• TKH
In summary, the Landau-Zener model is an approximation for predicting diabatic passage in a two-level system. It is based on the concept of the Landau-Zener velocity, which is a measure of the probability of state crossing. This velocity is directly related to the time derivative of the perturbation that is coupling the states. The model was originally proposed by Zener and later treated quantum mechanically by Landau and Stueckelberg.
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

TKH said:
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.

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.

## 1. What is the Landau-Zener model?

The Landau-Zener model is a theoretical model used to study the quantum mechanical phenomenon of avoided crossings. It describes the probability of a quantum system undergoing a transition from one energy state to another as a function of the rate at which the energy levels are changing.

## 2. How does the Landau-Zener model relate to adiabaticity?

The Landau-Zener model is often used to study the adiabaticity of a quantum system. Adiabaticity refers to the process of a system remaining in its initial energy state as it evolves slowly in time. The Landau-Zener model helps determine the conditions under which a system will remain adiabatic or transition to a different energy state.

## 3. What factors affect the Landau-Zener transition probability?

The transition probability in the Landau-Zener model is influenced by several factors, including the rate of change of energy levels, the energy difference between the levels, and the coupling strength between the levels. The presence of external fields or noise can also affect the transition probability.

## 4. How is the Landau-Zener model applied in research?

The Landau-Zener model has been used in various fields of research, including quantum optics, condensed matter physics, and chemical reactions. It has been used to study phenomena such as quantum tunneling, Bose-Einstein condensation, and the behavior of atoms and molecules in external fields.

## 5. What are some limitations of the Landau-Zener model?

While the Landau-Zener model is a useful tool for studying avoided crossings and adiabaticity, it has some limitations. For example, it assumes that the energy levels are changing at a constant rate, which may not always be the case in real systems. It also does not take into account the effects of decoherence or non-adiabatic transitions, which can also affect the behavior of a system.

Replies
0
Views
494
Replies
5
Views
2K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
4
Views
3K
Replies
1
Views
957
Replies
4
Views
2K
Replies
24
Views
1K
Replies
6
Views
2K
Replies
0
Views
1K