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arXiv:quant-ph/0501040 · Physical Review A

Geometric phase around exceptional points

Alexei A. Mailybaev, Oleg N. Kirillov, Alexander P. Seyranian

Published Jul 20, 2005• 181 citations
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Abstract

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.

DOI
10.1103/physreva.72.014104

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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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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.
We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.
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.
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.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
We consider the space of $n\ifmmode\times\else\texttimes\fi{}n$ non-Hermitian Hamiltonians (n=2n=2n=2, 3, ...) that are equivalent to a single $n\ifmmode\times\else\texttimes\fi{}n$ Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale $T=1/\ensuremath{\varepsilon}$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}$ times an expansion in powers of ${\ensuremath{\varepsilon}}^{1/n}$. The exponential of the term of nthn\mathrm{th}nth order (which is equivalent to the ``geometric'' or Berry phase modulo $2\ensuremath{\pi}$) is thus independent of $\ensuremath{\varepsilon}$ as $\ensuremath{\varepsilon}\ensuremath{\rightarrow}0$; it depends only on the homotopy class of the loop and is an integer power of ${e}^{2\ensuremath{\pi}i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.
Unitary evolution in P\mathcal{P}PT\mathcal{T}T-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase associated with a P\mathcal{P}PT\mathcal{T}T-symmetric two-level system is derived and is interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with nontrivial phase contributions. To gain more insight, the Hermitian analog of our non-Hermitian problem is also analyzed, which results in an intriguing class of geometric-phase problems in conventional QM as well, where the Hamiltonian includes a perturbative term that is proportional to the rate of change in adiabatic parameters.