I'm trying to understand the basics of the master equation formalism for the two-level atom in the presence of a dissipative environment.

After straightforward calculations, the master equation for the density operator reads: $$ \frac{d \hat{\rho}}{d t}=-\frac{i}{2}\left(\omega_{A}+\Delta+2 \Delta^{\prime}\right)\left[\hat{\sigma}_{z}, \hat{\rho}\right]+\frac{\gamma}{2}(\bar{n}+1)\left(2 \hat{\sigma}_{-} \hat{\rho} \hat{\sigma}_{+}-\hat{\sigma}_{+} \hat{\sigma}_{-} \hat{\rho}-\hat{\rho} \hat{\sigma}_{+} \hat{\sigma}_{-}\right) +\frac{\gamma}{2}\bar{n}\left(2 \hat{\sigma}_{+} \hat{\rho} \hat{\sigma}_{-}-\hat{\rho} \hat{\sigma}_{-} \hat{\sigma}_{+}-\hat{\sigma}_{-} \hat{\sigma}_{+} \hat{\rho}\right), $$ where if the environment is modeled as thermalized electromagnetic field modes, $\bar{n}$ is the Bose-statistics, $\gamma$=constant, $$ \Delta\propto\text{P}\int_0^\infty d\omega\,\omega^3\left(\frac{1}{\omega_A-\omega}+\frac{1}{\omega_A+\omega}\right), $$ and $$ \Delta'\propto\text{P}\int_0^\infty d\omega\,\omega^3\left(\frac{1}{\omega_A-\omega}+\frac{1}{\omega_A+\omega}\right)\bar{n}(\omega). $$ so that (in natural units) $$ \Delta+2 \Delta^{\prime}\propto\int_0^\infty d\omega\,\omega^3\left(\frac{1}{\omega_A-\omega}+\frac{1}{\omega_A+\omega}\right)\coth(\beta\omega/2). $$

My question is about the $\Delta$ and $\Delta'$ functions. Even with Cauchy's principal value, those integrals are divergent. Is it due to the simplifications in formalism? Is there any way to find the constants without that divergences?

I had read that for average temperatures, the terms $\Delta+2 \Delta^{\prime}$ are negligible, and that is why it is not taken into account. Nevertheless, I was wondering if there is a formal treatment where the temperature effects are not minor. Do you know any reference?


  • 1
    $\begingroup$ Nice question! Which treatment are you using for "thermalized electromagnetic field modes"? While the integral equations are usually similar in form, the answer to your questions depends a bit on the precise scenario and on details such as where the $\omega^3$ prefactor originates. If you are interested in the general case, the practical answer is usually along the lines of "absorb the infinite part into the definition of the bare frequency $\omega_A$". $\endgroup$ Jul 9 at 14:23

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