Adiabatic Evolution and Geometric Phases in a non-Abelian non-Hermitian system

In this work, we investigate the adiabatic evolution and geometric phase in non-Abelian non-Hermitian systems. By analyzing the adiabatic condition for non-Hermitian systems exhibiting energy degeneracy, we derive the non-Abelian geometric phases for such systems. By studying an example, we show how the adiabatic evolution can be realized in a four-level non-reciprocal system with two degenerate eigenenergies by choosing parameters to realize real eigenenergies, slowly enough varying parameters and Hermitian non-Abelian vector potentials. By analyzing the effects of nonreciprocity on adiabatic evolution, populations on energy levels, and geometric phases, we develop a scheme to manipulate the dynamic behavior of the population, geometric phases, and adiabatic conditions by nonreciprocity. Our results provide insights into potential applications in quantum technologies that exploit non-Abelian phases and non-Hermitian characteristics.