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Journal of South China Normal University

GEOMETRIC PHASES FOR HELICAL NEUTRON AND OPTICAL FIBER

Hu Lian

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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 this paper we define a non-dynamical phase for a spin-1/2 particle in a rotating magnetic field in the non-adiabatic non-cyclic case, and this phase can be considered as a generalized Berry phase. We show that this phase reduces to the geometric Berry phase, in the adiabatic limit, up to a factor independent of the parameters of the system. We could add an arbitrary phase to the eigenstates of the Hamiltonian due to the gauge freedom. Then, we fix this arbitrary phase by comparing our Berry phase in the adiabatic limit with the Berry's result for the same system. Also, in the extreme non-adiabatic limit our Berry phase vanishes, modulo 2π2π2π, as expected. Although, our Berry phase is in general complex, it becomes real in the expected cases: the adiabatic limit, the extreme non-adiabatic limit, and the points at which the state of the system returns to its initial form, up to a phase factor. Therefore, this phase can be considered as a generalization of the Berry phase. Moreover, we investigate the relation between the value of the generalized Berry phase, the period of the states and the period of the Hamiltonian.
This article summarizes the recent results about geometric and topologic quantum computation.Based on a rigorous theorem concerned on orthogonal superposition quantum state proved in this paper,an orthogonal state method is proposed to carry out the non-adiabatic geometric quantum computation.And these geometric quantum gates are demonstrated in detail by two two-level systems: one is the system of spin-1/2 particle under NMR uniform rotating magnetic field;the other is asymmetric nano-SQUID circuit.The paper presents a perfect explanation for the early controversial helical fiber experiment done by Chiao R.Y.,also proposes a scheme of constructing topologic gates and illustrates these gates still with above two two-level systems.Finally,a brief comment is given on the controllability of quantum gates with the asymmetric nano-SQUID circuit.
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.