Cooper-pair current in the presence of flux noise

We study the effect of flux noise on the Cooper pair current of a superconducting charge pump. We generalize the definition of the current in order to take into account the contribution induced by the environment. It turns out that this dissipative current vanishes for charge noise but it is finite in general for noise operators that do not commute with the charge operator. We discuss in a generic framework the effect of flux noise and present a way to engineer it by coupling the system to an additional external circuit. We calculate numerically the pumped charge through the device by solving the master equation for the reduced density matrix of the system and show how it depends on the coupling to the artificial environment.

Figure 1

(a) Circuit diagram of the Cooper-pair sluice (left) and the artificial environment circuit (right). The experimentally controlled parameters are $J_{L,R}$ and $n_g$, $\varphi =2\pi \Phi /{\Phi }_0$ is the phase difference across the device, and ${\Phi }_0$ is the flux quantum. The system is coupled to the artificial environment through mutual inductance $M$. The engineered environment is composed of a resistor $R$ associated with a voltage source $\delta V$, an inductor $L$, and an experimentally controllable SQUID. The SQUID allows us to change the effective noise spectrum of the environment. (b) Time dependence of the parameters $J_{L,R}$ and $n_g$ during a Cooper-pair pumping cycle.

Figure 2

(a) Spectral density function of the phase bias noise as a function of the magnetic flux through the control SQUID $\phi /{\Phi }_0$ for frequency ${\omega }_m=1.7\times {10}^{10}$ s${}^{-1}$. Inset: A zoom near the resonance point $\phi /{\Phi }_0=1/2$. Dashed lines show the region delimited by the minimum and maximum $S_{\varphi }({\omega }_m,\phi )$. Its width is given by $L_0/\left(\pi L\right)$. (b) The ratio $S_{\varphi }(\omega ,\phi )/S_{\varphi }(\omega ,0)$ for $\phi /{\Phi }_0=0.5$ (solid line), $\phi /{\Phi }_0=0.4991$ (dashed line), and $\phi /{\Phi }_0=0.5003$ (dot-dashed line). The shaded region denotes the frequency range spanned during the typical pumping cycle shown in Fig. 1b: $1.7\times {10}^{10}$s${}^{-1}⩽\omega ⩽7.9\times {10}^{10}$s${}^{-1}$. The parameters of the artificial noise are $R=30$ $\Omega$, $R_S=500$ $\Omega$, $C_S=50$ fF, $I_C=25$ $\mu$A, and $L=M=0.69$ nH.

Figure 3

(a) Dynamic dissipative currents $I^{D,\mathrm{diss}}$ (solid line) and geometric dissipative currents $I^{G,\mathrm{diss}}$ (dashed line) normalized to the maximum geometric current $I_{\max }^G$ during the pumping cycle. The maximum geometric current during one cycle is $I_{\max }^G=0.5$ nA and the control SQUID of the environmental circuit is open with $\phi /{\Phi }_0=1$. (b) Pumped charge as a function of the flux through the control SQUID of the artificial environment circuit. Inset: Behavior near one of the resonance points. Dashed lines are determined from Fig. 2a and they are located at the minimum $\phi /{\Phi }_0=1/2$ and at $\phi /{\Phi }_0=1/2\pm L_0/\left(\pi L\right)$ of the spectral density function. In numerical simulations, the phase bias is ${\varphi }_0=\pi /2$, the pumping frequency is $f=150$ MHz, and the device parameters are $J_i^M/E_{\mathrm{C}}=0.1$, $J_i^m/J_i^M=0.03$ (with $i=L,R$), $n_g^M=0.8$ $n_g^m=0.2$, and $E_{\mathrm{C}}/k_{\mathrm{B}}=1$ K [$E_{\mathrm{C}}/\left(2\pi \hslash \right)=21$ GHz].