Full counting statistics applied to dissipative Cooper pair pumping

We calculate the charge transport in a flux-biased dissipative Cooper pair pump using the method of full counting statistics (FCS). This is used instead of a more traditional technique of integrating a very small expectation value of the instantaneous current over the pumping period. We show that the rotating wave approximation (RWA), which fails in the traditional technique, produces accurate results within the FCS method.

Figure 1

(a) Flux-biased Cooper pair pump consisting of two SQUIDs enclosing a superconducting island. The SQUIDs are controlled by external fluxes ${\Phi }_i$ and ${\Phi }_{\mathrm{ext}}$. The charge on the island is controlled by the gate voltage $V_g$. (b) Evolution of parameters $E_{J1}$, $E_{J2}$, and $n_g$ during a pumping cycle of period $\tau =80\mathrm{ps}$.

Figure 2

Cooper pair pump as in Fig. 1. An infinite capacitor $C_m$ has been added as a gedanken measuring device of the transferred charge. The phase ${\phi }_m$ across $C_m$ serves as a counting field.

Figure 3

(a) Charge $Q_P$ pumped through the system as a function of time. The whole time span of the plot corresponds to a single pumping cycle. For all curves, $k_{B}T/\left(2\pi \hslash \right)=2\mathrm{GHz}$, while the coupling strengths are chosen to be $E_C^2{\gamma }_0=0.2$ (solid), $E_C^2{\gamma }_0=0.1$ (dashed), and $E_C^2{\gamma }_0=0.05$ (dotted). As expected, charge is only pumped when the gate voltage is varied. The weaker the coupling to the environment, the stronger is the decrease of the pumped charge. (b) The total charge $Q_{\mathrm{tot}}$ transferred through the system. The temperature is $k_{B}T/\left(2\pi \hslash \right)=2\mathrm{GHz}$, and it is plotted for different coupling strengths $E_C^2{\gamma }_0=0.2$ (solid), $E_C^2{\gamma }_0=0.1$ (dashed), and $E_C^2{\gamma }_0=0.05$ (dotted). The total charge also tends to decrease as the coupling strength is decreased.

Figure 4

(a) Energy splitting within the two-level approximation during a single pumping cycle. (b) The matrix element $|\langle {\varphi }_e\left(t\right)|{\sigma }_z|{\varphi }_g\left(t\right)\rangle |$ is plotted for the time span of a single pumping cycle. The dissipation rate is determined by this matrix element and reaches its maximum at times where the energy splitting reaches its minimum.

Figure 5

(a) Pumped charge $Q_P$ as a function of coupling strength $E_C^2{\gamma }_0$. The charge is plotted for different temperatures $k_{B}T/\left(2\pi \hslash \right)=0,1,2,3\mathrm{GHz}$ (from top to bottom). The solid red lines represent calculations for perfectly closable SQUIDs, i.e., $E_J^{\min }=0$, whereas the dashed black lines were made for realistic SQUIDs. The weaker the coupling to the environment, the more the pumped charge tends to decrease. In (b), the total charge is depicted as a function of the coupling strength for different temperatures $k_{B}T/\left(2\pi \hslash \right)=0,1,2,3\mathrm{GHz}$ (from top to bottom).