Which Hamiltonian to use?

[Pu ZHANG]

Dear all, 

I'm struggling with writing the Hamiltonian in QuTiP code. Particularly, it seems both interaction and Schrodinger picture Hamiltonian could be used. Here are some examples from a master thesis titled "simulation of electromagnetically induced transparency and absorption" (https://etd.ohiolink.edu/!etd.send_file?accession=miami1376438830&disposition=inline): 

#interaction hamiltonian

H_i=e1g1*omega2*(sigma_g1e1.dag()+sigma_g1e1)+

e1g2*omega1*(sigma_g2e1.dag()+sigma_g2e1)

H_b=(uB)*(-sigma_g1g1+0*sigma_e1e1+sigma_g2g2)

H=H_i+H_b

(from Appendix C)

#system hamiltonian

H=(-delta_min)*(sigma_ee-sigma_gg)+(phi)*(sigma_ee-sigma_uu)+

omega1*(sigma_ge+sigma_ge.dag())+omega2*(sigma_ue+sigma_ue.dag())

(from Appendix B)

The first piece of code uses interaction hamiltonian, and the second uses Schrodinger picture one. 

Question One: would the two hamiltonians (if they are describing the same system) give the same physical results (expectation values)? 

Question Two: In the second case, I suppose the operator sigma_ue is fast varying (not in the rotating frame). Shouldn't such fast varying behavior be cancelled by an exponential (provided by the external laser) in the Hamiltonian? I mean something like this: omega2 * sigma_ue * exp(i * \omega_2 * t). 

Thanks! 

Best regards, Pu Zhang

--

Faculty at School of Physics, Huazhong University of Science and Technology

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