  Let $M = \Gamma \backslash \text{SL}(2,\mathbb{R})$ be a compact quotient of
$\text{SL}(2,\mathbb{R})$ equipped with the normalized Haar measure
$\text{vol}$, and let $\{h_t\}_{t \in \mathbb{R}}$ denote the horocycle flow on
$M$. Given $p \in M$ and $W \in \mathfrak{sl}_2(\mathbb{R}) \setminus \{0\}$
not parallel to the generator of the horocycle flow, let $\gamma_{p}^W$ denote
the probability measure uniformly distributed along the arc $s \mapsto p
\exp(sW)$ for $0\leq s \leq 1$. We establish quantitative estimates for the
rate of convergence of $[(h_t)_{\ast} \gamma_{p}^W](f)$ to $\text{vol}(f)$ for
sufficiently smooth functions $f$. Our result is based on the work of Bufetov
and Forni [2], together with a crucial geometric observation. As a corollary,
we provide an alternative proof of Ratner's theorem on quantitative mixing for
the horocycle flow.
