  Nearly all thermal radiation phenomena involving materials with linear
response can be accurately described via semi-classical theories of light.
Here, we go beyond these traditional paradigms to study a nonlinear system
which, as we show, necessarily requires quantum theory of damping.
Specifically, we analyze thermal radiation from a resonant system containing a
$\chi^{(2)}$ nonlinear medium and supporting resonances at frequencies
$\omega_1$ and $\omega_2\approx 2\omega_1$, where both resonators are driven
only by intrinsic thermal fluctuations. Within our quantum formalism, we reveal
new possibilities for shaping the thermal radiation. We show that the
resonantly enhanced nonlinear interaction allows frequency-selective
enhancement of thermal emission through upconversion, surpassing the well-known
blackbody limits associated with linear media. Surprisingly, we also find that
the emitted thermal light exhibits non-trivial statistics ($g^{(2)}(0) \neq 2$)
and biphoton intensity correlations (at two distinct frequencies). We highlight
that these features can be observed in the near future by heating a properly
designed nonlinear system, without the need for any external signal. Our work
motivates new interdisciplinary inquiries combining the fields of nonlinear
photonics, quantum optics and thermal science.
