Geometric Phases

Yanxia Liu

The Weierstrass function approach is applied to investigate the fractal of geometric phase.A noninteracting 4-spin-12 system is considered,of which each spin is driven by a local and rotating magnetic field.A mixed state of this system is constructed.The adiabatic geometric phase for the mixed state is calculated.The mixed state geometric phase is studied under two conditions:① adiabatic approximation;②letting the geometric phases in the mixed state small.The geometric phase for the mixed state becomes an average of the geometric phases of the pure states,weighed by their eigenvalues.The mixed state geometric phase is constructed into the form of Weierstrass function.A novel fractal evolution of the geometric phase is found.The evolution curve of the geometric phase is presented in a special example.The box-counting dimension of the evolution curve of the geometric phase is calculated.

Juanjuan Chen, Jun‐Hong An, Qingjun Tong +2 more

We studied the geometric phase of a two-level atom coupled to an environment with Lorentzian spectral density. The non-Markovian effect on the geometric phase is explored analytically and numerically. In the weak coupling limit, the lowest order correction to the geometric phase is derived analytically and the general case is calculated numerically. It was found that the correction to the geometric phase is significantly large if the spectral width is small, and in this case the non-Markovian dynamics has a significant impact on the geometric phase. When the spectral width increases, the correction to the geometric phase becomes negligible, which shows the robustness of the geometric phase to the environmental white noises. The result is significant to the quantum information processing based on the geometric phase.

Luis A. Garza-Soto, Nathan A. Hagen, Dorilian Lopez-Mago +1 more

Since Pancharatnam's 1956 discovery of optical geometric phase and Berry's 1984 discovery of geometric phase in quantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches using the Jones calculus, or on spherical trigonometry approaches using the Poincaré sphere. The abstracted mathematics of the former and the abstracted geometry of the latter obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves and the resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by an interferogram, and provide the conditions under which the two match one another.

P. Hariharan