(cf. Fig.1). The two oscillating parameters we view as two out-of-phase gating potentials, one from the ac electric modulation, and the other from the ac magnetic modulation; on the other hand the quantum dot pump system experiences two oscillating gates. Electrons absorb or emit an energy quantum from the oscillating mesoscopic scatterer and those pumped to different directions contribute to the current. The external electric gate voltage oscillating at the spin-split quasistationary states excites pumping current and gives rise to the resonance- type peaks in the current. Within the two-gate picture, the DW quantum pump system can be viewed semiclassically as in the turnstile quantum dot—a classical analog of quantum pumping $^{34}$. 

Fig. 2 shows the tunneling current without external modulations as well as the current adiabatically pumped by the modulations (4) as functions of the Fermi energy. In Fig. 2 (b), the phase difference Δ ϕ is set to be − π/ 2 to produce maximal pump current and to demonstrate the pumping properties prominently. The conductance of the double DW structure possesses sharp resonances when the Fermi energy matches the spin quantum-well states as seen in Fig 2 (a) and Ref.24. Similarly, in the circumstance of the adiabatic pump, the electron can only tunnel through the spin-split levels between the two domains. Thus, the pump current, shot noise and heat flow show peak structures when the Fermi energy matches the spin quantum-well states (see also Fig. 3). 

Comparing panels (a) and (b) in Fig. 2 we find that the peak height (PH) of the pumped current decreases dramatically as the width of the DWs is increased. For narrower DWs, the peak width (PW) of the pumped current becomes more comparable with that of the linear-response conductance, while for wider DWs the former is much smaller than the latter. To further reveal the trend, the PH and the PW at half peak height of the pumped current versus δ are given in Figs. 4 and 5 respectively for the second quasistationary level counting from the spin quantum-well bottom. The peak height decreases exponentially with the increase of δ . In contrast to the linear-response process, the pumping process depends strongly on the lifetime of the quasistationary, spin-well states. When the period of the pump τ$_{p}$ = 2 π/ω exceeds the lifetime of the quasistationary states, the pumped current would be greatly suppressed. Here, we consider the limit of small modulation frequencies, i.e. an adiabatic pump 3 meaning that τ$_{p}$ is relatively large. Therefore, the strength of the pumped current decreases exponentially with the widening of the DW, i.e., the decrease of the lifetime of the quasistationary states. 

