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arXiv:2003.13737

Geometric phase of very slow neutrons

Erik Sjöqvist

Published Mar 30, 2020• 0 citations
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Abstract

The geometric phase (GP) acquired by a neutron passing through a uniform magnetic field elucidates a subtle interplay between its spatial and spin degrees of freedom. In the standard setup using thermal neutrons, the kinetic energy is much larger than the typical Zeeman split. This causes the spin to undergo nearly perfect precession around the axis of the magnetic field and the GP becomes a function only of the corresponding cone angle. Here, we perform a plane wave analysis of the GP of very slow neutrons, for which the precession feature breaks down. Purely quantum-mechanical matter wave effects, such as resonance, reflection, and tunneling, become relevant for the behavior of the GP in this low energy scattering regime.

DOI
10.48550/arxiv.2003.13737

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The geometric phase (GP) acquired by a neutron passing through a uniform magnetic field elucidates a subtle interplay between its spatial and spin degrees of freedom. In the standard setup using thermal neutrons, the kinetic energy is much larger than the typical Zeeman split. This causes the spin to undergo nearly perfect precession around the axis of the magnetic field and the GP becomes a function only of the corresponding cone angle. Here, we perform a plane wave analysis of the GP of very slow neutrons, for which the precession feature breaks down. Purely quantum-mechanical matter wave effects, such as resonance, reflection, and tunneling, become relevant for the behavior of the GP in this low energy scattering regime.
Geometric phase phenomena in single neutrons have been observed in polarimeter and interferometer experiments. Interacting with static and time dependent magnetic fields, the state vectors acquire a geometric phase tied to the evolution within spin subspace. In a polarimeter experiment the non-additivity of quantum phases for mixed spin input states is observed. In a Si perfect-crystal interferometer experiment appearance of geometric phases, induced by interaction with an oscillating magnetic field, is verified. The total system is characterized by an entangled state, consisting of neutron and radiation fields, governed by a Jaynes-Cummings Hamiltonian. In addition, the influence of the geometric phase on a Bell measurement, expressed by the Clauser-Horne-Shimony-Holt (CHSH) inequality, is studied. It is demonstrated that the effect of geometric phase can be balanced by an appropriate change of Bell angles.
Geometric phase phenomena in single neutrons have been observed in polarimeter and interferometer experiments. Interacting with static and time dependent magnetic fields, the state vectors acquire a geometric phase tied to the evolution within spin subspace. In a polarimeter experiment the non-additivity of quantum phases for mixed spin input states is observed. In a Si perfect-crystal interferometer experiment appearance of geometric phases, induced by interaction with an oscillating magnetic field, is verified. The total system is characterized by an entangled state, consisting of neutron and radiation fields, governed by a Jaynes-Cummings Hamiltonian. In addition, the influence of the geometric phase on a Bell measurement, expressed by the Clauser-Horne-Shimony-Holt (CHSH) inequality, is studied. It is demonstrated that the effect of geometric phase can be balanced by an appropriate change of Bell angles.
The energy eigenstates of a spin-12\frac{1}{2}21​ particle in a magnetic field confined to a plane, define a planar spin. If the particle moves adiabatically around a loop in this plane, it picks up a topological Berry phase that can only be an integer multiple of $\ensuremath{\pi}$. We propose a neutron-interferometry test of the Berry phase for a circulating planar spin induced by a magnetic field caused by a very long current-carrying straight wire perpendicular to the plane. This Berry phase causes destructive interference in the direction of the incoming beam of thermal neutrons moving through a triple-Laue interferometer.
Neutron resonance spin echo phenomena, produced by resonance coils with adiabatic passage of the neutron spin, are investigated experimentally and theoretically. The adiabatic passage of the neutron spin through a resonance coil requires a specific configuration of the magnetic field. The solution of the Schr\"odinger equation, obtained for the required configuration, shows the probability of a spin-flip process; it also shows the phase shift, which the neutron experienced, when it flies through it. The precession phase inside the coil consists of three contributions in the rotating frame approach. The first, biggest contribution is the phase of the rotating frame ${\ensuremath{\omega}}_{0}\ensuremath{\tau}.$ The second is the precession phase of the neutron spin in the rotating frame since it follows adiabatically the effective field as seen in this frame. The third, smallest contribution is Berry's phase since the magnetic field rotates over an angle approaching $\ensuremath{\pi}$ in this rotating frame. This rotation is followed adiabatically by the neutron spin. Finally, the amplitude of the interference pattern and the phase shift between the neutron-spin states are derived for a system consisting of two such flippers. The theoretical consideration is experimentally confirmed.
We consider the neutron-optical phenomena that emerge during the coherent interaction of a neutron with a sample when the neutron spin precesses in a magnetic field. As follows from general considerations, such an interaction gives rise to an extra precession phase, which is added to the Larmor precession phase. This phenomenon can be interpreted as a manifestation of the time delay due to a finite time of the neutron-sample interaction. The Larmor neutron spin precession with a constant frequency serves as a clock for measuring this time delay. We used such a clock to directly measure the difference between the neutron velocity in matter and its vacuum value. We also present the results of the first experiments in which Larmor clocks were used to measure the neutron tunneling time in the resonance of a quasi-bound state and the Bragg diffraction time. Prospects for further applications of the method are discussed.