I could not find any good reference deriving a quantum expression for the blackbody radiation.

Usually, people consider the photon spectrum as $\phi_N(E,T)=\frac{1}{2\pi^2\hbar^3 c^2}\frac{E^2}{\exp(\frac{E}{kT})-1}$, which is a classical description of the field.

Is the emitted radiation a superposition of Glauber states for each mode, with an average number of photon given by $\phi_N(E,T)$, something like

$$ |BB\rangle = \underset{\omega}{\Pi}\left(\exp\left(-\frac{\phi(\hbar\omega,T)}{2}\right)\underset{n}{\sum}\frac{\phi(\hbar\omega,T)^{n}}{2^{2}\times n!}|\omega:n\rangle\right) $$

where $|\omega:n\rangle$ is the Fock state of $n$ photons in the $\omega$ mode ?

If so, is there a proper derivation of such a formula ? If not, what is the correct form ?

  • $\begingroup$ No, that's a pure state and it therefore cannot be a thermal state. Have you tried looking for thermal states of a single mode? $\endgroup$ – Emilio Pisanty Feb 10 '17 at 9:36
  • $\begingroup$ Also, your $\phi_N$ is the quantum distribution, or semi-classical if you will. It is certainly not the classical description of the field. $\endgroup$ – AccidentalFourierTransform Feb 10 '17 at 11:16

So I eventually found a quantum description of the blackbody radiation in the book Optical Coherence by Mandel and Wolf, page 659.

Just to give the main results $$ \rho=\frac{\exp\left(-\frac{1}{kT}\hat{H}\right)}{{\rm Tr}\left(\exp\left(-\frac{1}{kT}\hat{H}\right)\right)}\\ =\sum_{n_{1}}...\sum_{n_{N}}\Pi_{l}\left(1-e^{-\frac{\hbar\omega_{l}}{kT}}\right)\exp\left(-\frac{\hbar\omega_{l}}{kT}n_{l}\right)\left|n_{l}\right\rangle \left\langle n_{l}\right|$$

which corresponds to a diagonal density matrix with population given by a Bose-Einstein distribution.

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