I would like to understand the relationship between the Quantum Langevin Equations, Input-Output formalism, and Master Equation. I already read these questions:
- Lindblad and Input-Output Formalism in Quantum Optics
- Input-output formalism of quantum optic for driven superconducting qubit in a waveguide
but I think my thread is a little bit different. My question is: how can I formally introduce dissipation and decoherence (using Lindblad operators) in a Quantum Langevin Equation?
In my case, I started with the Quantum Noise theory by Gardiner and Collet, to describe a system composed of a qubit, a cavity, and a transmission line. My idea is to study how the system evolves after a photon pulse (coherent state) travels from the transmission line to the cavity, but for a complete description I need also to introduce internal cavity dissipation, thermal qubit excitation, qubit dephasing, ... My Quantum Langevin Equation has this form $$ \frac{\text{d}\hat a}{\text{d}t} = i[\hat H_S, \hat a] - [\hat a, \hat c^\dagger]\left[\frac{\gamma}{2}\hat c + \sqrt{\gamma}\hat b_\text{in}(t)\right] + \left[\frac{\gamma}{2}\hat c^\dagger + \sqrt{\gamma}\hat b^\dagger_\text{in}(t)\right][\hat a, \hat c] $$ Is it correct to go straightforward and add the Lindblad operator or is lacking formality? For example, using this form $$ \frac{\text{d}\hat a}{\text{d}t} = i[\hat H_S, \hat a] - [\hat a, \hat c^\dagger]\left[\frac{\gamma}{2}\hat c + \sqrt{\gamma}\hat b_\text{in}(t)\right] + \left[\frac{\gamma}{2}\hat c^\dagger + \sqrt{\gamma}\hat b^\dagger_\text{in}(t)\right][\hat a, \hat c] + \sum_n \left(\hat L_n^\dagger\hat a \hat L_n - \frac 12\hat a \hat L_n^\dagger \hat L_n - \frac 12 \hat L_n^\dagger \hat L_n \hat a\right) $$
