Jaynes–Cummings model

The Jaynes -Cummings model (after Edwin Thompson Jaynes and Fred Cummings, also dressed -atom model (English as: " Model of the clothed ' atom ' ) ) describes the interaction of an atom with a monochromatic, resonant light field ( without consideration a polarization). It is a purely quantum mechanical approach to determine the energy values ​​and states of the total system atom - light field and to physical phenomena that occur in this interaction to explain. The Jaynes -Cummings model is the simplest non-trivial model the interaction of an atom describes an electromagnetic field.

In the Jaynes -Cummings model effects are evident, which are not explained in the semiclassical Rabi model. These include the change of the Landé factor in a high-intensity and high-frequency radio -frequency field, as well as a physical intuition for the Mollow triplet and the dipole force.

In the model described both the atom and the light field is treated quantum mechanically. The atom is here considered as two-state system, while the field is quantized according to the rules of quantum field theory. The consideration of the interaction between the atom and field in the Hamiltonian means that the states of the atom and the light field must be represented as a unit and can not be considered independently (hence the name of the " dressed " atom ).

Detailed Description

Energy values ​​without considering the interaction between the atom and field

The atom has two possible energy levels, the laser field infinite number corresponding to the number of photons. Without coupling, the total energy of the system is simply the sum of the two subsystems. The Hamiltonian is found to be in accordance with

Where and denote the Hamiltonian of the atom and the laser field.

In the ground state the atom has the energy and the excited state, the energy at the atomic resonance frequency. The energy of the laser field increases at a light frequency for each photon. The Hamiltonians are as follows:

With the creation and annihilation operators and. Plotting all possible energy levels on a scale on, the result is a printed circuit having discrete values ​​.

Taking into account the atom -field interaction

Shift by taking into account the interaction of the energy levels, this effect is called the Stark shift. In addition, the eigenstates of the atom, which can now be represented as a linear combination of the original ground and excited state of change. This coupled states is referred to as dressed states or dressed states. The fact that now both eigenstates contain an admixture of the original states, there is a new absorption and emission behavior, which explains, for example, the occurrence of the Mollow triplet.

If one assumes that neighboring states interact with one another in each case only on the energy ladder, the energy levels of the coupled atom - laser system can be determined by diagonalizing its Hamiltonian. This consists of the operator for the uncoupled system and an interaction term. The latter results in the dipole approximation, i.e., the wavelength of light is large compared to the wavelength of the atom:

With the quantized field

With the mode volume and the laser polarization and the dipole operator, which combines the two atomic states

This results in a frequency detuning of the laser with respect to the atomic resonance frequency, an energy shift of

With the Rabi frequency and the new eigenstates

With photons and a mixing angle,. For a derivation of the energy eigenvalues ​​and - states, see ( page 10) and for a derivation of the operators see.