Floquet theory of Cooper pair pumping

We derive a general formula for the charge pumped in a superconducting nanocircuit. Our expression generalizes previous results in several ways; it is applicable in both the adiabatic and in the nonadiabatic regimes and it takes into account also the effect of an external environment. More specifically, by applying Floquet theory to Cooper pair pumping, we show that under a cyclic evolution the total charge transferred through the circuit is proportional to the derivative of the associated Floquet quasi-energy with respect to the superconducting phase difference. In the presence of an external environment the expression for the transferred charge acquires a transparent form in the Floquet representation. It is given by the weighted sum of the charge transferred in each Floquet state, the weights being the diagonal components of the stationary density matrix of the system expressed in the Floquet basis. To test the power of this formulation we apply it to the study of pumping in a Cooper pair sluice. We reproduce the known results in the adiabatic regime and we show new data in the nonadiabatic case.

Figure 1

A generic setup of a Cooper pair pump. Two superconducting leads, kept at a phase bias $\phi ={\phi }_R-{\phi }_L$, are connected through a Josephson network. The system is operated in a regime where quantum effects are important. To have a pumped charge, some external parameters (e.g., gate voltages or magnetic fluxes) are varied periodically.

Figure 2

Circuit scheme of the Cooper pair sluice.

Figure 3

The variation in a cycle of $n_g$, $J_L$, and $J_R$. These latter are expressed in units of $E_C$. In this graph $n_g$ varies between $n_{g\text{min}}=0.2$ and $n_{g\text{max}}=0.8$, $J_L$ and $J_R$ vary between $J_{\text{min}}=0.003E_C$ and $J_{\text{max}}=0.1E_C$.

Figure 4

Pumped charge vs. $\phi$ in the adiabatic case computed with the Floquet theory (red crosses) confronted with the same quantity computed with the analytical expression (24) (black solid line), $0⩽\phi ⩽2\pi$. It is $J_{\text{min}}/J_{\text{max}}=0.03$, $J_{\text{max}}=0.1E_C$, ${\mathrm{TE}}_C/\hslash =8400$. Charge is in units of $e$. The slight scattering of the points is a numerical artifact due to intrinsic difficulties in looking at the system dynamics for very long times.

Figure 5

Pumped charge vs. $\phi$ with ${\mathrm{TE}}_C/\hslash =2.1$ for different values of $J_{\text{min}}$, $0⩽\phi ⩽2\pi$. Charge is in units of ${10}^{-5}e$.

Figure 6

Population $\rho {}_{\alpha \alpha }^{\text{st}}$ of the Floquet state with the lower quasi-energy vs. temperature $\Theta$ (in its dimensionless form) for different values of $\phi$. We are in the adiabatic limit (${\mathrm{TE}}_C/\hslash =8400$ and $\Delta E_{\text{min}}\simeq E_C/10$). Temperature is expressed in units of $E_C/k_B$. Inset: The same graph magnified to show the dependence on $\phi$.

Figure 7

Modulus of the pumped charge vs. temperature for different values of $\phi$. We are in the adiabatic limit (${\mathrm{TE}}_C/\hslash =8400$ and $\Delta E_{\text{min}}\simeq E_C/10$). Temperature is expressed in units of $E_C/k_B$ and charge in units of $e$.