In quantum mechanics, the Berry phase is a phase acquired by quantum states when subjected to adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. It was named after Sir Michael Berry, and is also known as Pancharatnam-Berry or geometric phase. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. Generally speaking, it occurs whenever one can externally control at least two parameters affecting a wave.

Waves are characterized by amplitude and phase, and both may vary as a function of those parameters. The Berry phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could e.g. involve rotations but also translations of particles, which are apparently undone at the end. Intuitively one expects that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursion is a cyclic loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the Berry phase, and its occurrence typically indicates that the system's parameter dependence is singular (undefined) for some combination of parameters.

To measure the Berry phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the Berry phase. This mechanics analogue of the Berry phase is known as the Hannay angle.

Further reading

• Connections to other physical phenomena (such as the Jahn-Teller effect) are discussed here: [1]
• For the connection to mathematics, see curvature tensor.