Rabi model9
The Rabi model is a semi-classical model. This means that we use a classical representation of the field
For the atom we use a quantized description, so the atom has two states; the ground state and the excited state
The atom-field interaction is described by the interaction Hamiltonian
where
The interaction Hamiltonian can be seen as the energy flow between the atom and the field. The total Hamiltonian of a quantum mechanical atom-field interacting system is
The free-atom Hamiltonian is
We choose the energy of the ground state of the atom to be zero, so the free-atom Hamiltonian becomes
Because we do not use a quantum field, there is no free-field Hamiltonian. So the total Hamiltonian becomes
The state vector of the system is
Substituting this expansion in the time-dependent Schrödinger equation
gives
leads to the set of coupled first-order differential equations for the amplitudes
As initial conditions we assume that all the population is in the ground state at
After we have expanded
At this point we introduce the detuning Δ
By integrating the differential equation 3.14, the solution is
where
is the Rabi frequency. The probability that the atom is in its excited state is
Equation 3.18 is plotted in figure 9 for various values of the detuning Δ. The smaller the detuning the
larger the probability of finding the atom in its excited state. If you want to see Rabi oscillations, you need an
atom in its excited state (see figure 8). For the case of exact resonance, Δ
all the atomic population has been transferred to the excited state. This is the ideal situation for Rabi oscillations.
Click here to learn more about the Jaynes-Cummings model. |