Expectation Values (RWA Model)

We would like to use our Master equation Eq.(7.48)

$$\frac{d}{dt}\rho(t) = -i\bar{\Omega}[a^{\dagger}a,{\rho}] - \kappa \Big\{ a^{\dagger} a {\rho} + \rho a^{\dagger} a - 2 a \rho a^{\dagger} \Big\}$$

$$- 2\kappa n_B(\Omega) \Big\{ a^{\dagger} a {\rho} +\rho a a^{\dagger} - a \rho a^{\dagger} -a^{\dagger} \rho a \Big\}$$

and calculate some `useful' quantities as, for examples, expectation values of System ($=$ oscillator) observables $\hat{\theta}$. Let us do this for the number operator, $\hat{\theta}=\hat{n}=a^{\dagger}a$. Multiplying with $n$ and taking the trace, we obtain

$$\frac{d}{dt}\langle n \rangle (t) = -i\bar{\Omega} \left( n[a^{\dagger}a,{\rho}]\right) - \kappa \Big\{ a^{\dagger}a a^{\dagger} a {\rho} + \rho a^{\dagger} aa^{\dagger}a - 2 a \rho a^{\dagger}a^{\dagger}a \Big\}$$

$$- 2\kappa n_B(\Omega) \Big\{ a^{\dagger} a a^{\dagger} a {\rho} +\rho a a^{\dagger} a^{... ... a - a^{\dagger} a a \rho a^{\dagger} - a^{\dagger} a a^{\dagger} \rho a \Big\}$$

$$= -i\bar{\Omega} \left(a^{\dagger} a a^{\dagger}a{\rho} -a^{\dagger}a{\rho}a^{\dagger}a\right) - \kappa \Big\{2 a^{\dagger}a a^{\dagger} a {\rho} - 2 \rho a^{\dagger}(aa^{\dagger}-1) a \Big\}$$

$$- 2\kappa n_B(\Omega) \Big\{ a^{\dagger} a a^{\dagger} a {\rho} +\rho ( a^{\dagger}a+1)... ...-a^{\dagger}(a a^{\dagger} -1) a \rho - a a^{\dagger} a a^{\dagger} \rho \Big\}$$

$$= -2\kappa \Big\{ \rho a^{\dagger} a \Big\}$$

$$- 2\kappa n_B(\Omega) \Big\{ a^{\dagger} a a^{\dagger} a {\rho} +\rho ( a^{\dagger}a+1)... ...ger}(a a^{\dagger} -1) a \rho - ( a^{\dagger}a+1) ( a^{\dagger}a+1) \rho \Big\}$$

$$= -2\kappa \Big\{ \rho a^{\dagger} a \Big\} +2\kappa n_B(\Omega)$$

$$= -2\kappa\left(\langle n \rangle (t) - n_B(\Omega)\right).$$ (50)

This now is a simple first order differential equation which has the solution

$$\langle n \rangle (t) = \langle n \rangle (t=0) e^{-2\kappa t} + \kappa n_B(\Omega) \left( 1- e^{-2\kappa t}\right).$$ (51)

In particular, one has

$$\langle n \rangle (t\to \infty) = n_B(\Omega).$$ (52)

For large times, the occupation number is thus given by the thermal equilibrium Bose distribution, regardless of the initial condition $\langle n \rangle (t=0)$.