Landau–Zener Avoided Crossing
Separate an instantaneous avoided crossing from its infinite-sweep transition probability.
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Landau–Zener Avoided Crossing
In quantum mechanics, the Landau–Zener formula calculates the probability that a system transitions between energy states when the parameters of its Hamiltonian are varied over time. An avoided crossing occurs when two energy levels approach each other but do not cross, instead repelling each other due to a non-zero coupling between the states.
Worked Example
Consider a two-level system described by the Hamiltonian H(t) = [[vt/2, g], [g, -vt/2]] in units where ℏ = 1.
The instantaneous energy eigenvalues are given by:
E(t) = +/- sqrt((vt)^2 + 4g^2)/2
Let's calculate the asymptotic transition probability for g = 0.5 and v = 1.
- Calculate the probability of staying in the same diabatic state: P_stay = exp(-2π g²/v) = exp(-2π (0.5)²/1) ≈ 0.20787957635076193.
- Calculate the probability of changing to the other diabatic state: P_change = 1 - P_stay ≈ 0.7921204236492381.
- At t=0, the gap is 2g = 1.0. The energies are ± g = ± 0.5.
Experiment
Observe how the coupling strength g and the sweep speed v affect the transition probabilities and the energy gap.
- Set the Coupling g to different values and observe how the minimum energy gap changes. When g=0, it is a true crossing.
- Increase the Sweep speed v and observe how the probability of remaining in the initial diabatic state changes. A faster sweep leads to a higher probability of staying in the same diabatic state.
- Change the Probe time to see the instantaneous energy gap at different points during the sweep.
Limitations
This module demonstrates the basic Landau-Zener transition for an infinite linear sweep. It does not simulate finite-duration sweeps or non-linear parameter variations. Additionally, it only considers a simple two-level system without dissipation or decoherence.
References
The exact asymptotic derivation of the two-level diabatic transition probability exp(-2πg² / ℏv) across a linear avoided crossing is presented in the following lecture notes:
Experience it interactively
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