`Monte Carlo' Procedure

The decomposition of histories can now be simulated on a computer in order to actually solve the Master equation. Here, we only describe the simplest version (spontaneous emission, no driving field), starting from a pure state $\vert\Psi\rangle$ of the total system. For more details, see Carmichael or Plenio/Knight.

Step 1: Fix a time step $\Delta t$. Calculate the probability $\Delta P$ of photon emission;

$$\Delta P \equiv \gamma \Delta t \langle \Psi \vert a^{\dagger} a \vert \Psi \rangle.$$ (147)

Step 2: Compare $\Delta P$ with a random number $0 \le r \le 1$

• For $\Delta P > r$: `emission', replace
  

  $$\vert\Psi\rangle \to \frac{a \vert\Psi\rangle}{\Vert a \vert\Psi\rangle \Vert}$$ (148)

  
  

• For $\Delta P \le r$: no emission but time-evolution under effective Hamiltonian $H_{\rm eff}$,
  

  $$\vert\Psi\rangle \to \frac{(1-i\Delta t H_{\rm eff})\vert\Psi\rangle }{(1-\Delta P)^{1/2}}$$ (149)

  
  

Step 3: Go back to Step 1.

This procedure (performed with small time-steps $\Delta t$ up to a final time $t_{\rm final}$) yields a `curve' of simulated states $\vert\Psi(t)\rangle$, $t\in [0,t_{\rm final}]$ in the system Hilbert space ${\cal H}_S$. The procedure is then repeated many times in order to obtain time-dependent averages $\langle \Psi\vert \hat{\theta} \vert \Psi \rangle$ of observables $\hat{\theta}$.

The entire procedure yields a density operator $\rho(t)=\vert\Psi\rangle \langle \Psi \vert$ that solves the original Master equation, Eq.(7.143): in one time step $\Delta t$, we have

$$\rho (t+\Delta t) = \Delta P \frac{a \vert\Psi(t)\rangle \langle \Psi (t)\vert a^{\da... ...ert (1+i\Delta t H_{\rm eff}^{\dagger}) }{(1-\Delta P)^{1/2}(1-\Delta P)^{1/2}}$$

$$= \gamma \Delta t a \rho(t) a^{\dagger}+ \rho(t) -i \Delta t [H,\rh... ...a t \left( a^{\dagger} a \rho(t) + \rho(t)a^{\dagger} a \right) + O(\Delta t)^2$$

$$\leadsto \frac{d}{dt}\rho(t) = -i [H,\rho(t)] - \kappa \Big\{ a^{\dagger} a {\rho} + \rho a^{\dagger} a - 2 a \rho a^{\dagger} \Big\} .$$ (150)

Remarks:

• The splitting of ${\cal L}$ as ${\cal L}={\cal L}_0+{\cal L}_1$ is not unique, there are ususally several ways of how to `unravel' the Master equation.
• For more complicated Master equations, one has to extend and modify the above procedure.