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When using the Monte-Carlo wave function method to simulate spontaneous decay in a two-level system one typically uses a non-Hermitian Hamiltonian, e.g.:

$$H_\mathrm{tot} = H_\mathrm{sys} - i\Gamma dt/2 \, |e\rangle\langle e|,$$

where $\Gamma$ is the decay rate out of the excited state ($|e\rangle$) and where $dt$ is the timestep. This reduces the excited state probability by a factor $1-\Gamma dt/2$. (See, for instance, the original Letter.)

I'm confused as to why this is necessary; after all, don't we draw a random number after the time step $dt$ to determine whether a jump will be made? Why do we need this extra term in the Hamiltonian to account for probability 'leaking out of' the excited state, only to renormalize right after?

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