The quantum–classical crossover of a field mode

On the ArXiv today, there’s a paper that answers my earlier question about the classical limit of the Jaynes–Cummings model. Apparently, if I just tried a little harder, I could have had another paper. Such is life.

Basically, in the Jaynes–Cummings model, the dynamics of a qubit and a single photon is given by

That is, the energy of the qubit, the energy of the photon, and the interaction between the two.  This predicts energy exchange between the photon and qubit.  The qubit will oscillate between zero and one.  But the theory predicts a decay of this oscillation as well as revivals.

A classical field interacting is described by .  This predicts oscillation of the qubit state between zero and one, but no decay.

It seems that the timescale for the oscillation in both cases is inversely proportional to the electric field strength.  The decay time is independent of the field strength.  So, in the large photon limit, we expect there to still be a decay of the oscillation, but we should still see many oscillations before.

The paper also has another correspondence, that I haven’t quite finished yet.  But it involves the Lindblad equation, which should make it fun.