The Geometric Phase

Michael Berry

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.

P. Hariharan

Arun Kumar Pati, Amitabh Joshi

We cast the nonadiabatic geometric phase in terms of the geometric distance function and the geometric length of the curve for arbitrary cyclic evolution of the quantum states. An interpretation is given to the geometric phase as the value of the integral of the contracted length of the curve along which the system traverses. It is found that for arbitrary cyclic evolution of the quantum states the geometric phase \ensuremath{\beta}(C) acquired by the system cannot be greater than the total length of the curve l(C). We have argued that the geometric phase arises because of the fundamental inequality between the length of the curve and the distance function. Finally, we have illustrated the calculation of the geometric phase based on the geometric distance function and the geometric length of the curve.

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.

J. Anandan