Measuring an adiabatic geometric phase

Take a smoothly varying Hamiltonian with a nondegenerate selected eigenstate and a gap to all other levels along the entire protocol. In the adiabatic limit the evolving state follows this eigenvector up to a phase. Substituting this form into the Schrödinger equation yields the negative time integral of energy divided by Planck's reduced constant, together with the connection integral already defined. The dynamical term changes when the time spent on the path changes; the geometric term depends on the oriented path itself.

The approximation needs more than a long total duration written on a stopwatch. Off-diagonal couplings generated by the changing Hamiltonian must be small relative to the squared energy separation, and the protocol must be sufficiently smooth with controlled endpoints. A small gap can defeat a seemingly slow sweep. The common local coupling ratio is a useful diagnostic, not a universal rigorous error bound for arbitrary histories, especially near crossings or with rapidly oscillating controls that repeatedly accumulate transitions.

Use $H=-(\Delta/2)\mathbf n\cdot\boldsymbol\sigma$ with a positive constant energy gap $\Delta$ and a unit vector at fixed polar angle. The ground-state spinor points along that vector and has energy minus half the gap. Increasing its azimuth through a full turn produces the negative half-solid-angle geometric phase under our convention. Its dynamical phase is positive because the ground-state energy is negative. Specifying the Hamiltonian sign is necessary to identify which eigenstate has which geometric phase.

Let the azimuth increase at angular speed $\omega$. The off-diagonal derivative overlap has magnitude $|\omega|\sin\theta/2$. Dividing by the gap angular frequency gives a dimensionless measure of unwanted interlevel coupling. At the poles the direction is actually fixed as azimuth changes, so this measure vanishes and no enclosed solid angle remains. Sending the gap toward zero does the opposite: the ratio diverges and the nondegenerate adiabatic description becomes unreliable, even if the ray diagram appears smooth.

Imagine one coherent interferometer arm follows the closed eigenstate protocol and the other supplies a reference. The internal states must match again at recombination, or distinguishability reduces the fringe visibility. If the arms have equal amplitudes and a calibrated relative phase, the output probabilities depend on the cosine of their total phase difference. A common phase applied to both arms cannot be read out. This is why merely following one normalized eigenvector and measuring its population does not establish a geometric-phase experiment.

Dynamical phases can be characterized by changing the traversal duration, or suppressed using carefully designed reference and echo sequences. Such procedures have their own control assumptions; they are not automatic consequences of drawing a closed path. Reversing the same path reverses the geometric term while leaving an equal energy-time integral unchanged, offering a useful comparison when coherence and timing are maintained. Finite-speed leakage, calibration errors, and environmental decoherence can alter the signal and must not be misidentified as an unexpected topological phase.