Self induced transparency
Self induced transparency is a phenomenon that can also be observed for the cubic nonlinear Schrödinger equations.
It is due to a balance between the natural dispersion of the wave and the nonlinearity: the wave thus propagates unchanged through the material.
This balance is obtained for the Maxwell–Bloch equation taking a wave frequency equal to the transition frequency of the material (supposed to be a twolevel medium).
The Rabi frequency (E.p/hbar) should also be an integer multiple n of this frequency.
The incident pulse, which should have the form of a sech is the called a nπpulse.
If the initial state of the material is an equilibrium state, the material undergoes n total inversions before comming back to its equilibrium state.
We present three results to illustrate this phenomenon.
We plot the time evolution of the field in red (its amplitude is normalized) and the population of the excite level in black.



Frequency matching by changing intensity.
1, 2, and 4πpulses. 
Frequency matching by changing the pulse duration.
1, 2, and 4πpulses. 
The shaped is unchanged
4πpulse in two different space locations. 
Frequency doubling
Frequency doubling is a phenomenon that could also be modelled coupling nonlinear Schrödinger equations.
We display here two examples based on threelevel atoms.
The first restcase is not physicallyrelevant since it is performed in 1 space dimension and we have to assume that the dipole moment are nonzero between all the levels wich is rigorously impossible for parity reasons of the wave functions,… but numerics allow to do a lot of things.
The second testcase also assumes thaht dipole moment are nonzero but, since we can take two different polarization directions, this is now reasonable.
The double frequency is observed in a perperdicular direction to that of the inicident wave.
Optical rectification (zero frequency) also takes place in this polarization.


1D doubling. 
2D doubling. 
Raman diffusion
Raman diffusion is the fact that a incident field with frequency ω_{0} gives raise to two new frequencies called Stokes (ω_{0}ω_{r}) and antiStokes (ω_{0}+ω_{r}) frequencies, where the Raman frequency ω_{r} only depend on matter parameter and not on the incident wave.
In a first testcase, we check that the shift ω_{r} is indeed constant.
This frequency is not easy to determine.
The advantage of Maxwell–Bloch simulations is that this frequency is a result of the simulations and does not have to be a priori known.
A coupled nonlinear Schrödinger simulation is also possible but necessitates to know the value of ω_{r} beforehand.
Using very strong coupling parameters and letting the wave and the matter interact a relatively long time, we can not only observe Raman diffusion but also multiple diffusions (Raman diffusions of the Stokes and antiStokes waves) as well as Raman diffusion on triple frequencies.
On the second testcase, we see the strength of a full model where the involved frequencies are not determined a priori.
The corresponding Schrödinger model would need to couple at least 16 nonlinear equations, with the difficulty of determining the Raman Frequency.


Raman frequency is constant. 
Multiple Raman diffusions. 