Quantum dot Quantum optics Nanocavity
Jaynes-Cummings model9

We now turn to the quantum electrodynamic version of the Rabi model. In this model we consider the same two level quantized atom as the Rabi model. However, we do not consider a classical field, but a quantized field of the form

  Equation 3.19 (3.19)

The interaction Hamiltonian is now

  Equation 3.20 (3.20)


  Equation 3.21 (3.21)

The total Hamiltonian of a quantum mechanical atom-field interacting system is

  Equation 3.22 (3.22)

We choose the zero energy level of the atom to be exactly halfway between the ground and excited state, so the free-atom Hamiltonian becomes

  Equation 3.23 (3.23)


  Equation 3.24 (3.24)

is the inversion operator.

The free-field Hamiltonian is

  Equation 3.25 (3.25)

In this case we can drop the zero-point energy term, because it does not contribute to the dynamics of the system.

  Equation 3.26 (3.26)

This approximation is known as the Jaynes-Cumming model.

We now introduce the atomic transition operators

  Equation 3.27 (3.27)

Because of parity consideration, one can say

  Equation 3.28 (3.28)

Now we may write

  Equation 3.29 (3.29)

where we have assumed that d is real. The total Hamiltonian of the system becomes now

  Equation 3.30 (3.30)


  Equation 3.31 (3.31)

Now we apply the Rotating Wave Approximation (RWA).

  Equation 3.32 (3.32)

We may break this Hamiltonian into two commuting parts

  Equation 3.33 (3.33)


  Equation 3.34 (3.34)


  Equation 3.35 (3.35)

respectively the electron ‘number’ and the excitation number, such that

  Equation 3.36 (3.36)

All the essential dynamics is contained in HII. HI contains only irrelevant phase factors.

We now consider a simple experiment with Δ=0. The atom is initially in the excited state |e> and the field initially in the number state |n>. The initial state of the system is |i>=|e>|n> and the final state is |f>=|g>|n+1>.

In the same way presented in the Rabi model, we obtain with the given initial conditions

  Equation 3.37 (3.37)

With a probability the system makes a transition to the state |f>

  Equation 3.38 (3.38)


  Equation 3.39 (3.39)

is the quantum electrodynamic Rabi frequency. In this case there are Rabi oscillations even for the case when n=0. These are the vacuum-field Rabi oscillations.