Spin Correlations and Finite-Size Effects in the One-dimensional Kondo Box

We analyze the Kondo effect of a magnetic impurity attached to an ultrasmall metallic wire using the density matrix renormalization group. The spatial spin correlation function and the impurity spectral density are computed for system sizes of up to L=511 sites, covering the crossover from $L<{\mathrm{\ell }}_K$ to $L > \ell_K$,with$\ell_K$ the spin screening length. %Strong mesoscopic variations of the Kondo temperature $T_K$ %and of the spectral features override, to some extent, the %even/odd effect predicted earlier for averaged quantities. We establish a proportionality between the weight of the Kondo resonance and ${\mathrm{\ell }}_K$ as function of $L$. This suggests a spectroscopic way of detecting the Kondo cloud.