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.

Required background. Superposition and Relative Phase and Eigenvalues and Eigenvectors are used to distinguish amplitudes from populations in two changing bases. Landau–Zener Worked Example develops finite-endpoint propagation, and the simulation supplies numerical implementations.

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 $H(t)$. For this sign convention, the lower-energy state approaches the first diabatic basis vector at large negative time and the second at large positive time. Following the lower branch therefore changes the diabatic label; staying in the first diabatic state ends on the upper branch.

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

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

The same exponential is the probability to remain in the initial diabatic state in the infinite-endpoint limit. At finite endpoints, diabatic and adiabatic states differ, so these are different measurement probabilities. This formula also gives only a population: repeated passages require the scattering phases as well as the propagation phase between crossings.

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 uses the dimensionless diagnostic $\hbar|\langle e|\dot H|g\rangle|/(E_e-E_g)^2$, or equivalently $\hbar|\langle e|\dot g\rangle|/(E_e-E_g)$. 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 exact formula assumes an isolated two-level crossing, linear detuning for the entire ideal sweep, constant coupling, coherent unitary evolution, and asymptotic endpoints. Local linearization of a real protocol can approximate it only when the transition region fits inside that linear regime and the preparation/readout are sufficiently far from the crossing. Noise, dissipation, nearby levels, and nonlinear sweeps require separate analysis.

Semiclassical Dynamics of Bloch Electrons maps an applied field to a band-gap sweep. Nonadiabatic Coupling explains the additional nuclear-motion assumptions in molecular applications.

Exercises

Fast passage from a Fresnel integral

Treat the off-diagonal coupling $\Delta$ to first order in the diabatic basis. Show that the probability to change diabatic state is $2\pi\gamma+O(\gamma^2)$, consistently with $1-P_{\mathrm{LZ}}$. You may use $\left|\int_{-\infty}^{\infty}e^{-iat^2}dt\right|^2=\pi/a$ for $a>0$, with a convergence regulator understood.

Solution

The diabatic energy difference for a transition from the first to second state is $-vt$. Its accumulated phase is $-vt^2/(2\hbar)$, up to a reference phase. Thus

$$c_{2}^{(1)}=-\frac{i\Delta}{\hbar} \int_{-\infty}^{\infty}e^{-ivt^2/(2\hbar)}dt, \qquad |c_2^{(1)}|^2=\frac{2\pi\Delta^2}{\hbar v}=2\pi\gamma.$$

The exact probability $1-e^{-2\pi\gamma}$ has this leading term. It remains bounded when the first-order result fails. This probability is adiabatic-branch following, despite being a change of diabatic label.

Translate a gap convention

Another author writes $H=(\alpha t\,\sigma_z+g\sigma_x)/2$. What exponent follows, and what is the minimum gap?

Solution

Here $v=|\alpha|$ and $\Delta=|g|/2$. The minimum gap is $|g|$ and the nonadiabatic probability is $\exp[-\pi g^2/(2\hbar|\alpha|)]$. Confusing $g$ with the off-diagonal matrix element changes the exponent by four.

References

• L. D. Landau, “Zur Theorie der Energieübertragung. II,” Physikalische Zeitschrift der Sowjetunion 2, 46–51, 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.
• C. Zener, “Non-adiabatic crossing of energy levels,” Proceedings of the Royal Society A 137, 696–702, 1932.