Complex magnetic monopoles, geometric phases and quantum evolution in vicinity of diabolic and exceptional points

TL;DR

This paper explores the behavior of geometric phases and quantum tunneling near diabolic and exceptional points, revealing complex monopole structures and topological transitions in non-Hermitian two-level systems influenced by dissipation.

Contribution

It introduces a novel framework for calculating complex geometric phases in non-Hermitian systems and analyzes the topological and dynamical effects near exceptional points.

Findings

Complex geometric phase is linked to flux of complex magnetic monopoles.

Near exceptional points, the geometric phase exhibits step-like behavior.

Dissipation induces pulses in the geometric phase, disappearing when dissipation ceases.

Abstract

We consider the geometric phase and quantum tunneling in 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 monopole. In weak-coupling limit the leading contribution to the real part of geometric phase is given by the flux of the Dirac monopole plus quadrupole term, and the expansion for its imaginary part starts with the dipolelike field. For a two-level system governed by the generic non-Hermitian Hamiltonian, we derive a formula to compute the non-adiabatic complex geometric phase by integral over the complex Bloch sphere. We apply our results to to study a two-level dissipative system driven by periodic electromagnetic field and show that in the vicinity of the exceptional point the complex geometric phase behaves as step-like function. Studying tunneling process near and at exceptional point, we find two different regimes: coherent and incoherent. The coherent regime is characterized by the Rabi oscillations and one-sheeted hyperbolic monopole emerges in this region of the parameters. In turn with the incoherent regime the two-sheeted hyperbolic monopole is associated. The exceptional point is the critical point of the system where the topological transition occurs and both of the regimes yield the quadratic dependence on time. We show that the dissipation brings into existence of pulses in the complex geometric phase and the pulses are disappeared when dissipation dies out. Such a strong coupling effect of the environment is beyond of the conventional adiabatic treatment of the Berry phase.