4 the probe and the transport terminals I = L, R exactly factorizes when the probe is single-channeled. This re- markable property is apparently mentioned here for the first time and is discussed in some detail in Appendix A. Consequently, the chemical potentials in Eq. (9) remain the same, regardless of the transparency of the tunnel barrier between the dot and the probe, when the lat- ter carries only a single channel. Our aim being to ob- tain the current-induced spin polarization and thus the spin-dependent chemical potentials on the dot, we are quite naturally interested in the limit Γ → 0, where the probe does not perturb the dot and accordingly the spin- dependent chemical potentials on the dot equal those on the probe. The observation we just made allows us to access that limit via an analytically simpler calculation at Γ$_{P}$ = 1 which we present in the next section. the spin current vector respectively, and introduced the spin-dependent transmission coefficients [19] T ( αβ ) ij = ∑ m ∈ i,n ∈ j Tr[ t † $_{mn}$σ$^{α}$t$_{mn}$ σ $^{β}$] , (7) where σ $^{α}$, α = x, y, z are Pauli matrices and σ 0 is the identity matrix. The trace in Eq. (7) is taken over the spin degree of freedom and t$_{mn}$ is a 2 × 2 matrix of spin- dependent transmission amplitudes from channel n in lead j to channel m in lead i . We are interested in the mesoscopic counterpart of the magnetoelectric geometry of Ref. [2], and consider a quantum dot connected to two external transport ter- minals L and R . An electric bias voltage V is applied between the terminals, µ (0) L − E$_{F}$ = − µ (0) R + E$_{F}$ = eV/ 2, where we further assume that there is no spin accumu- lation, µ ( β ) L,R = 0, β = 0. To measure local chemical potentials, we introduce an external probe with control- lable chemical potentials, which scans the dot surface. The situation is sketched in Fig. 1. The chemical poten- tials in the probe are determined by the condition that no charge nor spin current flows through it, i.e. they are solutions of To get the total spin polarization, we scan the probe over the whole sample. The transmission amplitudes t$_{mn}$ in Eq. (7) are elements of a scattering matrix that de- pends parametrically on the probe position. For a wide probe, N$_{P}$ ≫ 1, we argue semiclassically that the corre- lator 〈 T ( αβ ) PL T ( αβ ) P ′ L 〉 − 〈 T ( αβ ) PL 〉〈 T ( αβ ) P ′ L 〉 decays to zero when the distance between P and ′ is of order the width of the probe. This is so, because beyond that, correlated escapes of pairs of trajectories no longer contribute [32]. We next make the leap of faith that this semiclassical argument can be extended to the case of present inter- est of a single-channel probe, for which we accordingly conjecture that two measurements performed a distance ∼ k − 1 F apart deliver uncorrelated outcomes. To the best of our knowledge, this problem has not been investigated yet, and we will numerically confirm the validity of this conjecture below. Further assuming a sharp decay of the correlator at k − 1 F , we obtain that the sample-integrated chemical potentials are given by the sum µ ( β ) tot = ∑ P µ ( β ) P over k 2 $_{F}$A/ 4 π uncorrelated probe positions on the surface of our two-dimensional quantum dot. III. RANDOM MATRIX THEORY Having expressed local and sample-integrated polar- izations in terms of transmission coefficients, Eq. (9), we next proceed to evaluate them quantitatively. We focus on ballistic chaotic cavities, and accordingly use RMT to calculate the average and variance of µ ( β ) P and µ ( β ) $_{tot}$. This calculation is easier for Γ$_{P}$ = 1, since then the ensemble-average scattering matrix vanishes [24]. Be- cause of the factorization property of the Γ$_{P}$ -dependence of transmission coefficients mentioned above, the spin- dependent potentials on the dot are equal to those on the probe for Γ$_{P}$ → 0, which in their turn remain un- changed as Γ$_{P}$ 1. Thus we can extract the dot’s spin-dependent potentials µ ( β ) x$_{P}$ at point x$_{P}$ underneath the probe from a calculation at Γ$_{P}$ = 1, because µ ( β ) x$_{P}$ = µ ( β ) P (Γ$_{P}$ → 0) = µ ( β ) P (Γ$_{P}$ = 1). Because we are inter- ested in systems with time reversal symmetry but totally We next make the key observation that the Γ$_{P}$ - dependence of the transmission coefficients T ( αβ ) PI between which reproduces a result first derived in Ref. [31]. In diffusive samples, this gives a linear average decay of µ (0) P from eV/ 2 to − eV/ 2 as the probe is moved across the sample from regions more easily accessible from the left lead (where T (00) PL > T (00) PR ) to regions which are more easily accessible from the right. µ (0) P = [ T (00) PL − T (00) PR ] eV / 4Γ$_{P}$ , (10) In particular, for N$_{L,R}$ ≫ 1, the electrochemical potential in the probe is given by µ ( β ) P = [ T ( β 0) PL − T ( β 0) PR ] eV / 2[ T (00) PL + T (00) PR ] . (9) and depend a priori on the transparency of the contact between the probe and the dot. Eq. (8) determines the local chemical potentials µ ( β ) P as a function of the elec- trochemical potentials µ (0) L,R imposed on the longitudi- nal leads. We consider a single-channel probe N$_{P}$ = 1, which not only delivers the sharpest resolution of the dot’s chemical potentials landscape, but moreover en- sures that the spin reflection matrix T ( αβ ) PP from the probe to itself is exactly (i.e. not only on average) diagonal in spin indices. This follows from time-reversal symmetry, current conservation and gauge invariance [30], which ad- ditionally require T ( αα ) PP = T (00) PP , ∀ α . One then straight- forwardly obtains the chemical potentials in the probe as I ( α ) P = e h ∑ j = L,R,P ∑ β (2Γ$_{P}$ N$_{P}$ δ$_{αβ}$δ$_{P}$$_{,j}$ − T ( αβ ) P ,j ) µ ( β ) j ≡ 0 , (8)