A crossing sweep separates diabatic survival from ground-state following

Choose a fixed diabatic basis in which the diagonal energies are vt/2 and −vt/2, with positive sweep slope v, and take a constant real coupling g after a fixed basis-phase choice. The uncoupled diagonal difference grows at rate v. The exact instantaneous energies are plus and minus the square root of (vt/2)²+g². Their smallest separation is 2g at time zero. The two diabatic energy lines cross, while the adiabatic eigenvalues avoid crossing when g is nonzero.

At large negative time the first diabatic basis state is the lower-energy state. At large positive time that same fixed basis state is the higher-energy state. Therefore remaining in the same diabatic state through the sweep is a nonadiabatic outcome when the system began on the lower branch. Following the lower adiabatic branch instead transfers the population to the other diabatic state. Confusing these labels can turn a correct exponential into the opposite physical prediction about success or failure.

For unitary evolution from asymptotically negative to asymptotically positive time in this linear constant-coupling model, the probability to remain in the original diabatic state is exp(−2πg²/(ħv)). If the initial state is the lower adiabatic state, this is also the final excited-adiabatic-state probability. The lower-branch following probability is one minus that exponential. The result is exact for the ideal model, not a first-order perturbation estimate in the crossing region.

Slower sweep lowers the diabatic survival probability, while increasing coupling suppresses it through the square of g. Written in terms of the minimum gap, the same exponent contains πΔ_min²/(2ħv). This shows why using the gap where a formula expects the off-diagonal coupling changes the exponent by four. In the zero-coupling limit the fixed basis states never mix and diabatic survival is one; the instantaneous ground label then switches across a true crossing where the simple gapped adiabatic theorem no longer applies.

A finite experimental sweep can approximate the asymptotic formula when the endpoint detuning magnitudes greatly exceed both g and √(ħv), and the relevant crossing region is fully traversed. Endpoint preparation errors and coherent finite-time corrections may otherwise matter. Multiple crossings can interfere through accumulated phases, and more than two nearby levels can invalidate the reduction. Dephasing, relaxation, nonlinear sweep profiles, and time-dependent coupling require their own models. A local avoided-crossing plot alone does not authorize using the ideal exponential for every driven system.

The quantum lab can display the endpoint distinction without simulating the sweep. Reset and set both controls to θ=0° and φ=0°, predicting P(1)=0. Reset again and set θ=180° and φ=0°, predicting P(1)=1. These are the opposite fixed-basis populations that ideal lower-branch following connects across the crossing. Playback only adds relative phase and cannot change either population. There is no slope, coupling, avoided-gap, or Landau–Zener transition control, so these endpoint states must not be presented as a measured sweep trajectory.