  We construct non-equilibrium steady states in the Klein-Gordon theory in
arbitrary space dimension $d$ following a local quench. We consider the
approach where two independently thermalized semi-infinite systems, with
temperatures $T_{\rm L}$ and $T_{\rm R}$, are connected along a
$d-1$-dimensional hypersurface. A current-carrying steady state, described by
thermally distributed modes with temperatures $T_{\rm L}$ and $T_{\rm R}$ for
left and right-moving modes, respectively, emerges at late times. The
non-equilibrium density matrix is the exponential of a non-local conserved
charge. We obtain exact results for the average energy current and the complete
distribution of energy current fluctuations. The latter shows that the
long-time energy transfer can be described by a continuum of independent
Poisson processes, for which we provide the exact weights. We further describe
the full time evolution of local observables following the quench. Averages of
generic local observables, including the stress-energy tensor, approach the
steady state with a power-law in time, where the exponent depends on the
initial conditions at the connection hypersurface. We describe boundary
conditions and special operators for which the steady state is reached
instantaneously on the connection hypersurface. A semiclassical analysis of
freely propagating modes yields the average energy current at large distances
and late times. We conclude by comparing and contrasting our findings with
results for interacting theories and provide an estimate for the timescale
governing the crossover to hydrodynamics. As a modification of our Klein-Gordon
analysis we also include exact results for free Dirac fermions.
