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Physical Review A

Geometrical phase imprinted on eigenfunctions near an exceptional point

Soo-Young Lee

Published Dec 16, 2010• 14 citations
Publisher

Abstract

We illustrate how to get the geometric phase from eigenfunctions in the vicinity of an exceptional point in a dielectric microcavity whose non-Hermitian character comes from the outgoing-wave boundary condition. It is shown that the geometrical phase $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ can be obtained either from total variation of the inner product of eigenfunctions or from a continuous change of phase plot, not of intensity plot, during a double cyclic parameter variation encircling the exceptional point. One can use either of the two ways by properly choosing the arbitrary phase of the calculated eigenfunctions.

DOI
10.1103/physreva.82.064101

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We illustrate how to get the geometric phase from eigenfunctions in the vicinity of an exceptional point in a dielectric microcavity whose non-Hermitian character comes from the outgoing-wave boundary condition. It is shown that the geometrical phase \ifmmode±\else\textpm\fi\ensuremathπ\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}\ifmmode±\else\textpm\fi\ensuremathπ can be obtained either from total variation of the inner product of eigenfunctions or from a continuous change of phase plot, not of intensity plot, during a double cyclic parameter variation encircling the exceptional point. One can use either of the two ways by properly choosing the arbitrary phase of the calculated eigenfunctions.
Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.
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 $\ensuremath{\pi}$ 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 $\ensuremath{\pi}$ 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.
For an arbitrary possibly non-Hermitian matrix Hamiltonian H that might involve exceptional points, we construct an appropriate parameter space M and line bundle Ln over M such that the adiabatic geometric phases associated with the eigenstates of the initial Hamiltonian coincide with the holonomies of Ln. We examine the case of 2×2 matrix Hamiltonians in detail and show that, contrary to claims made in some recent publications, geometric phases arising from encircling exceptional points are generally geometrical and not topological in nature.
We consider the geometric phase and quantum tunneling in the vicinity of diabolic and exceptional points. We show that the geometric phase associated with the degeneracy points is defined by the flux of complex magnetic monopoles. In the limit of weak coupling, the leading contribution to the real part of the geometric phase is given by the flux of the Dirac monopole plus a quadrupole term, and the expansion of the imaginary part starts with a dipole-like field. For a two-level system governed by a generic non-Hermitian Hamiltonian, we derive a formula to compute the non-adiabatic, complex, geometric phase by integrating over the complex Bloch sphere. We apply our results to study a dissipative two-level system driven by a periodic electromagnetic field and show that, in the vicinity of the exceptional point, the complex geometric phase behaves like a step-function. Studying the tunneling process near and at the exceptional point, we find two different regimes: coherent and incoherent. The coherent regime is characterized by Rabi oscillations, with a one-sheeted hyperbolic monopole emerging in this region of the parameters. The two-sheeted hyperbolic monopole is associated with the incoherent regime. We show that the dissipation results in a series of pulses in the complex geometric phase which disappear when the dissipation dies out. Such a strong coupling effect of the environment is beyond the conventional adiabatic treatment of the Berry phase.
In the standard (Hermitian) formalism of quantum mechanics, usually when a potential parameter is varied the crossing of energy levels with the same symmetry is avoided. However, within the framework of the non-Hermitian formalism it is possible that two (or even more) complex eigenvalues with the same symmetry will cross. At the crossing point the eigenvalues are degenerate and this is accompanied by the coalescence of the eigenfunctions (or eigenvectors). Therefore, we may term this special situation as a non-Hermitian degeneracy. This special situation is associated with a branch point in the complex energy plane which is commonly termed an “exceptional point” in the spectrum of the non-Hermitian Hamiltonian. With respect to the c-product defined for non-Hermitian operators (matrices) in Chapter 6, the degenerate eigenstate is self-orthogonal. Since a branch point in the spectrum is removed by any infinitesimally small external perturbation, it seems to be inaccessible experimentally and may be considered just as a mathematical object rather than a physical one. However, as we will show here, by varying the potential parameters the existence of a branch point is reflected in the measurement of the geometrical phases also known as Berry phases. It should be stressed here that while in our case the geometrical phase results from a coalescence of eigenfunctions of a non-Hermitian Hamiltonian, the so-called Berry phase phenomenon occurs also within the Hermitian formalism of quantum mechanics when the eigenvalues of the molecular Hamiltonian in the Born–Oppenheimer approximation are degenerate for specific geometry of the poly-atomic molecule.
Exceptional points are singularities in the spectrum of non-Hermitian systems in which several eigenvectors are linearly dependent and their eigenvalues are equal to each other. Usually it is assumed that the order of the exceptional point is limited by the number of degrees of freedom of a non-Hermitian system. In this letter, we refute this common opinion and show that non-Markovian effects can lead to dynamics characteristic of systems with exceptional points of higher orders than the number of degrees of freedom in the system. This takes place when the energy returns from reservoir to the system such that the dynamics of the system are divided into intervals in which it describes by the product of the exponential and a polynomial function of ever-increasing order. We demonstrate that by choosing the observation time, it is possible to observe exceptional points of arbitrary high orders.