Adiabatic Dynamics Numerics

Adiabatic-dynamics numerics compares an actual propagated state with a slowly changing instantaneous eigenspace. It must separate physical transitions from errors in time integration, eigensolvers, and phase conventions. The adiabatic approximation states the assumptions for following an eigenstate; here we test the resulting dynamics computationally.

Helpful background. Excited-state targeting supplies eigenspace tracking, and the numerical Schrödinger equation supplies trajectory refinement.

Instantaneous projectors and leakage

At each relevant time, diagonalize $H(t)$ or compute the target eigencluster. For an isolated normalized eigenvector $u_n(t)$, define

$$P_n(t)=|u_n(t)\rangle\langle u_n(t)|,\qquad p_n(t)=\langle\psi(t)|P_n(t)|\psi(t)\rangle.$$

The leakage $1-p_n$ is independent of the arbitrary phase of $u_n$. For a degenerate or nearly degenerate target cluster, use the sum of its projectors. Individual populations inside such a cluster can be basis-dependent.

Resolve the smallest relevant gap and the matrix elements of $\dot H$. A common local indicator for distinct levels is

$$\eta_{mn}(t)= \frac{\hbar|\langle u_m|\dot H|u_n\rangle|} {|E_m-E_n|^2}.$$

A small indicator is useful but is not a universal theorem for every drive, endpoint, and long-time interference pattern. Direct convergence in sweep duration and comparison with propagated leakage remain necessary.

Phase tracking and geometric information

Adjacent numerical eigenvectors may differ by arbitrary phases. When their overlap is nonzero, align the new vector so that the overlap with the previous one is real and positive. For a cluster, use the polar factor or SVD of the overlap matrix to align frames without selecting unstable individual vectors.

The dynamical phase of an isolated level is $-\hbar^{-1}\int E_n(t)\,dt$. Remove it only after converging both the eigenvalues and quadrature. For a closed parameter loop, a discrete overlap product

$$\gamma_{\rm geom} =-\operatorname{Arg}\prod_{j=0}^{M-1} \langle u(t_j)|u(t_{j+1})\rangle, \qquad u(t_M)=u(t_0),$$

approximates the Berry phase in the convention $\gamma=i\int\langle u|\dot u\rangle dt$. The closing overlap is essential for gauge invariance. Coarse sampling with nearly orthogonal adjacent states makes the estimate unstable.

Geometric phase extraction is a separate observable from population following: small leakage does not automatically guarantee a precise accumulated phase over a long sweep.

An avoided-crossing benchmark

Use

$$H(t)=\frac12\begin{pmatrix}vt&\Delta\\\Delta&-vt\end{pmatrix}, \qquad v>0,\quad \Delta>0,$$

where $v$ has units of energy per time and the minimum gap is $\Delta$. For the infinite-time Landau–Zener passage, starting in the lower adiabatic state at $t\to-\infty$, the probability to end in the upper adiabatic state is

$$P_{\rm na}=\exp\!\left[-\frac{\pi\Delta^2}{2\hbar v}\right].$$

A finite simulation interval need not equal this asymptotic formula. Increase the endpoint magnitude until boundary preparation and finite-time interference are negligible, then refine the time step at fixed endpoints. Finally vary $v$ to test the physical adiabatic limit. Mixing these three changes obscures which error is being reduced.

At $t=0$, the local indicator between the two states is $\hbar v/(2\Delta^2)$. This agrees with the expected gap-squared scale but does not reproduce the full transition probability on its own.

Checking leakage and accumulated phase

Record the gap curve, maximum leakage, final leakage, eigenpair residuals, and time-step convergence. Check several intermediate times: transitions can occur and partially reverse, leaving a deceptively small final error. For a closed loop, compare the overlap-product phase with the propagated phase after removing the dynamical contribution.

If the target gap closes, a single isolated-eigenstate comparison loses its premise. Enlarge the tracked subspace or state explicitly that the calculation explores nonadiabatic dynamics through a degeneracy.

Exercises

In the Landau–Zener model, how must the sweep rate change to keep the asymptotic transition probability fixed when the minimum gap is halved?

Solution

The exponent depends on $\Delta^2/v$. Halving $\Delta$ requires reducing $v$ by a factor of four. Endpoint and numerical-resolution requirements must also be reconsidered.

References

• Blanes, S., F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Physics Reports 470, 151–238 (2009), doi:10.1016/j.physrep.2008.11.001 — controlled driven propagation.
• Vitanov, N. V., and B. M. Garraway, “Landau–Zener model: Effects of finite coupling duration,” Physical Review A 53, 4288–4304 (1996), doi:10.1103/PhysRevA.53.4288 — finite-duration versus asymptotic transitions.