Appearance of Topological Phases in Superconducting Nanocircuits

We construct non-Abelian geometric transformations in superconducting nanocircuits, which resemble in properties the Aharonov-Bohm phase for an electron transported around a magnetic flux line. The effective magnetic fields can be strongly localized, and the path is traversed in the region where the energy separation between the states involved is at maximum, so that the adiabaticity condition is weakened. In particular, we present a scheme of topological charge pumping.

Figure 1

(a) The system composed of two “charge qubits,” where the state is encoded in the number of Cooper pairs on the superconducting islands (black nodes), i.e., $n_1,n_2$. The voltages and the magnetic fluxes can be controlled in a cyclic way in order to perform non-Abelian geometric transformations. (b) A simplified setup for topological charge pumping.

Figure 2

(a) Ground-state charging diagram and (b) electrostatic energies corresponding to the thick line in (a). The black node denotes hereafter the starting point. Inset: nonzero Josephson coupling mixes the states so that there is an avoided level crossing near the “triple points” in (a).

Figure 3

(a) Parameter-dependent spectrum and (b) effective magnetic field. The minimum in the energy spacing corresponds to the maximum in the field strength. Variation of the parameters around the peak results in a phase shift between $|\overline{0}⟩$ and $|\overline{1}⟩$. (c),(d) Similar characteristics corresponding to the phase shift between $|+⟩$ and $|-⟩$. The contours denote the most robust paths.

Figure 4

Dependence of the phase acquired during one turn around the region of enhanced field (a) and localization of the field (b) as functions of $J_u/J_d$.