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Physica Scripta

Exactly-solvable dynamics of neutron in a helical magnetic field and geometric effects in the limit of strong field

Xiaoguang Wang, Chang-Pu Sun

Published Jun 1, 1995• 0 citations
Publisher

Abstract

By invoking the use of a space-spin coupling transformation, the exact Schrödinger evolution wavefunction is fortunately obtained for a neutron moving in a helical magnetic field. This result just enjoys the effect of Berry's phase or its induced gauge field in Bitter-Dubbers experiment as its limiting case with strong field. It is also predicted from the exactly-dynamical analysis that a pulse of polarized neutrons with monotonous momentum will be split into two pulses with different momentum shifts accompanying the spin-down and spin-up states respectively.

DOI
10.1088/0031-8949/51/6/001

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By invoking the use of a space-spin coupling transformation, the exact Schrödinger evolution wavefunction is fortunately obtained for a neutron moving in a helical magnetic field. This result just enjoys the effect of Berry's phase or its induced gauge field in Bitter-Dubbers experiment as its limiting case with strong field. It is also predicted from the exactly-dynamical analysis that a pulse of polarized neutrons with monotonous momentum will be split into two pulses with different momentum shifts accompanying the spin-down and spin-up states respectively.
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.
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.
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.
By invoking the Rabi rotation technique, the Schrödinger wavefunction of a neutral particle with a magnetic moment and a high-spin is exactly solved in a harmonically-changing magnetic field. This result is then applied to illustrate what the accuracy of the quantum adiabatic approximation is exactly and how the non-adiabatic transition vanishes in accompaniment with the appearence of Berry's geometrical phase. The discussion of this paper verifies C. N. Yang's point of view about the generalization of the quantum adiabatic approximation in connection with Berry's phase.
By making use of the improved Born-Oppenheimer approximate method, we deal with the separation of spin variable and space variable of neutral particle with arbitrary spin in inhomogeneous magnetic field and non-adiabatic corrections for the problems are thereby computed as perturbations. We also analysis the geometrical properties of loop phase of the induced gauge potential and point out its observable effects relating to Bitter and Dubbers' experiment. Finally, we discuss the transitions between the instantaneous spin states as non-;-adiabatic effects resulting from inhomogeneity of the magnetic field.