Alexei A. Mailybaev

Multiparameter stability theory with mechanical applications

Fundamentals of Stability Theory Bifurcation Analysis of Eigenvalues Stability Boundary of a General System Depending on Parameters Bifurcation Analysis of Roots and Stability of a Characteristic Polynomial Depending on Parameters Vibrations and Stability of a Conservative System Depending on Parameters Stability of a Linear Hamiltonian System Depending on Parameters Stability of Linear Gyroscopic Systems Depending on Parameters Mechanical Effects Related to Bifurcation of Eigenvalues and Singularities of the Stability Boundary Stability of Periodic Systems Depending on Parameters Stability Boundary of a General Periodic System Depending on Parameters Instability Domains of Oscillatory Systems with Small Parametric Excitation and Damping Stability Domains of Nonconservative Systems Under Small Parametric Excitation.

On the observability and asymmetry of adiabatic state flips generated by exceptional points

In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This ‘adiabatic flip’ effect is an outcome of the appearance of a degeneracy known as an ‘exceptional point’ inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.

Dynamically encircling an exceptional point for asymmetric mode switching.

Physical systems with loss or gain have resonant modes that decay or grow exponentially with time. Whenever two such modes coalesce both in their resonant frequency and their rate of decay or growth, an ‘exceptional point’ occurs, giving rise to fascinating phenomena that defy our physical intuition. Particularly intriguing behaviour is predicted to appear when an exceptional point is encircled sufficiently slowly, such as a state-flip or the accumulation of a geometric phase. The topological structure of exceptional points has been experimentally explored, but a full dynamical encircling of such a point and the associated breakdown of adiabaticity have remained out of reach of measurement. Here we demonstrate that a dynamical encircling of an exceptional point is analogous to the scattering through a two-mode waveguide with suitably designed boundaries and losses. We present experimental results from a corresponding waveguide structure that steers incoming waves around an exceptional point during the transmission process. In this way, mode transitions are induced that transform this device into a robust and asymmetric switch between different waveguide modes. This work will enable the exploration of exceptional point physics in system control and state transfer schemes at the crossroads between fundamental research and practical applications.

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 an 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.

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.

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.

Filtration Combustion in Wet Porous Medium

We consider the filtration combustion for configuration where air is injected behind the wave into a porous medium containing a solid fuel. The simplest flow contains planar combustion and thermal waves, each propagating with its own speed. In this work, we study such a flow, in the case where the porous medium contains initially also some amount of liquid; therefore, vaporization and condensation occur too, giving rise to a wave structure richer than in dry combustion. We find two possible sequences of waves, and we characterize the internal structure of all waves. In an example for typical parameters of in-situ combustion, we compare the analytical results with direct numerical simulations.

On Singularities of a Boundary of the Stability Domain

This paper deals with the study of generic singularities of a boundary of the stability domain in a parameter space for systems governed by autonomous linear differential equations

Transformation of Families of Matrices to Normal Forms and its Application to Stability Theory

\dot y=Ay

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

or x(m) + a1x(m-1) + . . . + amx=0. It is assumed that elements of the matrix A and coefficients of the differential equation of mth order smoothly depend on one, two, or three real parameters. A constructive approach allowing the geometry of singularities (orientation in space, magnitudes of angles, etc.) to be determined with the use of tangent cones to the stability domain is suggested. The approach allows the geometry of singularities to be described using only first derivatives of the coefficients ai of the differential equation or first derivatives of the elements of the matrix A with respect to problem parameters with its eigenvectors and associated vectors calculated at the singular points of the boundary. Two methods of study of singularities are suggested. It is shown that they are constructive and can be applied to investigate more complicated singularities for multiparameter families of matrices or polynomials. Two physical examples are presented and discussed in detail.