anticrossing

Anticrossing, also known as avoided crossing, is a phenomenon in quantum mechanics and related fields where two energy levels that would cross as a system parameter is varied instead repel each other due to coupling between the states. The result is a characteristic minimum separation, or gap, at the point where the crossing would occur if the states were uncoupled.

A common way to describe anticrossing is with a two-level model. Let E1(p) and E2(p) be the energies of two bare states as functions of a control parameter p, and let V be the coupling between them. The Hamiltonian can be written as a 2×2 matrix: [ E1(p) V ; V E2(p) ]. The eigenvalues are E±(p) = (E1+E2)/2 ± sqrt( ((E1−E2)/2)^2 + V^2 ). If E1(p0) = E2(p0) at some p0, the energies do not cross; instead they split by ΔE = 2|V|. The corresponding eigenstates are mixtures of the bare states, with a mixing angle determined by tan(2θ) = 2V/(E1−E2). As p passes through p0, the character of the eigenstates can interchange.

In dynamics, whether the system follows an energy level smoothly (adiabatic) or jumps between levels (nonadiabatic) depends on the sweep rate across the anticrossing. The Landau–Zener model provides a probabilistic description of such transitions.

Anticrossings occur in various settings, including molecular spectroscopy, quantum dots, superconducting qubits, and photonic systems. They often reveal coupling strength, symmetry considerations, and the structure of interacting subsystems. The presence or absence of a gap reflects whether the states can mix under the system’s symmetries.