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I have system which has states two spin states and an optically excited states. Everytime optically excited states relaxes, a photon is emitted, which I can detect a fraction of. If I simulatenously drive the transitions between spin states and optically excited states, and both drives resonant with their respective transitions, then systems enters a dark state and the optically excited states cannot be populated. However, if one of the drives is slightly detuned, the A will be populated and photons will be emitted

Because the system is coupled to a nucleus, the driving frequency at which the dark state is formed, shifts by the hyperfine coupling. Therefore when you drive the system at a specific frequency, based on the number of photons you receive you measure the nuclear state.

I want to write Kraus operators for the dark state. I guess this corresponds to the weak measurement examples of Kraus operators. I found two documents(one is Jonathon Gross's thesis: Weak measurements for quantum characterization and control and the other is: Weak-measurement-induced phases and dephasing: Broken symmetry of the geometric phase ) on it yet, I could not write the Kraus operator of the system

Any helps, sources, advises are very welcome

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