= Berry phase
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A <Berry phase> is the <geometric phase> acquired when a nondegenerate <energy eigenstate> follows a closed cycle of parameters of a <Hamiltonian operator> in the ideal limit of the <adiabatic theorem>. The <energy eigenstate> must stay separated from other <energy eigenstates> by a nonzero <spectral gap>, and the parameters must change sufficiently slowly. Returning the <Hamiltonian operator> to its starting value returns the state to the same ray, but its <quantum phase> can remember the journey. The total <quantum phase> contains both the <dynamical phase> and the <Berry phase>.
The beauty of the <Berry phase> is that, within the <adiabatic theorem>'s limit, it depends on the oriented path of the parameters rather than how long the journey takes. For a <spin one-half> <qubit>, the <Berry phase of a Bloch-sphere latitude> turns that memory into half a <solid angle>. <Berry-phase interference with a reference path> converts this geometry into a detection <probability>.
Michael Berry's 1984 paper formulated the <geometric phase> for cycles governed by the <adiabatic theorem> and derived the <solid angle> result for <spin>. Its bibliographic details are Proceedings of the Royal Society of London A, volume 392, pages 45–57, DOI 10.1098/rspa.1984.0023. https://michaelberryphysics.wordpress.com/publications/[Berry's authored publication list, entry 120: Quantal phase factors accompanying adiabatic changes (1984)]{ref}; https://stuff.mit.edu/afs/athena/course/8/8.06/spring06/handouts/Berry_1984.pdf[Berry's original paper, available through MIT]{ref}. The wider concept of <geometric phase> does not require every cycle to obey the <adiabatic theorem>: Aharonov and Anandan formulated a <geometric phase> for general cyclic quantum evolution in 1987. https://doi.org/10.1103/PhysRevLett.58.1593[Aharonov and Anandan, Phase change during a cyclic quantum evolution, Physical Review Letters 58, 1593–1596 (1987)]{ref}.
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