Beyond Adiabatic: Landau-Zener and Applications

Consider a two-level system where the diabatic (uncoupled) energies cross linearly in time:

where $\alpha$ is the sweep rate (energy per unit time) and $\delta$ is the coupling between the two states. The diabatic levels $\pm\alpha t$ cross at $t=0$, but the coupling $\delta$ creates an avoided crossing with minimum gap:

If the system starts in the lower adiabatic state as $t \to -\infty$, the probability of a diabatic transition (jumping to the upper adiabatic state) as $t \to +\infty$ is:

Equivalently, the probability of remaining in the same adiabatic state (following the avoided crossing) is:

Two important limits:

• Slow sweep ($|\alpha| \to 0$): $P_{\text{diabatic}} \to 0$. The system follows the adiabatic state (no transition). The adiabatic theorem is recovered.
• Fast sweep ($|\alpha| \to \infty$): $P_{\text{diabatic}} \to 1$. The system stays in the diabatic state, ignoring the coupling. This is the sudden limit.

The Landau-Zener formula can be derived by several methods. The essential steps in the contour integral approach (Stokes phenomenon):

• Step 1: The Schrodinger equation for the two-level system reduces to a second-order ODE (Weber's equation or parabolic cylinder equation).
• Step 2: The exact solution involves parabolic cylinder functions, which have known asymptotic behavior.
• Step 3: The transition amplitude is related to the Stokes multiplier of the differential equation -- a quantity that measures how the dominant asymptotic behavior switches across anti-Stokes lines in the complex time plane.
• Step 4: The result is $|c_2|^2 = e^{-2\pi\delta^2/(\hbar|\alpha|)}$, where the exponential arises from the imaginary part of the action evaluated at the complex crossing point $t^* = i\delta/\alpha$.

The key dimensionless parameter is $\pi\delta^2/(\hbar|\alpha|) = \pi\Delta E_{\min}^2/(4\hbar|\alpha|)$, which compares the gap squared to the sweep rate.

In molecular collisions, the inter-nuclear distance $R$ varies in time as the nuclei approach and separate. If two electronic potential energy curves have an avoided crossing at some distance $R_c$:

where $v$ is the nuclear velocity at the crossing point and $|\Delta F|$ is the difference in slopes of the diabatic curves. The system traverses the crossing twice (incoming and outgoing), so the total non-adiabatic transition probability is:

This "surface hopping" formula (due to Tully) is widely used in non-adiabatic molecular dynamics simulations of photochemical reactions, charge transfer, and energy transfer processes.

Modern quantum control techniques allow achieving the same final state as adiabatic evolution, but in finite (even short) time. The main approaches are:

1. Counterdiabatic (Transitionless) Driving

Add an auxiliary Hamiltonian $\hat{H}_{\text{CD}}$ that exactly cancels all non-adiabatic transitions:

The total Hamiltonian $\hat{H}_0(t) + \hat{H}_{\text{CD}}(t)$ drives the system along the exact adiabatic path at any speed. The challenge is that $\hat{H}_{\text{CD}}$ typically requires knowing the instantaneous eigenstates and may involve non-local operators.

2. Optimal Control

Design the time-dependent parameters $\vec{R}(t)$ to minimize transitions. Using variational calculus or numerical optimization, find the optimal schedule that achieves the desired state transfer in minimum time subject to experimental constraints.

3. Invariant-Based Inverse Engineering

Construct a Lewis-Riesenfeld invariant $\hat{I}(t)$ satisfying $d\hat{I}/dt + (i/\hbar)[\hat{H}, \hat{I}] = 0$. Design $\hat{H}(t)$ so that the invariant eigenstates interpolate between the desired initial and final states.

The adiabatic theorem provides an alternative paradigm for quantum computation, distinct from the standard gate model:

Step 1: Encode the computational problem in a "problem Hamiltonian" $\hat{H}_P$ whose ground state encodes the solution.

Step 2: Prepare the system in the ground state of a simple "initial Hamiltonian" $\hat{H}_0$ (easy to prepare, e.g., all spins in $|+x\rangle$).

Step 3: Slowly interpolate:

Step 4: By the adiabatic theorem, the system ends in the ground state of $\hat{H}_P$ -- the solution to the problem.

The runtime is determined by the minimum gap encountered during the evolution:

Key results:

• Adiabatic quantum computation is polynomially equivalent to gate-based quantum computation
• For NP-hard problems, the gap typically closes exponentially with system size, requiring exponential runtime
• For certain problems (e.g., Grover search), optimal gap engineering gives quadratic speedup
• D-Wave quantum annealers implement a version of this approach with ~5000 qubits

Quantum annealing is a practical implementation of adiabatic quantum computation for optimization problems. The problem Hamiltonian is typically an Ising model:

The initial Hamiltonian introduces quantum fluctuations via a transverse field:

The transverse field is gradually reduced to zero while the problem Hamiltonian is turned on. Quantum tunneling through energy barriers (enabled by the transverse field) can help the system find the global minimum more efficiently than classical simulated annealing in some cases.