Super-Operators

We have another look at the (Markoffian) Master equation of the damped harmonic oscillator at zero temperature $T=0$ (spontaneous emission only);

$$\frac{d}{dt}\rho(t) = -i\bar{\Omega}[a^{\dagger}a,{\rho}] - \kappa \Big\{ a^{\dagger} a {\rho} + \rho a^{\dagger} a - 2 a \rho a^{\dagger} \Big\} .$$ (139)

Suppose we started from a pure state $\rho(0)=\vert\Psi\rangle \langle \Psi\vert$ at time $t=0$: after a short time $\Delta t$, this would evolve according to

$$\vert\Psi\rangle \langle \Psi\vert \to \vert\Psi\rangle \langle \Psi\vert + \Delta t \Big\{ \left(-i \ba... ...rt \left(i \bar{\Omega} a^{\dagger}a - \kappa a^{\dagger} a {\rho}\right)\Big\}$$

$$+2\kappa a \vert\Psi\rangle \langle \Psi\vert a^{\dagger}$$

$$\equiv \vert\Psi\rangle \langle \Psi\vert + \Delta t \Big\{ {\cal L}_0 \... ...rangle \langle \Psi\vert + {\cal L}_1\vert\Psi\rangle \langle \Psi\vert \Big\},$$ (140)

where we defined the super-operators via

$${\cal L}_0 \rho \equiv -iH_{\rm eff} \rho + \rho iH_{\rm eff}^{\dagger},\quad H_{\rm eff... ...v H -i \kappa a^{\dagger} a = \bar{\Omega} a^{\dagger}a -i \kappa a^{\dagger} a$$

$${\cal L}_1\rho \equiv 2\kappa a \rho a^{\dagger}.$$ (141)

For pure states $\vert\Psi\rangle \langle \Psi\vert$: ${\cal L}_0$ generates time-evolution with non-hermitian Hamiltonian $H_{\rm eff}$, but ${\cal L}_1$ generates a quantum jump;

$${\cal L}_0 : \vert\Psi\rangle \to -iH_{\rm eff}\vert\Psi\rangle$$

$${\cal L}_1 : \vert\Psi\rangle \to \sqrt{2\kappa} a \vert\Psi\rangle.$$ (142)

The state $a \vert\Psi\rangle$ corresponds to a state with one photon less.