  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.
