# Find the eigenvalues of Liouvillian

1. Aug 16, 2011

### emind

The master equation of the damped harmonic oscillator is
$\frac{d}{dt}\rho_S(t) = -i\omega_0 [a^\dagger a,\rho_S(t)] + \gamma_0(\bar n+1) \{ a\rho_S(t) a^\dagger -\frac{1}{2} a^\dagger a \rho_S(t) -\frac{1}{2} \rho_S(t) a^\dagger a \} + \gamma_0\bar n \{ a^\dagger \rho_S(t) a -\frac{1}{2} a a^\dagger \rho_S(t) -\frac{1}{2} \rho_S(t) a a^\dagger \} \equiv \mathcal{L}\rho_S(t).$

Is there any method to analytically find out some eigenvalues, like the smallest three eigenvalues, of the liouvillian?

I am an engineering graduate, but now a rookie in physics.