# How to determine Collapse operator for Lindblad equation for a specific case

The general form of a Lindblad equation has the form

$$\frac{d\rho}{dt}=-i[H,\rho]+\sum\gamma(A{\rho}A^{\dagger}-\frac{1}{2}A^{\dagger}A\rho-\frac{1}{2}\rho{A}^{\dagger}A)$$

How can I find Collapse operator for a specific case, For example, for a single Bosonic mode, For a purely gaussian process the Lindbladian is taken to be of the form given in this link. Here for the Amplitude damping effect $$A$$ is taken as $$\sqrt{2}a$$ and $$L^{\dagger}=\sqrt{2}a^{\dagger}$$ and for the Purely Dephasing case $$A=A^{\dagger}=\sqrt{2}a^{\dagger}a$$. But I don't understand how can I get these Collapse operators for these specific cases.

• You may want to go the Hamiltonian $\rightarrow$ Redfield eq. (aka 2nd order in perturbation approximation) $\rightarrow$ Lindblad eq. way. Start with the purely dephasing Hamiltonian for a bosonic mode interacting with some thermal bath and have a look at physics.stackexchange.com/questions/229319/… for general pointers on how to proceed.
– udrv
Commented Dec 25, 2016 at 0:34