Landau-Zener Transition

The Landau-Zener transition is the canonical exactly solvable model of an avoided crossing swept in time. It explains when a state follows an instantaneous eigenstate adiabatically and when it jumps nonadiabatically across the avoided crossing. For a lighter introduction, see Landau-Zener Problem: First Encounter.

The abrupt opposite limit, in which the state remains fixed while the Hamiltonian basis changes, is developed in Sudden Approximation.

For the static wave-mechanics picture of localized states, tunneling coupling, and the avoided-crossing gap, see Coupled Wells and Avoided Crossings.

Nonadiabatic Coupling explains how this ideal sweep is embedded in molecular wavepacket dynamics, derivative couplings, and surface-hopping approximations.

Conical Intersections explains why a one-dimensional avoided-crossing cut can be a displaced slice through a multidimensional zero-gap seam.

For a convention translator, explicit adiabatic-state mapping, complex-time exponent check, and independently converged finite-window propagation, see Landau–Zener Transition Worked Example.

For downloadable Magnus and RK4 propagators, endpoint-basis audits, phase-sensitive convergence, and retained data, see Landau–Zener Simulation.

Use the two-level Hamiltonian

$$H(t) = \begin{pmatrix} vt/2 & \Delta \\ \Delta & -vt/2 \end{pmatrix}, \qquad v>0, \qquad \Delta>0.$$

The diabatic energy difference is $vt$, and the minimum adiabatic gap is $2\Delta$.

Diabatic and Adiabatic Bases

The diabatic basis is the fixed basis in which the Hamiltonian has diagonal entries $\pm vt/2$. If $\Delta=0$, the diabatic energies cross at $t=0$.

For $\Delta\ne0$, the instantaneous eigenvalues are

$$E_\pm(t) = \pm \frac12 \sqrt{v^2t^2+4\Delta^2}.$$

They do not cross. The minimum gap is

$$E_+(0)-E_-(0)=2\Delta.$$

The adiabatic basis is the instantaneous eigenbasis of $H(t)$.

Landau-Zener Probability

Suppose the system starts in one instantaneous adiabatic eigenstate at $t\to-\infty$. In the convention above, the probability to make a nonadiabatic transition to the other adiabatic branch after passing through the avoided crossing is

$$P_{\mathrm{LZ}} = \exp\left( - \frac{2\pi\Delta^2}{\hbar v} \right).$$

The probability to follow the adiabatic branch is therefore

$$P_{\mathrm{ad}} = 1-P_{\mathrm{LZ}}.$$

Different books may describe the same exponential as the probability to remain in the same diabatic state. Always check whether the probability is being quoted in the diabatic or adiabatic basis.

Slow and Fast Passage

The dimensionless adiabaticity parameter is

$$\gamma = \frac{\Delta^2}{\hbar v}.$$

Slow passage means $v$ is small, so $\gamma\gg1$. Then

$$P_{\mathrm{LZ}}\ll1,$$

and the system follows the adiabatic eigenstate with high probability.

Fast passage means $v$ is large, so $\gamma\ll1$. Then

$$P_{\mathrm{LZ}}\approx1,$$

and the system is unlikely to follow the changing adiabatic eigenstate.

Relation to the Adiabatic Condition

The adiabatic approximation is controlled by the ratio of nonadiabatic coupling to the square of the instantaneous gap. For a two-level avoided crossing, the most dangerous region is near the minimum gap. The Landau-Zener formula is valuable because it gives the exponentially small nonadiabatic probability for the ideal linear sweep, not just a qualitative condition.

The result shows that adiabaticity improves exponentially when the gap is large or the sweep is slow:

$$P_{\mathrm{LZ}} \sim e^{-2\pi\gamma}.$$

Semiclassical Interpretation

The exponential resembles a tunneling factor. In complex time, the adiabatic energy gap vanishes at branch points away from the real axis. The transition amplitude can be interpreted semiclassically as controlled by an action accumulated along a contour to those complex turning points.

This connection is one reason Landau-Zener physics sits naturally between time-dependent perturbation theory, adiabatic methods, and WKB-style asymptotics.

Applications

Landau-Zener transitions appear in:

• atomic and molecular avoided crossings;
• spin dynamics in swept magnetic fields;
• superconducting and semiconductor qubits;
• band transitions in driven crystals;
• quantum annealing and adiabatic passage protocols;
• Stückelberg interferometry from repeated passages.

The simple formula assumes an isolated two-level crossing, linear detuning near the crossing, constant coupling, and coherent unitary evolution. Noise, dissipation, many levels, and nonlinear sweeps require extensions.

For a Bloch packet driven through a small band gap, Semiclassical Dynamics of Bloch Electrons owns the field-to-sweep mapping and decides when isolated-band motion fails. This page retains the two-level probability, basis convention, finite-window audit, and coherent-crossing physics.

Common Mistakes

• Quoting $P_{\mathrm{LZ}}$ without saying whether it is diabatic or adiabatic.
• Forgetting that $v$ is the slope of the diabatic energy difference in this convention.
• Applying the two-level formula when nearby levels also participate.
• Treating slow passage as exact adiabaticity for finite gaps and finite sweep rates.
• Ignoring dephasing or relaxation in experimental applications.

Worked Calculation

Landau–Zener Transition Worked Example applies this formula to one dimensionless sweep. It distinguishes the same-diabatic-state probability from the adiabatic-branch transition probability, translates common factors-of-two conventions, and separates step-size error from finite-endpoint error in direct Schrödinger propagation.

Landau–Zener Simulation extends that calculation into a reproducible numerical laboratory with three propagators, an endpoint sweep, full-state error, norm diagnostics, and downloadable artifacts.

Exercises

1. For the Hamiltonian above, find the instantaneous eigenvalues.

Solution

The characteristic equation is

$$\det \begin{pmatrix} vt/2-E & \Delta \\ \Delta & -vt/2-E \end{pmatrix} = 0.$$

This gives

$$E^2 = \frac{v^2t^2}{4} + \Delta^2.$$

Therefore

$$E_\pm(t) = \pm \frac12 \sqrt{v^2t^2+4\Delta^2}.$$

2. What happens to $P_{\mathrm{LZ}}$ as $\Delta$ increases with $v$ fixed?

Solution

The probability is

$$P_{\mathrm{LZ}} = \exp\left( - \frac{2\pi\Delta^2}{\hbar v} \right).$$

Increasing $\Delta$ makes the exponent more negative, so $P_{\mathrm{LZ}}$ decreases. A larger avoided-crossing gap improves adiabatic following.

3. Why does the formula become convention-dependent when stated in words?

Solution

The same physical evolution can be described in the diabatic basis or in the instantaneous adiabatic basis. A transition in one basis can correspond to staying on the same labeled state in another basis. The exponential is unambiguous only after the Hamiltonian convention and the meaning of the quoted probability are stated.

References

• L. D. Landau, “Zur Theorie der Energieübertragung. II,” Physikalische Zeitschrift der Sowjetunion 2, 46-51, 1932.
• C. Zener, “Non-adiabatic crossing of energy levels,” Proceedings of the Royal Society A 137, 696-702, 1932.
• E. C. G. Stueckelberg, “Theorie der unelastischen Stösse zwischen Atomen,” Helvetica Physica Acta 5, 369-422, 1932.
• N. V. Vitanov and B. M. Garraway, “Landau-Zener model: Effects of finite coupling duration,” Physical Review A 53, 4288-4304, 1996.