Between February 24-25, 2009, the EIS spectrometer onboard the Hinode spacecraft performed special "sit & stare" observations above the South polar coronal hole continuously over more than 22 hours. Spectra were acquired with the 1" slit placed off-limb covering altitudes up to 0.48 R ⊙ (3.34 × 10 2 Mm) above the Sun surface, in order to study with EIS the non-thermal spectral line broadenings. Spectral lines such as Fe xii λ186.88, Fe xii λ193.51, Fe xii λ195.12 and Fe xiii λ202.04 are observed with good statistics up to high altitudes and they have been analyzed in this study. Results show that the FWHM of Fe xii λ195.12 line increases up to ≃ 0.14 R ⊙ , then decreases higher up. EIS stray light has been estimated and removed. Derived electron density and non-thermal velocity profiles have been used to estimate the total energy flux transported by Alfvén waves off-limb in polar coronal hole up to ≃ 0.4 R ⊙ . The computed Alfvén wave energy flux density f w progressively decays with altitude from f w ≃ 1.2 · 10 6 erg cm −2 s −1 at 0.03 R ⊙ down to f w ≃ 8.5 · 10 3 erg cm −2 s −1 at 0.4 R ⊙ , with an average energy decay rate ∆ f w /∆h ≃ −4.5 · 10 −5 erg cm −3 s −1 . Hence, this result suggests energy deposition by Alfvén waves in a polar coronal hole, thus providing a significant source for coronal heating.