# How to determine the collapse operator for a Lindblad equation

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 $A$.

• Please define your terms "collapse" operator etc. very carefully in the post. This is a great question and deserves to be written well. Aug 5, 2015 at 2:08
• @DanielSank I have edited it, is it better? Aug 5, 2015 at 2:18
• What environmental effect to do you want to model? This equation is somewhat general and can be applied to a variety of situations. Aug 5, 2015 at 3:01
• @JohnM pretty much any environmental effect.. But what I want is a procedure that I can use to find the collapse operator for this environmental effect to use in the Lindblad equation Aug 5, 2015 at 22:22
• I don't think there's a simple procedure to do this. The collapse operator should be derived by considering the specific physical processes concerned. You could check Carmichael's book "statistical methods in quantum optics 1". Oct 10, 2016 at 14:17

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 $$|\psi\rangle\langle\chi|$$ then the action of the Lindblad equation will be to kill all the probabilities related to states including $$|\chi\rangle$$ and move them to corresponding states including $$|\psi\rangle$$.
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.