Crossover from Adiabatic to Antiadiabatic Quantum Pumping with Dissipation

Quantum pumping, in its different forms, is attracting attention from different fields, from fundamental quantum mechanics, to nanotechnology, to superconductivity. We investigate the crossover of quantum pumping from the adiabatic to the antiadiabatic regime in the presence of dissipation, and find general and explicit analytical expressions for the pumped current in a minimal model describing a system with the topology of a ring forced by a periodic modulation of frequency $\omega$. The solution allows following in a transparent way the evolution of pumped dc current from much smaller to much larger $\omega$ values than the other relevant energy scale, the energy splitting introduced by the modulation. We find and characterize a temperature-dependent optimal value of the frequency for which the pumped current is maximal.

Figure 1

Schematic of the minimal ring model, realized by three quantum dots $a$, $b$, $c$ with externally controlled potentials ${ϵ}_a$, ${ϵ}_b$, and ${ϵ}_c$ used for pumping. The ${\gamma }_{ij}$ are here interdot hoppings, generally fixed. An alternative realization could be a molecular trimer [10, 11] where three atomic orbitals $|a⟩$, $|b⟩$, and $|c⟩$ have identical energies, while the hopping integrals ${\gamma }_{ij}$ can actuate the pumping by modulating atomic displacements. In either case the relevant states of the system can be mapped to the pseudospin-$1/2$ model of Eq. (1).

Figure 2

Steady-state dc circulating current $I$ as a function of the pumping frequency $\omega$. Solid line: $T=0$ current, obtained by Eq. (2) with $P\equiv 1$. Note that this coincides with the purely quantum result (no dissipation) when $|\tilde{\psi }⟩=|\widehat{\mathbf{n}};-⟩$ is chosen for the initial state. Dashed and dot-dashed curves: Pumped dc current for several temperatures, as obtained by the exact expression (6) for $P$.

Figure 3

Time evolution of the current $I(t)=I_0\mathrm{Tr}[{\sigma }^y{\tilde{\rho }}_S(t)]$ for $\omega =2\Delta$, in the dissipationless case ($\alpha =0$, solid curve), and in the transient induced by weak dissipation ($\alpha =0.005$), at low (dashed curve) and intermediate temperature (dot-dashed curve), starting from the initial $|z,-⟩$ state. Inset: Temperature dependence of the steady-state dc current $I$.