Problem set 2: Jaynes-Cummings Hamiltonian (10 Points) Consider the Jaynes-Cummings Hamiltonian H = huyởa + hoa a + A (6

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Problem set 2: Jaynes-Cummings Hamiltonian (10 Points) Consider the Jaynes-Cummings Hamiltonian H = huyởa + hoa a + A (6,4 + 6_a") with the operators ôz = e)(el-19X91,ô+ = leg and ở_ = 19Xe] Show that the respective Hamilton matrix can be written in the basis of bare states {l 9, n+1), e,n)} in the following form 2n +1 H. - **") (n + 1/ ) ( 7 ) + 1/3 (22+ 21VA FI = hw n 2 where w is the frequency of the electromagnetic field, n the number of photons, A=wo-w the detuning of the light of the atomic transition between the states 9) and e) and X is the dipole coupling strength of this transition. Diagonalize this matrix and determine the eigenstates, i.e. find the dressed states of the system.