In a open quantum system, one can easily derive the Heisenberg-Langevin equation of motion which describes the time evolution of creation/annihilation operators (in say, a cavity) $$\dot{a}(t) = i[H,a(t)]-\frac{\kappa}{2}a(t)+\sqrt{\kappa}a^{(in)}(t)$$ The first term describes the usual coherent evolution, the second term corresponds decay that comes from coupling to the bath (with decay rate $\kappa$), and $a^{(in)}(t)$ corresponds to the input into the cavity (which could be, for example, just vacuum noise, in which case $\sqrt{\kappa}a^{(in)}(t)$ can be though of as a Langevin force. I've used this https://arxiv.org/abs/0810.4729 reference, see appendix E, section 2, page 73

My question is, how do we go from this formalism to the Lindblad master equation? $$\dot{\rho}(t) = -i[H,\rho(t)] + L \rho(t) L^\dagger -\frac{1}{2}\{L^\dagger L, \rho\}$$ I don't really understand where these operators $L$ come into play when passing from the Heisenberg equation to the master equation. Any help woul be greatly appreciated.


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