HOW SCIENCE WORKS · PHYSICS
Landau–Zener Transition: What Determines Passage Through an Avoided Crossing
When two coupled quantum levels approach, they avoid crossing and leave a minimum gap. Sweeping a control parameter through that gap creates a contest between coupling strength and sweep speed.
See what changes
Read the measurement logic
Find where the model breaks
Read in order for the full explanation, or use a route to enter at the question that matters most.
Scientific note: The equations below describe idealised regimes. Real measurements require calibration, uncertainty analysis and tests of competing mechanisms.
Chapter index
Core idea
Evidence and calculation
Limits and transfer
CHAPTER 1 OF 8
The puzzle: two levels approach but do not simply cross
Uncoupled states can have energies that cross as a field or bias changes. Coupling hybridises them and opens an avoided crossing with a finite minimum gap.
A system driven through this region may follow an instantaneous eigenstate or retain its original diabatic character. Landau–Zener theory assigns probabilities to those outcomes.
CHAPTER 2 OF 8
The mechanism: time available for state mixing
Near the avoided crossing, coupling rotates amplitude between the two diabatic states. A slow sweep gives the system time to follow the changing eigenstate; a rapid sweep leaves less time to adjust.
The same passage can therefore look like state transfer in one basis and branch retention in another. Naming the measurement basis prevents verbal confusion.
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CHAPTER 3 OF 8
The model: gap squared divided by sweep rate
For coupling g and linear rate v of the diabatic energy difference, the ideal probability of remaining on the diabatic branch is PD = exp[-2πg²/(ħv)].
A larger gap suppresses diabatic passage exponentially. A faster sweep increases it. This elegant result assumes an isolated two-level system and a sufficiently broad, approximately linear sweep.
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CHAPTER 4 OF 8
A dimensionless worked example
Let g²/(ħv) = 0.50. Then PD = exp(-π) ≈ 0.043, so the ideal adiabatic-transfer probability is about 0.957.
If the sweep rate doubles with the same gap, the ratio becomes 0.25 and PD = exp(-π/2) ≈ 0.208. The outcome is highly sensitive to timing near the gap.
| Parameter change | Effect on P_D | Interpretation |
|---|---|---|
| Gap/coupling g increases | Decreases exponentially | More adiabatic following |
| Sweep rate v increases | Increases | More diabatic passage |
| Repeated crossing | Interference possible | Phase becomes observable |
| Decoherence increases | Fringes wash out | Open-system model needed |
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CHAPTER 5 OF 8
Experimental evidence at a controlled crossing
A 1981 experiment with lithium Rydberg states used a pair of Stark levels to test the diabatic transition probability under conditions close to the Landau–Zener assumptions.
Scientists vary sweep speed and coupling, prepare a known initial state, and measure final populations. Agreement across the full curve is stronger evidence than one well-fitted point.
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CHAPTER 6 OF 8
Interference from repeated passages
Cross the avoided region twice and the amplitudes from alternative histories can interfere. The resulting Stückelberg fringes depend on the phase accumulated between passages.
Those fringes verify coherence but also make final population depend on timing outside the immediate crossing. A one-pass probability cannot describe every multi-pass experiment.
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CHAPTER 7 OF 8
Limits: noise, decoherence and extra levels
Environmental noise can randomise phase or move the crossing. Relaxation changes populations during the sweep, and neighbouring levels can turn the problem into a multistate network.
Researchers change temperature, waiting time and sweep span, then compare two-level, open-system and multilevel models. The simplest formula remains a baseline, not a universal fit.
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CHAPTER 8 OF 8
Why the model travels so widely
Landau–Zener logic appears in atomic physics, molecular collisions, superconducting qubits, spin control and band transport. It is a standard language for controlled non-adiabaticity.
Its broader lesson is that a quantum outcome depends not only on endpoints but on the route and speed used to move between them.