Landau-Zener Avoided Crossing (2D)

Avoided Crossing

Coupling C (half min. gap) 0.50

Sweep rate v 0.30

Live State

Sweep progress

Instantaneous gap Δ(t)

—

P(stay in start state)

100.0%

Min. gap (2C)

—

Final numeric vs analytic Landau-Zener P_LZ = exp(−πC²/v)

not run yet

How this differs from the 3D version

The 3D simulator numerically diagonalizes an N-spin Ising Hamiltonian (up to 16×16) to track a many-body ground state through an adiabatic sweep. This 2D companion isolates the single mechanism underneath every avoided crossing in that bigger problem — the exactly-solvable Landau-Zener two-level model — and checks the live numeric integration against its closed-form textbook result.

H(t) = [ v·t    C  ]
       [  C   −v·t ]     (ħ = 1)

E±(t) = ± sqrt((v·t)² + C²)     adiabatic energies
Δ(t)  = E+(t) − E−(t) = 2·sqrt((v·t)² + C²)

The state ψ(t) is evolved from t = −T (deep in the lower diabatic branch) to t = +T by real-time RK4 integration of i·dψ/dt = H(t)ψ — a genuinely independent 2×2 complex solve, not a projection of the 3D engine. The probability the system is still in its original diabatic branch at t = +T (a "diabatic transition", i.e. it failed to adiabatically follow the ground state through the crossing) has an exact closed form:

P_LZ = exp(−π C² / v)

Slow sweep (small v) → P_LZ → 0: the adiabatic theorem holds, the state smoothly follows the ground state across the gap and ends up in the other diabatic branch — exactly the "success" regime of the 3D annealer.
Fast sweep (large v) → P_LZ → 1: the sweep is too quick for the state to adjust (sudden approximation), so it is left behind in the original branch — the "diabatic transition" failure mode.
Coupling C sets the minimum adiabatic gap (2C) at t = 0 — a bigger gap makes the crossing easier to follow adiabatically at any given sweep rate, mirroring how a larger Δ_min helps a real annealer.

This exact formula is the v→0 / single-avoided-crossing limit of the same adiabatic theorem the 3D simulator's many-body Δ_min obeys only approximately (T ≫ 1/Δ_min²) — this page verifies that limit numerically rather than assuming it.

Diabatic energies ±v·t Adiabatic branches E±(t) Instantaneous state energy

P(start state) — numeric P(other state) — numeric P_LZ analytic (final)

Landau-Zener Avoided Crossing: Exact Two-Level Solver