↗