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In this paper, we generalize the known results on the super circles $S^{1|1}$and $S^{1|2}$. We construct the fine equivariant quantization on the supercircle $S^{1|n}$ for $n\geqslant 3$. The equivariant Lie superalgebra is$\spo(2|n)$ which is constituted of the contact projective vector fields on$S^{1|n}$. In order to construct the fine equivariant quantization on$S^{1|n}$, we use the model developed, in the purely even case, by Charles H.Conley and Valentin Ovsienko in \textit{[Linear Differential Operators onContact manifolds, http://www.arxiv:math-Ph/1205.6562v1,24p, 2012]}. We alsouse the technical of Casimir operators to prove the uniqueness of the finequantization on $S^{1|n}$. The technical of Casimir operators used here is thesame as the one used by P. Mathonet and F. Radoux in [\textit{Lett. Math. Phys.98 (2011),311-331}] to prove the existence of a $\pgl(p+1|q)$-equivariantquantization on $\R^{p|q}$.

The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$