An inverse Cartier transform via exponential in positive characteristic

Let $k$ be a perfect field of odd characteristic $p$ and $X_0$ a smooth connected algebraic variety over $k$ which is assumed to be $W_2(k)$-liftable. In this short note we associate a de Rham bundle to a nilpotent Higgs bundle over $X_0$ of exponent $n\leq p-1$ via the exponential function. Presumably, the association is equivalent to the inverse Cartier transform of A. Ogus and V. Vologodsky for these Higgs bundles. However this point has not been verified in the note. Instead, we show the equivalence of the association with that of Sheng-Xin-Zuo in the geometric case. The construction relies on the cocycle property of the difference of different Frobenius liftings over $W_2(k)$, which plays the key role in the proof of $E_1$-degeration of the Hodge to de Rham spectral sequence of $X_0$ due to P. Deligne and L. Illusie.