A Physical explanation of time-correlation function and spectrum

[Cedric Chia]

TL;DR Summary

Why is time-correlation function related to emission/absorption spectrum? What is a good physical explanation behind this?

From my reading of several quantum optics textbooks and spectroscopy texbooks, the emission and absorption spectrum of an atom or molecule are always given in terms of the time-correlation function, for example the emission spectrum of a two level atom is given by:
$$
S(\omega_0)=\frac{1}{\pi}Re\int_0^{\infty}d\tau\left<E^{(-)}(0)E^{(+)}(\tau)\right>e^{i\omega_0\tau}
$$
where $E^{(-)}$ and $E^{(+)}$ are electric field operators of the negative and postivie frequency part respectively. The absorption spectrum of a molecule is said to be given by:
$$
\alpha(\omega_0)=\frac{1}{\pi}Re\int_0^{\infty}d\tau\left<\mu(0)\mu(\tau)\right>e^{-i\omega_0\tau}
$$
where $\mu$ is the transition dipole moment. I have looked at many places but none of them gives a good "physical explanation" of these formulae. Also, is somehow the emission spectrum and absorption spectrum are related to each other? Can we simply switch:
$$
\left<E^{(-)}(0)E^{(+)}(\tau)\right>\longrightarrow\left<E^{(+)}(0)E^{(-)}(\tau)\right>
$$
in the integral and we obtain the absorption spectrum? If this is true then why is this? I have just started my postgrad and it just seems like this is already well-established in the literature so people just accept it and apply it without giving any explanations. Hopefully someone can provide a good explanation or simply a good resource for me to look into. Thank you.

 

[vanhees71]

The formula with the correlation function is derived from the dipole approximation for the interaction between the quantized radiation field and the atom or molecule in 1st-order perturbation theory. You find a treatment of this in almost all textbooks on quantum optics. For this topic I like

M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).

Sects. 5 (semi-classical theory) and 6 (fully quantized theory)

 

[WernerQH]

I have looked at many places but none of them gives a good "physical explanation" of these formulae. Also, is somehow the emission spectrum and absorption spectrum are related to each other?

Kirchhoff's law relates emission and absorption; the principle of detailed balance ensures that there can be thermal equilibrium. For a homogeneous medium (with a refractive index close to 1) the emissivity (in $\rm W m^{-3} Hz^{-1} sr^{-1}$) can be written as $$
\epsilon = \frac {\mu_0 \omega^2} {8 \pi^2 c} \sum_{\mu\nu} e_\mu^* e_\nu
\int_{-\infty}^\infty dt \int d^3x \ e^{-i(kx-\omega t)} \langle j_\nu(0,0) j_\mu(x,t) \rangle ,
$$ a Fourier integral of the current density fluctuations in the medium. Similarly, the absorption coefficient (in $\rm m^{-1}$) for a wave with polarization vector $e_\mu$ can be expressed as $$
\kappa = \frac {\mu_0 c} {2 \hbar\omega} \sum_{\mu\nu} e_\mu^* e_\nu
\int_{-\infty}^\infty dt \int d^3x \ e^{-i(kx-\omega t)} \langle j_\mu(x,t) j_\nu(0,0) - j_\nu(0,0) j_\mu(x,t) \rangle .
$$ The derivation is just an application of the fluctuation/dissipation theorem (linear response theory), and if you want to dig deeper you can look into Landau-Lifshitz vol IX, section 75 on the photon Green's function in a medium. (I don't have Scully-Zubairy at hand.) But perhaps my paper in J. Phys. A (Math. Gen.) 21, 407-418 is sufficient. :-)

I believe that your hunch about "switching" the correlation functions to change absorption into emission is correct. The current commutator in the formula for the absorption coefficient can be interpreted as absorption minus stimulated emission. (And it can become negative in a laser!)

 

[yucheng]

J. Phys. A (Math. Gen.) 21, 407

https://doi.org/10.1088/0305-4470/21/2/020