# Quantum optical master equation

In Quantum Noise by Gardiner and Zoller (p. 86f) they derive and write a version of the quantum optical master equation.

$\dot{\rho}_S (t) = -\frac{i}{\hbar} [H_\mathrm{sys} ,\rho_S] - \sum_m \frac{\pi \omega_m}{2\hbar } \bigg( \bar{N}(\omega_m) +1 \bigg) \kappa (\omega_m)^2 [\rho_S X_m^+ -X^{-}_{m} \rho_S , X] + \mathrm{ ... more\, terms ... }$

Eq. 3.6.64 (and eq. 3.6.67) do not make dimensional sense to me. The units of the derivative of the density operator should be frequency.

The units of terms in the sums are those of $\omega_m \kappa^2 X^2/\hbar$ where $\kappa^2$ was defined (on p. 45) to be the spring constant of a harmonic oscillator.

On that page they also write energy terms as $\kappa^2 X^2$. This means that the terms have dimension of frequency-squared.

Is there a mistake in the book or am I misunderstanding the meaning of some part of the equation? (the derivation of the equation is not given in the book; they write instead "one finds, after some labour".

## More definitions from the book

X is a system operator and $X = \sum_m (X_m^+ + X_m^- )$ where

$[H_\mathrm{sys}, X_m^\pm ] = \pm \hbar \omega_m X_m^\pm$

and the total system Hamiltonian is written (eq. 3.1.5) as

$H = H_\mathrm{sys} + \frac{1}{2} \sum_n \big[ (p_n-\kappa_n X)^2 + \omega_n^2 q_n^2 \big]$

• Please do not rely on external links for content which is crucial to the question. I know it is a long equation but you should still transcribe it in full, or at the very least provide a screenshot. – Emilio Pisanty Jun 22 '15 at 10:24