  Microwave photons are important carriers of quantum information in many
promising platforms for quantum computing. They can be routinely generated,
controlled, and teleported in experiments, indicating a variety of applications
in quantum technology. However, observation of quantum statistical properties
of microwave photons remains demanding: The energy of several microwave photons
is considerably smaller than the thermal fluctuation of any room-temperature
detector, while amplification necessarily induces noise. Here, we present a
measurement technique with a nanobolometer that directly measures the photon
statistics at millikelvin and overcomes this trade-off. We apply our method to
thermal states generated by a blackbody radiator operating in the regime of
circuit quantum electrodynamics. We demonstrate the photon number resolvedness
of the nanobolometer, and reveal the n(n+1)-scaling law of the photon number
variance as indicated by the Bose--Einstein distribution. By engineering the
coherent and incoherent proportions of the input field, we observe the
transition between super-Poissonian and Poissonian statistics of the microwave
photons from the bolometric second-order correlation measurement. This
technique is poised to serve in fundamental tests of quantum mechanics with
microwave photons and function as a scalable readout solution for a quantum
information processor.
