Nonadiabatic fluctuation in the measured geometric phase

We study how the nonadiabatic effect causes the observable fluctuation in the “geometric phase” for a two-level system, which is defined as the experimentally measurable quantity in the adiabatic limit. From the Rabi exact solution to this model, we give a reasonable explanation to the experimental discovery of phase fluctuation in the superconducting circuit system [P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. Göppl, J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J. Schoelkopf, and A. Wallraf, Science 318, 1889 (2007)], which seemed to be regarded as the conventional experimental error.

Figure 1

(Color online) (a) Schematic of a TLS in a static magnetic field along with a microwave field. (b) Realized effective Hamiltonian rotating in the parametric space.

Figure 2

(Color online) (a) Comparison between the Berry’s phase (blue solid line) ${\varphi }_B$ and nonadiabatic phase (red cross) ${\varphi }_{na}$ with $n=1$ circular rotation in each round. (b) The discrepancy between them with red solid line for the numerical result and black dashed line for the second-order approximation.

Figure 3

(Color online) Comparison between deviations induced by a low-frequency noise (${\sigma }_{\varphi }$, red lines) and the nonadiabatic effect ($\left|\Delta \varphi \right|$, green lines), with solid lines for $n=1$ and cross lines for $n=1.5$.

Figure 4

(Color online) Comparison between different numbers of rotation loops $n$, with black dotted line for $n=1$, blue dashed line for $n=2$, red dashed-dotted line for $n=3$, and green solid line for $n=4$.