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 $A$.

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    $\begingroup$ Please define your terms "collapse" operator etc. very carefully in the post. This is a great question and deserves to be written well. $\endgroup$ – DanielSank Aug 5 '15 at 2:08
  • $\begingroup$ @DanielSank I have edited it, is it better? $\endgroup$ – TanMath Aug 5 '15 at 2:18
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    $\begingroup$ What environmental effect to do you want to model? This equation is somewhat general and can be applied to a variety of situations. $\endgroup$ – John M Aug 5 '15 at 3:01
  • $\begingroup$ @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 $\endgroup$ – TanMath Aug 5 '15 at 22:22
  • $\begingroup$ 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". $\endgroup$ – Pu Zhang Oct 10 '16 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.


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