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- If one looks carefully, there is a slight displacement of the centre from zero detuning. This is due to the difference in the free Hamiltonian in the two Hamiltonians. By transformation to the interaction picture, the only difference is in the time-dependent detuning of the interaction Hamiltonian involving gb. The latter version gives eiνbtbˆ†bˆeiνvtvˆ†vˆbˆvˆe−iνbtbˆ†bˆe−iνvtvˆ†vˆ→gbbˆvˆe−iΔbt since [bˆ†bˆ,bˆ]=bˆ and [vˆ†vˆ,vˆ]=vˆ. In the earlier version, the transformation ei(νb+νv)bˆ†vˆ†bˆvˆbˆvˆe−i(νb+νv)bˆ†vˆ†bˆvˆ does not give gbbˆvˆe−iΔbt since [bˆ†vˆ†bˆvˆ,bˆvˆ]≠bˆvˆ. Thus the difference is due to the transformation on the exponential term. Therefore the two results are different slightly. The latter is correct as it follows the standard unitary transformation.
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