Let's assume that states |1> and |2> are degenerate states and the system is prepared in state |1>. Also, the matrix element of electric dipole moment is not zero between these two states (=!0). If we interact this system with vacuum field, does this system remain in its initial state? (I know from Wigner Weisskopf theory that if these two levels were not degenerate and level |1> was the excited state, the system would decay with Einstein rate.)
The question, as posed, doesn't make sense, since the fact that the states are degenerate, meaning that H|1>=E|1> and H|2>=E|2>, doesn't have anything to do with the electric dipole moment-nor is there any notion as that of a vacuum field.
What may be meant is that, if one introduces an electric field, then this is enough to raise the degeneracy. However, if the system is, initially, in an eigenstate of the full Hamiltonian, it will remain in an eigenstate of the full Hamiltonian. So it's not enough that the degeneracy be lifted, to induce a transition; the initial state shouldn't be an eigenstate of the full Hamiltonian.
This is, once more, a homework assignment.
@Stam Nicolis
Thank you so much for your kind reply. Maybe I made the problem so simple that it looks like an assignment. In fact, let's look at the problem in a deeper but still a simple view. I plot energy levels of the system against a parameter(alpha). The value of alpha can be changed practically. In alpha=alpha_c, I have a energy level crossing. It means that for alpha is the ground state and state |2> is the first excited state. But for alpha>alpha_c these two states change their roles. For this case state |1> is first excited state and |2> is the ground state. I prepare the system in state |1> for alpha
Stam Nicolis, to the state 2.
If vacuum can cause this decay to other state when the states are not degenerate, I don't know, maybe it can do the same to these two degenate states. If this was a normal problem I could say that this will not happen for degenerate states but I have a sentence for this problem that when I tend the energies of these to states to a certain value, this sentence does not vanish.
Hi Omid,
The spontaneous emission rate from an excited state |1> to a ground state |2> is proportional to | |^2 and w^3, where is the matrix dipole moment and w=w1-w2 is the atomic transition frequency (see Ficek, Quantum interference and coherence). It is clear that if both the states are degenerate, then w=0 since they have the same energy. Therefore, spontaneous emission does not take place between degenerate states.
When an atom decays from an excited state to a ground state, it emits a photon with frequency equal to the transition frequency. It does not make sense physically for spontaneous emission to take place between degenerate states. Please check your problem thoroughly.
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