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# 6.3. The Landau-Zener effect¶

Expected prerequisites

Before the start of this lecture, you should be able to:

- Solve a Hamiltonian.
- Explain the adiabatic theorem.

Learning goals

After this lecture you will be able to:

- Describe the role of the velocity in the evolution of a two level system.
- Describe the interplay between the gap and the velocity during the evolution.
- Extend the discussion to systems with more energy levels.

## 6.3.1. Adiabatic evolution of a quantum system¶

The problem of two-levels crossing is often encountered in quantum mechanics applications. The Landau and Zener effect describes how a two-level system evolves under a time-dependent Hamiltonian. Consider a system described as
The solution of this equation can be writen as,
From the adibatic theorem, we know that the system will remain in the eigenstate if it evolves adiabatically from to , that is . However, if the evolution is not sufficiently *slow*, the system will leak to high energy states as .

## 6.3.2. Two level system¶

Consider a two level system described by the Hamiltonian, Here is the coupling between the two levels, is time, and is a the velocity at which the Hamiltonian changes. We can find an explicit expression for the energy levels and eigenstates following the usual procedure. The time-dependent state of the system can be written as,

Landau and Zener found an explicit expression that describes the probability of tunnelling to a high energy state as . It is Here, , and . From the adiabatic theorem, we recall that when the system evolves in an adiabatic way, it will remain in the initial eigenstate as shown in the animation below.

import common from IPython.display import HTML, display common.configure_plotting() import landau_zener as lz times, ens1, probs1 = lz.landau_zener_data(L=2, v=0.0001, gap=1) anim = lz.animate_landau_zener(times=times, ens=ens1, probs=probs1, interval=100, L=2, title=r'adiabatic: $v=10^{-3}$') display(HTML(anim.to_jshtml(default_mode='loop')))