Landau-Zener model & adiabacity

[TKH]

When we talk about rapid adiabatic passage in a two-level system, then what is the Landau-Zener model? I gues it has something to do with perturbation theory... but I'm not sure what.

Hope the question makes sense...

TKH

 

[SpectraCat]

When we talk about rapid adiabatic passage in a two-level system, then what is the Landau-Zener model? I gues it has something to do with perturbation theory... but I'm not sure what.

Hope the question makes sense...

TKH

The Landau-Zener model is an approximation for predicting diabatic passage (i.e. state crossing) in a two-level system. You define a phenomenological quantity called the Landau-Zener velocity, which is a measure of the probability that a state crossing will occur. You can think of it in terms of the following (non-physical) analogy ... imagine that the LZ velocity corresponds to the actual velocity of a classical particle approaching an avoided crossing. If it is moving slowly (i.e. low LZ velocity), then it "feels" the avoided crossing, and tends to stay on the adiabatic state and not cross. If the particle is moving fast (i.e. high LZ velocity), then it has a high probability to shoot through the avoided crossing before it even "knows" it is there, remaining on the diabatic state and "crossing" in the adiabatic picture.

The LZ velocity is directly proportional to the time derivative of the perturbation that is coupling the states .. the physical nature can be a magnetic or electric field, or a geometric parameter of a molecule, or anything else that gives rise to a coupling. The more quickly the perturbation is changing with time, the more likely a curve crossing is to be observed.

 

[DrDu]

Zener proposed the following Hamiltonian in which the nuclear motion is classical and
with constant velocity v. epsilon is the minimal energy splitting of the adiabatic surfaces:
$$H(t)=vt\sigma_z +\epsilon \sigma_x$$
The model was treated subsequently by Landau and Stueckelberg, who treated the nuclear motion quantum mechanically.