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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. ^[ref]; ^[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. ^[ref].

A geometric phase is a quantum phase associated with the geometry of a path of state rays, after separating the dynamical phase. The Berry phase is its cyclic case governed by the adiabatic theorem; more general cyclic evolution can also have a geometric phase.

A theoretical physicist whose work includes geometric phases in quantum mechanics.

The angular momentum commutators used in the frame transformation are representation-independent. Therefore $H_R=\Delta S_z+{\Omega }_R{S}_x$ still holds. Rotating its axis onto $z$ gives $2s+1$ eigenvalues and exact laboratory solutionsNow $R(T)=(-1{)}^{2s}I$, so $e^{i{\Phi }_{\mu }}=(-1{)}^{2s}{e}^{-i\mu \Omega T}$. In the adiabatic limit the dynamical phase is $-\mu BT$ andIndeed $2\pi s+2\pi \mu \cos \theta$ and $-2\pi \mu (1-\cos \theta )$ differ by $2\pi (s+\mu )$, an integer multiple of $2\pi$. Integer spins return with a positive rotation sign and half-integer spins with a negative one; the geometric phase need not vanish for either.

Set ${\vartheta }_{\alpha }(t)={\hslash }^{-1}{\int }^t{E}_{\alpha }(s)ds$. Differentiating the expansion produces ${\dot{a}}_{\beta }|{\phi }_{\beta }⟩+a_{\beta }|{\dot{\phi }}_{\beta }⟩-i{E}_{\beta }{a}_{\beta }|{\phi }_{\beta }⟩/\hslash$ for each term. The energy term cancels $H|\Psi ⟩$ in the Schrödinger equation. Projecting the remainder with $⟨{\phi }_{\alpha }|$ and using orthonormality yieldsThe diagonal term is purely imaginary and supplies the geometric phase. Off-diagonal instantaneous eigenstate couplings transfer population; differentiating the eigenvalue equation gives $⟨m|\dot{n}⟩=⟨m|\dot{H}|n⟩/(E_n-E_m)$ when the levels are distinct.

The primary paper derives the geometric phase accompanying adiabatic transport around a closed parameter loop. Section 4 treats a spinning particle in a slowly changing magnetic field, with quantum phase proportional to its magnetic quantum number and the enclosed solid angle.