**4. Quantum Adiabatic Pump**

For a non-interacting system, the response (particle transfer) to the slow modulation of the system's controlling parameters is well described by Brouwer's formula [19], which is expressed by the elements of the scattering matrix. The particles induced in the left lead in one cycle of the adiabatic modulation of two control parameters *g*1 and *g*2 is

$$m\_{\perp} = \sum\_{\sigma} n\_{\sigma \prime} \tag{31}$$

$$m\_{\sigma} = -\int\_{S} dg\_1 dg\_2 \Pi\_{\sigma}(g\_1, g\_2),\tag{32}$$

where *S* is the area in the two-dimensional control parameter space whose edge corresponds to the trajectory of the cycle. The Berry curvature <sup>Π</sup>*σ*(*g*1, *g*2) for spin *σ* is

$$\Pi\_{\mathcal{F}}(\mathcal{g}\_1, \mathcal{g}\_2) \quad = \quad \frac{1}{\pi} \mathbb{S} \left< \sigma \right| \left\{ \frac{\partial \mathbb{P}}{\partial \mathcal{g}\_2} \frac{\partial \mathbb{P}^\dagger}{\partial \mathcal{g}\_1} + \frac{\partial \mathbb{P}}{\partial \mathcal{g}\_2} \frac{\partial \mathbb{P}^\dagger}{\partial \mathcal{g}\_1} \right\} \left| \sigma \right>\,, \tag{33}$$

where *r*ˆ and ˆ*t* are given in Equations (27) and (25) and |*σ* is the spinor vector of spin *σ*.

If we choose the AB phase *φ* and parameters of the interferometers, for example, *X*0 or *X*1, but not the SOI strengths, we can show that the Berry curvature is finite in general as studied in Reference [26]. In the following, however, we focus on the situation that the control parameters are the AB phase *φ* and Rashba or Dresselhaus SOI strength that modulate the eigenvalue *λ*± as well as the scattering eigenstates |±*n*<sup>ˆ</sup>, |±*n*<sup>ˆ</sup>. To calculate the Berry curvature, we need to evaluate the derivatives of the scattering amplitude matrices, *r*ˆ and ˆ*t*. Then, as shown in Appendix D, after some manipulations, we have the Berry curvatures for spin components parallel to ±*n*ˆ,

$$\Pi\_{\mathfrak{h}}(\mathcal{g}\_1, \mathcal{g}\_2) \quad = \left( |r\_+ - r\_-|^2 - |t\_+|^2 + |t\_-|^2 \right) \mathcal{C}\_{\mathfrak{J}^1 \mathfrak{J}^2 \prime} \tag{34}$$

and

$$\Pi\_{-\mathsf{A}}(\mathsf{g}\_{1\prime}\mathsf{g}\_{2})\; :=\;\;\left(-\left|r\_{+}-r\_{-}\right|^{2}-\left|t\_{+}\right|^{2}+\left|t\_{-}\right|^{2}\right)\mathcal{C}\_{\mathsf{S}1\star\mathsf{S}2}\,\mathrm{.}\tag{35}$$

where the factor at the end is independent of spin and is defined as

$$\begin{split} \mathcal{C}\_{\mathbb{S}^{1},\mathbb{S}^{2}} &= \quad \frac{1}{4\pi n\_{z}} \frac{1}{1-\tau\_{z}^{2}} \Big[ \frac{\partial \tau\_{y}}{\partial \mathcal{G}\_{1}} \frac{\partial \tau\_{x}}{\partial \mathcal{G}\_{2}} - \frac{\partial \tau\_{y}}{\partial \mathcal{G}\_{2}} \frac{\partial \tau\_{x}}{\partial \mathcal{G}\_{1}} \\ &+ \quad \frac{\tau\_{z}}{1-\tau\_{z}^{2}} \Big\{ \tau\_{x} \left( \frac{\partial \tau\_{y}}{\partial \mathcal{G}\_{1}} \frac{\partial \tau\_{z}}{\partial \mathcal{G}\_{2}} - \frac{\partial \tau\_{y}}{\partial \mathcal{G}\_{2}} \frac{\partial \tau\_{z}}{\partial \mathcal{G}\_{1}} \right) + \tau\_{y} \left( \frac{\partial \tau\_{x}}{\partial \mathcal{G}\_{2}} \frac{\partial \tau\_{z}}{\partial \mathcal{G}\_{1}} - \frac{\partial \tau\_{x}}{\partial \mathcal{G}\_{1}} \frac{\partial \tau\_{z}}{\partial \mathcal{G}\_{2}} \right) \Big\} \Big]. \tag{36} \end{split} \tag{37}$$

This is one of the main results of this work.

The vector *τ* is independent of *φ*, but only depends on the SOI strength. Therefore, when one chose the AB phase, *g*1 ≡ *φ*, as one of the control parameters, <sup>C</sup>*φ*,*g*<sup>2</sup> is identically zero as is evident from Equation (36). Hence we do not expect QAP by modulating the AB phase and SOI strength. It is also obvious that if we chose *γb* or *γc* as one of the control parameters and the other by SOI strength, <sup>C</sup>*γb*,*c*,*g*<sup>2</sup>is zero since *τ* is independent of *γb* and *γc* and no pumping is expected.

Even for a fixed AB phase, there is still some freedom to choose two control parameters related to the SOI strength since we have two types of SOI interaction mechanisms, Rashba and Dresselhaus SOI. In the next section, we study Rashba-Dresselhaus interferometer in a simple diamond-shape structure made of four sites and choose the strengths of two types of SOI as control parameters.
