Maxwell–Bloch Image Gallery

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Self induced transparency
Coherence transfer
Frenquency doubling
Raman diffusion

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 two-level 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 nπ-pulse. 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 4π-pulses.
Frequency matching by changing the pulse duration.
1, 2, and 4π-pulses.
The shaped is unchanged
4π-pulse in two different space locations.

Coherence transfer

Contrarily to self-induced transparency, cohenrence transfer is typically an experiment which necessitates the full Maxwell–Bloch model since coherences have to be modelled. With a three-level model and choosing correctly the entries of the dipole moment matrix, we can transfer an initial coherence between levels 1 and 2 (which implies that tese levels should be populated) to a coherence betzeen levels 2 and 3.

Real part of the three populations.
Real part of the three coherences.

Frequency doubling

Frequency doubling is a phenomenon that could also be modelled coupling nonlinear Schrödinger equations. We display here two examples based on three-level atoms.

The first rest-case is not physically-relevant since it is performed in 1 space dimension and we have to assume that the dipole moment are non-zero 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 test-case also assumes thaht dipole moment are non-zero 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 ω0 gives raise to two new frequencies called Stokes (ω0r) and anti-Stokes (ω0r) frequencies, where the Raman frequency ωr only depend on matter parameter and not on the incident wave.

In a first test-case, we check that the shift ωr 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 ωr 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 anti-Stokes waves) as well as Raman diffusion on triple frequencies.
On the second test-case, 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.