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Communications in Theoretical Physics

Exactly-Analytical Approach for Neutrons in Magnetic Field and Berry's Phase Effect in Moving Translation Reference Frame

Xiaoguang Wang, Chang-Pu Sun

Published Oct 30, 1994• 0 citations
Publisher

Abstract

In connection with Bitter–Dubbers experiment, an exact solution for the quantum dynamics of neutrons in magnetic field which is harmonically inhomogeneous (helical) and changes harmonically with time is obtained by invoking a time-dependent transformation with spin–space coupling. It predicts that a beam of neutrons with monotonous momentum will be split into two beams by the magnetic field. As its limiting case, the Berry's geometric phase in moving reference frame appears while the field is very strong.

DOI
10.1088/0253-6102/22/3/383

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In connection with Bitter–Dubbers experiment, an exact solution for the quantum dynamics of neutrons in magnetic field which is harmonically inhomogeneous (helical) and changes harmonically with time is obtained by invoking a time-dependent transformation with spin–space coupling. It predicts that a beam of neutrons with monotonous momentum will be split into two beams by the magnetic field. As its limiting case, the Berry's geometric phase in moving reference frame appears while the field is very strong.
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The exact wave function and rigorous solution of Bloch equation for spin-1/2 under the rotating magnetic field are obtained in this paper. Berry's geometric phase can be obtained by taking the adiabatic limit on the exact wave function, and be compared with the slow helical neutron experiment done by Bitter et al. For general cyclic solution, the analytic expressions of AA geometric phase and dynamic phase are obtained, and a perfect explanation to a controversial helical optics fiber experiment done by Chiao et al is given. Another theorem in this paper shows that both Berry's dynamic phase and Berry's geometric phase can be obtained by taking the adiabatic limit on the AA dynamic phase. An orthogonal state method is proposed to reach parallel transport, which is practical for the non-adiabatic geometric quantum computation.
I derive the effective phase of the spin precession for a neutral particle with spin 1∕2 moving in a superposition of constant and radio frequency fields. The fields are perpendicular to each other at all times, and the radio frequency field is slowly rotating with angular speed ω. The derivation is accomplished with the help of the exact solution of the Schrödinger equation. I show that the geometrical (or Berry) phase is the linear term in the power series expansion of the effective phase in terms of ω. I also discuss the role of the Berry phase in the experimental search for an electric dipole moment of the neutron with ultracold neutrons.
Abstract We propose a novel framework to describe geometric phases in quantum systems under non-adiabatic conditions by introducing the concept of a hidden geometric phase. Conventional geometric phases, such as the Berry phase, rely on adiabatic evolution, limiting their applicability in rapidly changing systems. Here, we remove this constraint by reinterpreting the geometric phase as arising from a dynamically evolving reference basis, independent of the external topological features. The hidden phase is revealed through transitionless quantum control techniques, ensuring pure geometric phase accumulation even in non-adiabatic regimes. Our method offers an exact solution to the neutron spin rotation phase in the Bitter–Dubbers experiment, aligning more closely with experimental data without depending on adiabatic approximations. This unexpected result broadens our understanding of the geometric phase observed in neutron spin rotation beyond the adiabatic conditions that are conventionally required.
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.
This article obtained the exact wave function and accurate solution of Bloch equation for spin-1/2 particle under the rotating magnetic field. Berry's geometric phase was obtained by taking the adiabatic limit on the exact evolutional wave function and was compared with the slow helical neutron experiment done by Bitter and Dubbers. The analytic results of AA geometric phase and the total phase have been obtained for a general cyclic evolution in a conic magnetic field. This article also presented a general description for orthogonal state method. By taking the initial state as the two superposed orthogonal states with equal weight, we proved that the off-diagonals of geometric and dynamic phases cancel exactly at any time. The diagonal parts of dynamic phase also cancel automatically. For the uniform rotating magnetic field, the diagonal geometric phase will evolve as a linear function like the Berry's case. We also discussed the relationship between our studies and the experiment recently done by Du et al.