  We introduce the notion of a ${\cal PT}$-symmetric dimer with a $\chi^{(2)}$
nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity
should be accessible in a pair of optical waveguides with quadratic
nonlinearity and gain and loss, respectively. An interesting feature of the
problem is that because of the two harmonics, there exist in general two
distinct gain/loss parameters, different values of which are considered herein.
We find a number of traits that appear to be absent in the more standard cubic
case. For instance, bifurcations of nonlinear modes from the linear solutions
occur in two different ways depending on whether the first or the second
harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover,
a host of interesting bifurcation phenomena appear to occur including
saddle-center and pitchfork bifurcations which our parametric variations
elucidate. The existence and stability analysis of the stationary solutions is
corroborated by numerical time-evolution simulations exploring the evolution of
the different configurations, when unstable.
