How to determine the collapse operator for a Lindblad equation

Question

Given a Hamiltonian $H$, how can I relate the collapse operator for the Lindblad equation to a given environmental effect? Also, how can I relate the constant $gamma$ in front of the sum of the collapse operators to the full Hamiltonian?

For reference, the lindblad equation is:

$$dot rho = -i[H, rho] + left( gamma sum A rho A^dagger - frac{1}{2} A^dagger A rho - frac{1}{2} rho A^dagger A right) , .$$

When I say collapse operator, I am referring to the operator $A$.

Nikodem · Answer

There is probably an infinite number of possible environmental effects one can describe with the Lindblad equation. But one can gain some understanding of the collapse operators $A$ by considering some simple cases.
If $A$ is an orthogonal projecton-operator on a subspace $H_+$ then the action of the Lindblad equation will be to kill the coherences (off-diagonal terms in the density matrix $rho$) between the states in $H_+$ and $H_-$ (where the Hilbert space is $H = H_+ oplus H_-$).
If $A$ is a rotation-operator (I have no better name for it) of the form $|psiranglelanglechi|$ then the action of the Lindblad equation will be to kill all the probabilities related to states including $|chirangle$ and move them to corresponding states including $|psirangle$.
Since one can generate a great deal of different operators $A$ by combining the described rotations and projections this can help to interpret various environmental effects in terms of the collapse operators.
The constant $gamma$ (which must stand in front of the brackets!) describes only the strength of the collapse process. Essentially, it determines the time-scale of the collapse process.

How to determine the collapse operator for a Lindblad equation

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