Maxwell-Bloch equations

The Maxwell - Bloch equations describing the interaction of a quantum mechanical two-level system with an oscillating electric field. They are used for the description of absorption and emission of light in solids and gases, and in particular, play in the theoretical understanding of the gain in lasers a central role. Prerequisite is that the energy difference of the transition near the photon energy of the light is and that the other transitions of the system have significantly different transition energies.

Equations

Are the Maxwell - Bloch equations

With:

  • : Complex amplitude of the electric field
  • : Complex amplitude of the polarization
  • : Population inversion with and occupation number density of levels 1 and 2
  • : Number of Zweiniveusysteme per volume
  • : Frequency of the electrical field
  • : Frequency of the transition with
  • : Dephasing, coherence time of the polarization.
  • : Lifetime of the second state
  • Projection of Dipolübergangsmatrixelement to the direction of the electrical field
  • : Group velocity in the medium
  • : Phase velocity in the medium

Approximations

Coherent regime

In coherent regime, it is believed that the typical time derivatives and are much greater than the decay terms so

Applies. Thus, the Maxwell - Bloch equations take the form

Of. One can easily show that in this case

Applies. Therefore, there is the introduction of the so-called Bloch vector

With related. This movement equation

With the so-called Rabi frequency and the detuning.

In the case of so-called resonant coupling, i.e. and real one finds the equations

Are the solutions of this system of differential equations

With the so-called pulse area with

Thus lead and vibrations, which are driven by the electric field. This is called Rabi oscillations. With the third Maxwell Bloch equation, it is found on the assumption of a thin sample of length L, that For the re-emitted electric field

If now an incoming light pulse prepared so that you can invert with the medium completely. One then speaks of a pulse ( see figure). For the population inversion is zero, and the polarization is maximum. With this method, so you can bring a material in a well defined state.

Derivation

For the derivation of the Maxwell - Bloch equations describing the interaction between the electric field and atom in the so-called dipole approximation. The Hamiltonian of the system consists of two parts. The share of the atom without interaction with the electric field describes and the proportion of a dipole interaction between light and atom describes:

With

The wave function can be in the base of the unperturbed system as

Are shown. The Schrödinger equation is now

Multiplying by and inserting the base representation follows

It was exploited. The microscopic polarization of the system is now by

Given. For the time derivatives of the polar Ista Transportation components and follows

The equations were

Be used. The equation for simply results from the complex conjugate equation.

For the field-free case (), the polarization now oscillates harmonically. In real system, the polarization decays, however, which is why adding a decay term. The material constant called this phase relaxation. Furthermore, one uses the so-called rotating wave approximation. The standards they

And are neglected in the equation for and accordingly in the equation, since the neglected terms oscillate and thus in comparison to the terms with small. Thus follows for the polarization

Which by the approach still

Can be simplified. For the time derivative of the population inversion follows

Again, the population inversion would remain constant in the field-free case, why one adds a term with.

Here, the average lifetime of the excited state. Last you still need an equation for the electric field. This process starts from the wave equation

From. Substituting the already obtained correlations and approaches follows

And thus the last Maxwell - Bloch equation

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