Final Form of Master Equation

Using these definitions, we can now write

$$\frac{d}{dt}\rho(t) = -i[\Omega a^{\dagger}a,{\rho}(t)] %%\nonumber\\ -\frac{1}{2}\Big... ...2 i \Delta_+)a^{\dagger} a + (\gamma + 2i \Delta) a a^{\dagger}\right]{\rho}(t)$$ (46)

$$+ {\rho}(t) \left[ (\gamma - 2i \Delta) a a^{\dagger} + (\gamma_+ -... ...mber\\ -2\gamma_+ a{\rho}(t)a^{\dagger} -2\gamma a^{\dagger}{\rho}(t)a \Big\}.$$

We write $2i\Delta aa^{\dagger} = 2i\Delta (a^{\dagger}a+1)$ and obtain

$$\frac{d}{dt}\rho(t) = -i[(\Omega +\Delta_++\Delta) a^{\dagger}a,{\rho}(t)]$$ (47)

$$- \frac{1}{2}\gamma_+ \Big\{ a^{\dagger} a {\rho}(t) + {\rho}(t) a^... ...a^{\dagger}{\rho}(t) + {\rho}(t) a a^{\dagger} -2 a^{\dagger}{\rho}(t)a \Big\}.$$

This can be further re-arranged into

$$\fbox{$ \begin{array}{rcl} \displaystyle \frac{d}{dt}\rho(t) &=& ... ... a a^{\dagger} - a \rho a^{\dagger} -a^{\dagger} \rho a \Big\}, \end{array}$\ }$$ (48)

where

$$\bar{\Omega} \equiv \Omega + P\int_{0}^{\infty}d\omega\frac{\rho(\omega)}{\Omega-\omega}, \quad \kappa\equiv \pi \rho(\Omega).$$ (49)

Remarks

• This is the `standard' Master equation for the damped harmonic oscillator, as discussed in many text books and used for many applications.
• Modifications appear if one uses the non-RWA model Hamiltonian instead of the RWA Hamiltonian.
• Eq.(7.48) is, of course, not exact because we have used 2nd order perturbation theory (in the system-bath coupling $\gamma_Q$), and the Markov approximation.
• The oscillator energy $\hbar \Omega$ is renormalised due to the coupling to the environment. The renormalised frequency $\bar{\Omega}$ is temperature independent.

• The integral for the renormalised frequency $\bar{\Omega}$ may diverge, depending on the form of the spectral density $\rho (\omega )$, Eq.(7.37), in which case this theory breaks down. We will make this statement more precise below.
• One can show that the Master equation Eq.(7.48) (and its non-RWA analogon, model 1) is indeed `wrong' in the sense that there is an exact solution for the density operator $\rho(t)$ within the same model, which is different from the solution of Eq.(7.48). This again will be discussed below.
• Comparing the exact $\rho(t)$ with that obtained from Eq.(7.48), one could now discuss the `validity of the entire Master equation approach'. However, the damped harmonic oscillator is (with very few exceptions) the only quantum dissipative system where an exact solution exists.