How to calculate the resonant florescent spectrum of 2-level

[zzihuai]

I am working on calculating the interaction between 2-level atoms and an coherent laser field. I can calculate the radiation intensity versus time from the atoms. I want to calculate the spectrum now. I know the spectrum can be calculated as the Fourier transform of ⟨$\langle \sigma^\dagger (t)\sigma (t+\tau)\rangle $,where σ(t)=|0><1| is the lower operator of the atom. I am not sure how to calculate the expectation value containing 2 different time. I heard that it is possible to do so by using the quantum regression theorem. But I don't really understand it, could someone tell me how to use quantum regression theorem using computer explicitly? I am using MATLAB and treating the operators as matrix. Thank you.

 

[eljose79]

Thank you for your question regarding calculating the spectrum of a coherent laser field interacting with 2-level atoms. You are correct in your understanding that the spectrum can be calculated using the Fourier transform of the expectation value of the lower operator of the atom at two different times. To do this, you will need to use the quantum regression theorem.

The quantum regression theorem is a powerful tool in quantum mechanics that allows us to calculate the time evolution of an operator by using the time evolution of its expectation value. In your case, you will need to use the following formula:

⟨$\langle \sigma^\dagger (t)\sigma (t+\tau)\rangle $ = Tr{ρ(t)U(t)σ†U(t+τ)σ}

Where ρ(t) is the density matrix at time t, U(t) is the time evolution operator, and σ is the operator of interest. In your case, you will need to replace σ with the lower operator of the atom, σ(t) = |0><1|.

To use this formula in MATLAB, you will need to first define your density matrix, time evolution operator, and the lower operator of the atom as matrices. Then, you can use the "trace" function in MATLAB to calculate the trace of the product of these matrices, which will give you the expectation value at two different times. You can then use this expectation value to calculate the spectrum using the Fourier transform.

I hope this helps you in your calculations. If you have any further questions, please do not hesitate to ask. Good luck with your research!