Decoherence in Adiabatic Quantum Evolution: Application to Cooper Pair Pumping

One of the challenges of adiabatic control theory is the proper inclusion of the effects of dissipation. Here we study the adiabatic dynamics of an open two-level quantum system deriving a generalized master equation to consistently account for the combined action of the driving and dissipation. We demonstrate that in the zero-temperature limit the ground state dynamics is not affected by environment. As an example, we apply our theory to Cooper pair pumping, which demonstrates the robustness of ground state adiabatic evolution.

Figure 1

An example of a quantum pump, the Cooper pair sluice, is shown on the left. A pumping cycle is sketched on the right. The time-dependent classical control parameters are the magnetic fluxes tuning the Josephson tunnel couplings $J_L$ and $J_R$ and the gate voltage controlling the offset charge $n_g$ of the island. They vary in time with period $T_p$, whereas the phase difference across the device, $\phi$, is stationary.

Figure 2

Pumped charge of the sluice under gate charge noise. (a) Phase dependence of the pumped charge $Q^G$ in a fully symmetric pumping cycle with respect to $J_L$ and $J_R$ of Fig. 1 as a function of the phase bias $\phi$. The dashed line shows the analytic result of Eq. (8) for adiabatic pumping. The solid lines are from the numerical calculations based on Eqs. (3, 4) for $f\equiv T_p^{-1}=100\mathrm{MHz}$, with $C_g/C_{\Sigma }=0.015$, 0.0175, 0.02, 0.025, and 0.3 from bottom to top. (b) Coupling dependence of the pumped charge at $\phi =\pi /2$. The dashed line shows the analytic result as in (a). The solid lines are for $f=10$, 100, 150, 200, and 300 MHz from top to bottom. The other parameters are $J_{\max }/E_C=0.1$, $J_{\min }/J_{\max }=0.03$, $n_{g\max }=0.8$, $n_{g\min }=0.2$, $E_C/k_B=1\mathrm{K}$ ($E_C/2\pi \hslash =21\mathrm{GHz}$), $R=300\mathrm{k}\Omega$, environment temperature $T=0$, $S({\omega }_0)=2\hslash {\omega }_{0}R$, $S(-{\omega }_0)=0$, and $S(0)=2k_B{T}_{0}R$, with $T_0=0.1\mathrm{K}$.