# Master equation approach only suitable for good cavity laser?

As I understand, master equation and quantum Langevin equation are equivalent formalisms. However, in a Review of Modern Physics article titled sub-Poissonian processes in quantum optics, the author says:

"The above analysis, based on the master-equation approach, assumes that different time scales govern the decay of the field in the cavity and the interaction time between the atom and the cavity mode. For lasers, this means that $t_{cav}\gg\tau_a,\tau_b$, while for micromasers one must have $t_{cav}\gg t_{int}$. This precludes consideration of effects associated with atomic dynamics, which will be considered in the next section, using the Heisenberg-Langevin approach."

So it seems Heisenber-Langevin formalism is more capable than master equation in certain cases. This is inconsistent with their equivalence. Could anyone point me out where I understand wrongly?

Thanks!

Note that the paper does not really claim that the Heisenberg-Langevin formalism is more powerful. All it says is that the

analysis (emphasis mine), based on the master-equation approach ... precludes consideration of effects associated with atomic dynamics

So, it is not the tool which is lacking, but how it has been used.

In section V, the author derives a master equation, while making the assumptions clear: $t_\mathrm{cav} \gg t_\mathrm{int}$. This does not include the dynamics of the atomic levels $a$ and $b$, and hence the master equation derived under this approximation does not yield the effects of short timescale behaviour. The Liouvillian in equation 5.9 has only the operators associated with the cavity in it.

In the analysis of section VI. A, the dynamics of the atomic operators $\sigma^j_a$, $\sigma^j_b$ and $\sigma^j$ are explicitly considered.

In short, yes, the master equation and Heisenberg-Langevin formalisms are equivalent, but in some situations it is more convenient to think in terms of one of them. Had the author included effects of the atomic dynamics in the master equation, the result might have been the same (I say "might" because I did not read the paper: I think the two methods are used to look at two different regimes, where an approximation for one may not hold in another. However, if it were done exactly, then in principle the equations of motion of observables would be identical).

• @PuZhang please mark the answer as accepted if it answers the question. Thanks! :) – Abhinav Jan 26 '17 at 21:07