**4. Spin Pumping**

### *4.1. A Minimum Model of Spin Pumping*

We consider a minimum model of spin pumping involving a quantum dot with dynamic magnetization and an electron lead (Figure 4A). The magnetization of the dot *M*(*t*) rotates around the *z*-axis with a period T . An electron in the quantum dot is spin polarized because of the *s–d* exchange interaction with magnetization and is represented by the two-component creation and annihilation operators *d*† = (*d*†↑, *<sup>d</sup>*†↓), and *d*, where ↑ and ↓ denote the direction of the electron's spin magnetic moment parallel and antiparallel, respectively, to the *z*-axis.

The Hamiltonian of the minimum model consists of three terms *H*(*t*) = *<sup>H</sup>*d(*t*) + *H*l + *H*t. *<sup>H</sup>*d(*t*), describing the dot, is defined by

$$H\_{\rm d}(t) = d^{\dagger}[\varepsilon\_{\rm d} - \mathcal{M}(t) \cdot \sigma]d,\tag{28}$$

where d is the unpolarized energy of a dot electron, *M*(*t*) ≡ *M*(sin *θ* sin *φ*(*t*), sin *θ* sin *φ*(*t*), cos *<sup>θ</sup>*), and *σ* = (*<sup>σ</sup>x*, *<sup>σ</sup>y*, *<sup>σ</sup>z*) the vector of Pauli matrices. Introducing the eigenstates |*j*↑, *j*↓ (with *j*↑(↓) = 0 or 1) of the number operator of the dot electron ∑*σ <sup>d</sup>*†*σd<sup>σ</sup>* as a basis, the dot Hamiltonian is represented by the matrix

$$H\_{\mathbf{d}}(t) = \begin{pmatrix} |0,0\rangle & |0,1\rangle & |1,0\rangle & |1,1\rangle \\ 0 & 0 & 0 & 0 \\ 0 & \epsilon\_{\mathbf{d}} + M\cos\theta & -M\epsilon^{+i\phi(t)}\sin\theta & 0 \\ 0 & -M\epsilon^{-i\phi(t)}\sin\theta & \epsilon\_{\mathbf{d}} - M\cos\theta & 0 \\ 0 & 0 & 0 & 2\epsilon\_{\mathbf{d}} \end{pmatrix}. \tag{29}$$

**Figure 4.** (**A**) The minimum model consists of a ferromagnetic quantum dot attached to an electron lead. The dot has a dynamic magnetization *M*(*t*) that rotates around the *z*-axis with a period T . The number of transferred electrons with spin magnetic moment ↑ (↓) is captured by the counting field. (**B**) Schematic of the spin current generation in the minimum model. The scheme can be summarized as follows: (1) an electron with ↓-spin enters from the lead onto the dot subject to the dot–lead interaction; (2) the spin of the electron is flipped by the precessing magnetization; (3) the electron with ↑-spin moves back from the dot to the lead.

The electron lead is described by the term

$$H\_{\rm l} = \sum\_{\sigma = \uparrow, \downarrow} \sum\_{k} \epsilon\_{k} \mathfrak{c}\_{\sigma, k}^{\dagger} \mathfrak{c}\_{\sigma, k} \tag{30}$$

where *<sup>c</sup>σ*,*<sup>k</sup>* and *<sup>c</sup>*†*σ*,*<sup>k</sup>* with *σ* =↑ or ↓ are annihilation and creation operators of a lead electron with energy  *k* and spin-*σ*. The dot–lead interaction is assumed to be spin conserving with

$$H\_{\rm t} = \sum\_{\sigma} \sum\_{k} \hbar \upsilon\_{k} (d\_{\sigma}^{\dagger} c\_{\sigma,k} + c\_{\sigma,k}^{\dagger} d\_{\sigma})\_{\prime} \tag{31}$$

where ¯*hvk*is the coupling strength, which we assume to be weak.

Intuitively, the generation of the spin current in the minimum model is summarized by the following scheme (see Figure 4B): (1) an electron with ↓-spin moves from lead to dot under the dot–lead interaction, (2) the spin of the electron is flipped by the precessing magnetization, and (3) an electron with ↑-spin moves back from dot to lead. For spin-current generation, the essential conditions required in setting parameter values are

$$
\epsilon\_{\rm d} - M < \mu < \epsilon\_{\rm d} + M \quad \text{and} \quad \beta^{-1} \le 2M,\tag{32}
$$

where *β* is the inverse temperature of the lead.

### *4.2. FCS Formalism of the Spin Pumping*

In the following, we apply the FCS outlined in Section 2 to evaluate the number of transferred electrons with spin *σ* from projective measurements of the electron number in the lead represented by *Nσ* ≡ ∑*k <sup>c</sup>*†*σ*,*kcσ*,*k*. By associating *<sup>H</sup>*d(*t*), *H*l, and *H*t with *H*S, *H*E, and *H*SE, respectively, and defining an outcome of the projective measurement at time *t* as *<sup>n</sup>σ*,*t*, we analyze the electron dynamics under spin pumping.

In order to explicitly examine the influence of the relaxation process on the spin current generation, we discretize the rotation of *<sup>M</sup>*(*t*): divide the period T into *N* intervals, *ti* ≤ *t* ≤ *ti*+<sup>1</sup> (*i* = 1, ···, N) with *t*1 = 0 and *tN*+<sup>1</sup> = T ; fix the direction of *M*(*t*) during each interval; and change *φ* at each *ti* discretely with substitution *φi* = *φ<sup>i</sup>*−<sup>1</sup> + *δφ* with *φ*0 = 0, *φN* = 2*π* and *δφ* ≡ 2*π*/*N* (see Note [57]). The net number of electrons with spin-*σ* during the *i*th interval can be evaluated from the difference in outcomes <sup>Δ</sup>*<sup>n</sup>σ*,*<sup>i</sup>* = *<sup>n</sup>σ*,*ti*+<sup>1</sup> − *<sup>n</sup>σ*,*ti* .

By introducing counting fields *λ*↑ and *λ*↓ corresponding to observables *<sup>N</sup>*↑ and *<sup>N</sup>*↓, respectively, we can evaluate the mean value of transferred electrons,

$$
\langle \Delta n\_{\uparrow(\downarrow),i} \rangle = \int\_{t\_i}^{t\_{i+1}} J\_{\uparrow(\downarrow)}(t), \tag{33}
$$

with an inertial flow of electrons

$$J\_{\uparrow(\downarrow)}(t) \equiv \langle \langle 1 | \left[ \frac{\mathfrak{B} \Xi^{(\lambda\_{\uparrow(\downarrow)})}(t)}{\mathfrak{B}(i\lambda\_{\uparrow(\downarrow)})} \right]\_{\lambda\_{\uparrow(\downarrow)}=0} | \rho^{(\lambda\_{\uparrow(\downarrow)}=0)}(t) \rangle \rangle. \tag{34}$$

The inertial flow of electrons provides an instantaneous spin current,

$$J\_{\rm spin}(t) \equiv J\_{\uparrow}(t) - J\_{\downarrow}(t),\tag{35}$$

and its time integration over one period provides a temporal average of the spin current,

$$I\_{\rm spin} \equiv \frac{1}{\mathcal{T}} \int\_0^{\mathcal{T}} J\_{\rm spin}(t)dt. \tag{36}$$

To discuss the role of nonadiabaticity in spin pumping, we focus the Born-Markovian (long-time) limit by taking the limit *t* → ∞ of the supermatrix Ξ(*λ*)(*t*) in each interval. In this limit, the matrix elements of Ξ(*λ*) are time-independent during each interval and determined by the direction of *M* in each interval.
