Non-Abelian geometric phases in ground-state Josephson devices

We present a superconducting circuit in which non-Abelian geometric transformations can be realized using an adiabatic parameter cycle. In contrast to previous proposals, we employ quantum evolution in the ground state. We propose an experiment in which the transition from non-Abelian to Abelian cycles can be observed by measuring the pumped charge as a function of the period of the cycle. Alternatively, the non-Abelian phase can be detected using a single-electron transistor working as a charge sensor.

Figure 1

(Color online) (a) Eigenenergy diagram of the so-called tripod Hamiltonian consisting of two bright states, $B$, and two degenerate dark states, $D$. (b) Eigenenergy diagram of the circuit Hamiltonian along the considered cycle. The energy difference between the ground, $g$, and excited, $e$, state is denoted by $\Delta$ and the ground state degeneracy splitting by $\delta$. In the ideal case, $\delta =0$. (c) Circuit diagram of the non-Abelian superconducting pump. Green (left) and blue (right) denote the superconducting islands and red the Josephson junctions.

Figure 2

(Color online) Parameter cycle inducing non-Abelian transformations with the stability diagram as the background. The charge state $|i,j⟩$ which minimizes the charging energy in each hexagonal area is denoted by $\left(i,j\right)$. The $z$ axis represents $E_i$ with blue in the front for $E_1$, purple on the right for $E_m$, and red on the left for $E_2$. The SQUID energies can be closed down to $E_i^{\text{min}}$, which is zero in the ideal cycle. The inset shows the parameter cycle projected into the gate-voltage plane.

Figure 3

(Color online) (a) Adiabaticity as a function of the cycle period in a cycle with $E_i^{\text{max}}=-0.4E_C$ and $E_i^{\text{min}}=E_i^{\text{max}}/k$. Red (blue) denotes a cycle starting from the initial state $|1,0⟩$ $\left(|0,1⟩\right)$ for $k=1000$ (dots) and $k=5000$ (dashed line). (b) The pumped charges corresponding to (a). (c) Adiabaticity of the ground (red) and excited states (blue) in a cycle with all the SQUID energies fixed to $E_i^{\text{max}}=E_i^{\text{min}}=-0.4E_C$. (d) The pumped charges corresponding to (c). In (b) and (d) the gray lines denote the geometric pumped charges in the perfectly adiabatic limits. The shaded areas indicate the cycle periods with which the evolution is adiabatic. The charging energy $E_{C_i}$ used is experimentally realizable 0.2 meV (Ref. 22) and the phase across the device is fixed to zero, $\phi =0$.