Coherent Cooper-pair pumping by magnetic flux control

We introduce and discuss a scheme for Cooper-pair pumping. The scheme relies on the coherent transfer of a superposition of charge states across a superconducting island and is realized by adiabatic manipulation of magnetic fluxes. Differently from previous implementations, it does not require any modulation of electrostatic potentials. We find a peculiar dependence of the pumped charge on the superconducting phase bias across the pump and that an arbitrarily large amount of charge can be pumped in a single cycle when the phase bias is $\pi$. We explain these features and their relation to the adiabatic theorem.

Figure 1

Pumping with magnetic fluxes. (a) Schematic circuit of a Cooper pair sluice. A superconducting island (green) is coupled to superconducting leads by two SQUIDs, acting as tunable Josephson junctions of energy $J_l$, $J_r$. A gate capacitively coupled to the island controls its polarization charge $n_g$. The superconducting phase of the two leads is held at a fixed difference $\phi$. (b) Representative time modulation of $J_l$, $J_r$ leading to FP. The gate position is kept fixed throughout the modulation.

Figure 2

Pumped charge in the adiabatic limit. (a) Pumped charge $Q_p$ versus offset charge $\delta n_g$ for different values of the superconducting phase bias $\phi$. (b) $Q_p$ versus $\phi$ for different values of $\delta n_g$.

Figure 3

Breakdown of adiabatic behavior. $Q_p$ versus $\delta n_g$ for $\phi =0$ (a) and $\phi =\pi$ (b), for different pumping frequencies $f$ (solid lines). The results are obtained by numerical integration of the master equation of Ref. 26. The adiabatic-limit predictions (10) and (13) are also plotted for comparison (dashed lines). Smooth pulses are used in place of those of Fig. 1b for improved adiabaticity. Relevant parameter values for the pump are $E_C=1\mathrm{K},J_{\max }=0.1E_C,J_{\min }=0.03J_{\max }$. For the fictitious environment (see Ref. 27 for details) $g=0.02$, $R=300\mathrm{k}\Omega$, $T=0$, $T_0=0.4\mathrm{K}$.

Figure 4

Instantaneous geometric currents. $I_{p,l}$ (solid line) and $I_{p,r}$ (dashed line) versus time for $\delta n_g=-0.02$ and $\phi =0$ (a), $\pi /2$ (b), and $\pi$ (c).

Figure 5

Pumping cycle and Berry phase. (a) Plots of the path drawn by the ground state of the sluice on the Bloch sphere along a pumping cycle, for $\phi =\alpha$ (blue), $\phi =\pi /2$ (red), and $\phi =\pi -\alpha$ (green), with $\alpha \ll 1$. The Berry phase is proportional to the solid angle spanned by the paths. (b) Berry phase ${\Theta }_B$ versus $\phi$ and $\delta n_g$. According to (9), the pumped charge $Q_p$ is proportional to the slope of the surface plot along the $\phi$ axis.

Figure 6

Optimal pumping frequency when $\phi =\pi$. (a) Average pumped current $\langle I_p\rangle$ versus pumping frequency $f$ with $\phi =\pi$ and $\delta n_g$ taking a set of values in the range of 0.005 and 0.1 (from left to right). The other parameters are the same as in Fig. 3. (b),(c) Optimal pumping frequency $f^{*}$ (b) and corresponding maximum pumped current $I^{*}$ (c) versus $\delta n_g$. The dashed lines in (b) and (c) are linear fits to the numeric data close to $\delta n_g=0$.