4.4.1. Electron Transfer Dynamics

Let us first examine electron transfer dynamics under the cyclic rotation of the magnetization to show the generation of the spin current in the nonadiabatic regime. For this purpose, we numerically evaluate the time evolution of the populations *<sup>ρ</sup>jj*(*t*) = *j*, *j*|*ρ*(0)(*t*)|*j*, *j* (*ρ*00: empty state, *ρ*10: half-filled state with spin-↑, *ρ*01: half-filled state with spin-↓, and *ρ*11: completely filled state) and corresponding instantaneous electron and spin currents, *J*↑(↓)(*t*) and *J*spin(*t*). In Figure 5A,B, we present the time evolution of populations and instantaneous currents for one cycle of the step-like rotation with division number *N* = 5 and time interval *δt* = 20. The change in angle at each subsequent *ti* is *δφ* = 2*π*/5, that is *φi* = *φ<sup>i</sup>*−<sup>1</sup> + 2*π*/5 with *φ*0 = 0.

**Figure 5.** (**A**) Time evolution of the populations in the dot under the step-like precession of the magnetization with *δ*¯*t* = 20 and *N* = 5. The populations deviate from their steady-state values just after a sudden change of the angle *φ*, but then they approach new steady state-values for each *φ<sup>i</sup>*. The figure shows the steady-state values of the populations to be invariant. This is because of the rotational symmetry of the system about the *z*-axis. (**B**) The instantaneous electron and spin currents *J*↑(*t*) (red line), *J*↓(*t*) (blue line) and *J*spin(*t*) (black line) corresponding to the population dynamics in panel (A). The time dependences of the instantaneous currents indicate that electrons starts moving between dot and lead just after the sudden change of *φ*, and *J*↑ and *J*↓ have opposing directions. The latter trend show that the instantaneous electron currents are balanced as a result of charge conservation in the lead. In contrast, the instantaneous spin current *J*spin always takes positive values indicating constant spin current generation. The parameters are set to ¯d = 10, *μ*¯ = 10, *β* ¯ = 100, *λ* = 0.01, *ω*¯ *c* = 4, *θ* = 5*π*/6, and *δφ* = 2*π*/5, which satisfies the condition (32).

In Figure 5A, we find that initially the populations deviate from their steady-state values by changing *φ* at *ti*, but then they approach new steady-state values for each *φi* with the populations remaining unchanged from their initial values because the steady-state populations are independent of *φ* (see the analytic expression of the steady state, Equation (A25)). In the figure, the time evolution of the components *ρ*01 and *ρ*10 (blue and red lines) exhibit oscillations caused by transitions between states |0, 1 and |1, 0 in consequence of the applied magnetization *M* (Larmor precession). Its period

is given by the inverse of the Larmor frequency *TL* ≡ *h*/2*M* = *tu*. The other two components *ρ*00 and *ρ*11 also exhibit transient behavior after changing *φ* but they do not exhibit a Larmor precession because the magnetization contributes transitions including neither |0, 0 nor |1, 1 (see Equation (29)).

In Figure 5B, the colored lines representing *Jσ* show that spin-↑ electrons (red line) and spin-↓ electrons (blue line) are moving in opposite directions; the former move from dot to lead, whereas the latter move from lead to dot. These trends show that the instantaneous electron currents *J*↑ and *J*↓ are balanced as a result of charge conservation in the lead. In contrast, the instantaneous spin current (black line) always takes positive values, *J*spin > 0, indicating the generation of positive spin current into the lead without an associated charge current, which we call pure spin current.
