For a general evolution of a quantal system, the geometric phase measured with reference to a given initial state is derived as an integral of a function of the pure state density operator by invoking the Pancharatnam connection continuously.

In mid-1950s, Pancharatnam [1] encountered the geometric phase associated with the evolution along a geodesic triangle on the Poincaré sphere. We generalize his 3-vertex phase and employ it as the fundamental building block, to geometrically construct a general ray-space expression for geometric phase. In terms of a reference ray used to specify geometric phase, we delineate clear geometric meanings for gauge transformations and gauge freedom, which are generally regarded as mere mathematical abstractions.

We present an overview of polarized neutron experiments observing SU(2) phases. The first experimental separation of geometric and dynamical phases, the explicit verification of Pauli anticommutation and the first observation of interference amplitudes and phases in noncyclic evolutions are described. These experiments elucidate the physics of phases and phase jumps propounded by the Pancharatnam connection.

A general gauge-invariant formalism for parallel transport, geodesics and geometric phase based on the pure state density operator is propounded. A single-query quantum search algorithm is proposed.

Charge neutrality, a spin 1/2 and an associated magnetic moment of the neutron make it an ideal probe of quantal spinor evolutions. Polarized neutron interferometry in magnetic field Hamiltonians has thus scored several firsts such as direct verification of Pauli anticommutation, experimental separation of geometric and dynamical phases and observation of non-cyclic amplitudes and phases. This paper provides a flavour of the physics learnt from such experiments.

Using a right-angled magnetic air prism, we have achieved a separation of ∼10 arcsec between ∼2 arcsec wide up and down-spin peaks of 5.4 Å neutrons. The polarised neutron option has thus been introduced into the SUSANS instrument. Strongly spindependent SUSANS spectra have been observed over ±1.3 × 10⁻⁴ Å⁻¹ range for several magnetic alloy samples. Spatial pair-distribution functions for the up and down-spins as well as the nuclear and magnetic scattering length density distributions in the micrometer domain, have been deduced from these spectra.

While studying neutron deflections produced by a magnetic prism, we have stumbled upon a simple ‘geometric’ formula. For a prism of refractive indexn close to unity, the deflection simply equals the product of the refractive powern − 1 and the base-to-height ratio of the prism, regardless of the apex angle. The base and height of the prism are measured respectively along and perpendicular to the direction of beam propagation within the prism. The geometric formula greatly simplifies the optimisation of prism parameters to suit any specific experiment.