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

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

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.Comment: 23 pages, 7 figure

Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry

A PT-symmetric optically active medium that lases at the threshold gain also acts as a complete perfect absorber at the laser wavelength. This is because spectral singularities of PT-symmetric complex potentials are always accompanied by their time-reversal dual. We investigate the significance of PT-symmetry for the appearance of these self-dual spectral singularities. In particular, using a realistic optical system we show that self-dual spectral singularities can emerge also for non-PT-symmetric configurations. This signifies the existence of non-PT-symmetric CPA-lasers.Comment: 11 pages, 3 figures, 1 table, accepted for publication in J. Phys.

Detecting level crossings without looking at the spectrum

In many physical systems it is important to be aware of the crossings and avoided crossings which occur when eigenvalues of a physical observable are varied using an external parameter. We have discovered a powerful algebraic method of finding such crossings via a mapping to the problem of locating the roots of a polynomial in that parameter. We demonstrate our method on atoms and molecules in a magnetic field, where it has implications in the search for Feshbach resonances. In the atomic case our method allows us to point out a new class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. In the case of molecules, it enables us to find curve crossings with practically no knowledge of the corresponding Born-Oppenheimer potentials.Comment: 4 pages, new title, no figures, accepted by Phys. Rev. Let

Perturbation theory of PT-symmetric Hamiltonians

In the framework of perturbation theory the reality of the perturbed eigenvalues of a class of \PTsymmetric Hamiltonians is proved using stability techniques. We apply this method to \PTsymmetric unperturbed Hamiltonians perturbed by \PTsymmetric additional interactions

Biorthogonal quantum mechanics

The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems. © 2014 IOP Publishing Ltd

Projective Hilbert space structures at exceptional points

A non-Hermitian complex symmetric 2x2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.Comment: 20 pages; discussion extended, refs added; bug correcte