  We characterize perfect state transfer on real-weighted graphs of the Johnson
scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$
and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory
results, that $X$ has perfect state transfer at time $\tau$ if and only if
$n=2k$, $m\ge 2^{\lfloor{\log_2(k)} \rfloor}$, and there are integers $c_1,
c_2, \cdots, c_m$ such that (i) $c_j$ is odd if and only if $j$ is a power of
$2$, and (ii) for $r=1,2,\cdots,m$, \[w_r = \frac{\pi}{\tau} \sum_{j=r}^m
\frac{c_j}{\binom{2j}{j}} \binom{k-r}{j-r}.\] We then characterize perfect
state transfer on unweighted graphs of $\mathcal{J}(n,k)$. In particular, we
obtain a simple construction that generates all graphs of $\mathcal{J}(n,k)$
with perfect state transfer at time $\pi/2$.
