As in the work of Moskalets and B¨ uttiker $^{3}$, in the weak pumping limit ( X$_{ω,j}$ ≪ X$_{j}$$_{0}$ ) and at zero temperature, the spin-polarized pump current, noise, and heat flow could be expressed in terms of the scattering matrix as follows. I$_{α}$ = eω 2 π ∑ βj$_{1}$j$_{2}$ X$_{ω,j}$$_{1}$X$_{ω,j}$$_{2}$ ∂S ∗ αβ ∂X$_{j}$$_{1}$ ∂S$_{αβ}$ ∂X$_{j}$$_{2}$ 2 sin ( ϕ$_{j}$$_{1}$ − ϕ$_{j}$$_{2}$ ) , (5) H$_{α}$ = ℏ ω 2 4 π ∑ βj$_{1}$j$_{2}$ X$_{ω,j}$$_{1}$X$_{ω,j}$$_{2}$ ∂S$_{αβ}$ ∂X$_{j}$$_{1}$ ∂S ∗ αβ ∂X$_{j}$$_{2}$ 2 cos ( ϕ$_{j}$$_{1}$ − ϕ$_{j}$$_{2}$ ) , (6) S$_{αβ}$ = e $^{2}$ω π [ δ$_{αβ}$ ∑ βj$_{1}$j$_{2}$ X$_{ω,j}$$_{1}$X$_{ω,j}$$_{2}$ ∂S$_{αβ}$ ∂X$_{j}$$_{1}$ ∂S ∗ αβ $_{∂X$_{j}$$_{2}$}$2 cos ( ϕ$_{j}$$_{1}$ − ϕ$_{j}$$_{2}$ ) − ∑ γ$_{1}$j$_{1}$γ$_{2}$j$_{2}$ S ∗ $_{βγ$_{1}$}$S$_{βγ}$$_{2}$ X$_{ω,j}$$_{1}$X$_{ω,j}$$_{2}$ ∂S ∗ αγ 1 ∂X$_{j}$$_{1}$ ∂S$_{αγ}$$_{2}$ ∂X$_{j}$$_{2}$ 2 cos ( ϕ$_{j}$$_{1}$ − ϕ$_{j}$$_{2}$ ) ] . (7) Here, we remark that capturing the sensitivity of the quantum levels and scattering matrix to general perturbations is a complicated problem entailing the treatment of a time-dependent Hamiltonian. In the limit of an adiabatic pump however, i.e., assuming that the scattering properties follow the time-dependent potentials instantaneously, it is sufficient to expand the time-dependent scattering matrix to first order in the frequency. In addition, the amplitudes X$_{ω,j}$ are chosen small with respect to their residual values (e.g., the amplitude of X$_{2}$ is smaller than Θ) such that only the terms linear in X$_{ω,j}$ are relevant in an expansion of the scattering matrix, which leads to a (bi)linear response in the amplitudes $^{2,3}$. The pumped current and the noise relate to the parametric derivatives of the scattering matrix of the system $^{2,3}$. For the consideration of the time-reversal symmetry (TRS) in this system we remark the following. The (non-diffusive) scattering region consists of non-collinear localized magnetic moments (that build the DWs) which reverse sign upon time-reversal. This operation leads in general to different scattering and pump properties $^{33}$. In our particular case however, upon time-reversal, the spin polarizations of the states left and right to the DWs is reversed and outgoing waves turn into incoming waves and vice versa. In total, even though the individual wave functions and the scattering region are modified upon a time-reversal, we observe no physical effect of this operation. The situation would be different, if for example the wire to the right side of the DWs were paramagnetic. An elaboration on this point is given below. Here, we start by specifying the single particle states and calculate with those the T 6