Sweep a two-level system through an avoided crossing—adjust speed to transition between adiabatic and diabatic behavior
The Landau-Zener problem is one of the most important exactly solvable models in quantum mechanics. When a two-level quantum system passes through an avoided crossing—where two energy levels approach but never touch due to coupling—the probability of transitioning between levels depends critically on how fast you sweep through the crossing. This is the heart of the adiabatic theorem: go slow enough, and the system stays in its instantaneous eigenstate (adiabatic); go fast, and it follows the diabatic (uncoupled) states.
The plot shows energy as a function of time (or equivalently, detuning δ(t) = αt). The solid curves are the adiabatic energy eigenvalues—they repel each other, forming an avoided crossing with minimum gap 2V. The dashed lines are the diabatic energies—the states the system would follow if there were no coupling. The crossing point is at t=0, where δ=0. The coupling V determines the size of the gap; the sweep rate α determines how fast you pass through.
The population plot shows the probability of being in each adiabatic eigenstate as a function of time. Starting in the lower diabatic state (far to the left), the system initially tracks the lower adiabatic state. As it passes through the avoided crossing, some probability leaks to the upper state. The final probability of remaining in the lower state is the Landau-Zener survival probability P_LZ = 1 - exp(-2πΔ²/ℏα), where Δ = V is the half-gap. For slow sweeps (small α), this is nearly 1 (adiabatic). For fast sweeps (large α), it's nearly 0 (diabatic).
The adiabatic regime (α → 0) means the system always stays in the instantaneous eigenstate—it follows the adiabatic curve. The diabatic regime (α → ∞) means the system ignores the coupling and follows the uncoupled diabatic states, jumping from lower-left to upper-right. The Landau-Zener formula interpolates between these limits. The transition probability is maximized when πΔ²/α ≈ 1—this is the "intermediate" regime where quantum interference between paths matters most.
The Hamiltonian for a two-level system with linear time dependence is:
H(t) = ½[δ(t)σ_z + Vσ_x]
where δ(t) = αt is the time-varying detuning (the diagonal energy difference), V is the off-diagonal coupling, and α is the sweep rate. At t → -∞, the diabatic states are well separated. As t → 0, they cross. The coupling V prevents a true crossing, creating an avoided crossing with minimum gap 2V. The instantaneous eigenvalues are E_± = ±½√(δ² + V²).
The probability of a diabatic transition (jumping from lower diabatic to upper diabatic) is:
P_diabatic = exp(-2πΔ²/ℏα)
where Δ = V is the half-gap at the avoided crossing. In natural units (ℏ=1), this becomes exp(-2πV²/α). The adiabatic probability (staying in the same adiabatic state) is P_adiabatic = 1 - P_diabatic. This formula is exact for linear sweeps and is derived by solving the Schrödinger equation in parabolic cylinder functions. It's one of the crown jewels of time-dependent quantum mechanics.
The adiabatic theorem states that if a system starts in an eigenstate and the Hamiltonian changes sufficiently slowly, the system remains in the instantaneous eigenstate (up to a phase). The quantitative condition is α ≪ Δ²—the sweep rate must be much slower than the square of the energy gap. At the avoided crossing, the gap is smallest (2V), so this is where the adiabatic condition is most stringent. If α is too large, the system can't "keep up" with the changing Hamiltonian, and diabatic transitions occur.
In quantum mechanics, energy levels of the same symmetry cannot cross—they must repel. This is the non-crossing rule (von Neumann-Wigner theorem). The coupling V lifts the degeneracy, creating an avoided crossing. If you could turn off V, the diabatic curves would cross at t=0. With V ≠ 0, they repel. The size of the gap is 2V at the closest approach. This avoided crossing structure appears everywhere: molecular physics (curve crossing), atomic physics (level repulsion), quantum optics (dressed states), quantum computing (driven qubits).
Perfect Adiabatic Evolution: Set α=0.01 (very slow) and V=0.5. Watch the population stay almost entirely in the lower adiabatic state as it passes through the crossing. The system follows the adiabatic curve smoothly. This is the textbook adiabatic limit: P_diabatic ≈ exp(-2π·0.25/0.01) ≈ 10⁻³⁴.
Diabatic Jump: Set α=0.5 (very fast) and V=0.5. The system jumps from lower diabatic to upper diabatic, almost completely ignoring the avoided crossing. P_diabatic ≈ exp(-2π·0.25/0.5) ≈ 0.28, so there's still some adiabatic character, but the jump is clearly visible. Increase α further to approach the pure diabatic limit.
Intermediate Regime: Adjust α until P_diabatic ≈ 0.5 (try α ≈ 0.08 for V=0.5). This is the regime where both paths (adiabatic and diabatic) contribute equally. The Landau-Zener formula predicts maximum "indecision" when 2πV²/α ≈ ln(2). This is where quantum mechanics is most non-classical.
Coupling Strength Effect: Fix α=0.1 and vary V from 0.1 to 2.0. Small V means a tiny avoided crossing—easy to jump diabatically. Large V means a wide gap—hard to jump, strong adiabatic following. The transition probability scales as exp(-V²), so doubling V drastically suppresses diabatic transitions.
Sweep Direction: Switch the initial state from lower to upper diabatic. Now you're sweeping from right to left on the energy diagram. The Landau-Zener probability is the same—the formula doesn't care about sweep direction. This symmetry is a consequence of time-reversal invariance.
Varying Start Time: Change t₀ (the starting time). Starting further back (more negative t₀) means you spend more time in the adiabatic state before reaching the crossing. The final transition probability is independent of t₀ (as long as you start far enough from the crossing), confirming that Landau-Zener is a local phenomenon—only the behavior near t=0 matters.
Oscillations in the Gap: Set α very small (0.02) and V=1.0. Watch closely as the populations evolve through the crossing. You'll see rapid oscillations (Rabi-like) within the avoided crossing region. These are Stückelberg oscillations—quantum interference between the two paths through the crossing. They're washed out in the fast sweep limit but visible when α is small.
Measure the Landau-Zener Formula: Pick V=0.5. For various α values (0.02, 0.05, 0.1, 0.2, 0.4), read off the final P_upper from the population plot. Plot ln(P_upper) vs α. You should get a straight line with slope -2πV² ≈ -1.57. This confirms the exponential form of the Landau-Zener formula empirically.
Landau-Zener transitions are not a curiosity—they're a fundamental building block of quantum control. Here's why:
Adiabatic quantum computing: AQC relies on the adiabatic theorem to solve optimization problems. You encode the solution in the ground state of a final Hamiltonian and slowly evolve from an easy initial state. Landau-Zener transitions at avoided crossings are the main source of errors—if you sweep too fast through a small gap, you leak to excited states and lose the solution.
STIRAP (stimulated Raman adiabatic passage): In atomic physics, you can transfer population between states by sweeping through an avoided crossing in the dressed-state picture. By using counterintuitive pulse ordering, you suppress spontaneous emission. This is Landau-Zener physics applied to light-atom interactions, and it's how we do robust quantum state transfer in the lab.
Molecular collisions and curve crossing: When molecules collide, their potential energy curves cross. Landau-Zener theory predicts the probability of hopping from one curve to another (e.g., from covalent to ionic configuration). This determines reaction rates, energy transfer, and chemical outcomes. It's how we model non-adiabatic chemistry.
Majorana fermions and topological qubits: In topological quantum computing, braiding operations involve sweeping parameters through avoided crossings. Landau-Zener transitions set the error rate for these operations. To achieve fault tolerance, you need exponentially suppressed errors, which means large gaps or slow sweeps.
Quantum annealing and optimization: D-Wave's quantum annealers work by slowly ramping a transverse field while turning on a problem Hamiltonian. The success probability depends on avoiding diabatic transitions at small gaps (where the problem is hard). Landau-Zener theory tells you how slowly you need to anneal.
Shortcuts to adiabaticity: Counter-diabatic driving and transitionless quantum driving use additional control fields to suppress Landau-Zener transitions, allowing you to achieve adiabatic outcomes with fast sweeps. These techniques require understanding exactly how LZ transitions arise, then engineering Hamiltonians to cancel them.
Landau and Zener's 1932 papers (derived independently) solved a problem that seemed hopelessly difficult: time-dependent quantum mechanics with a level crossing. Their formula has become a cornerstone of quantum physics, appearing in atomic, molecular, condensed matter, and quantum information contexts. Every time you sweep through an avoided crossing—whether with lasers, magnetic fields, or gate voltages—you're doing a Landau-Zener experiment.