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American Journal of Physics

A plane wave approach to particle beam magnetic resonance

Robert Golub, R. Gähler, T. Keller

Published Sep 1, 1994• 115 citations
Publisher

Abstract

We present a quantum mechanical treatment of particle beam magnetic resonance and related experiments, in terms of static plane waves, applicable to experiments with oscillating fields. The results yield an alternate view of Larmor precession, showing it as a beating between different waves, and help to shed new light on several aspects of the problems discussed. We discuss magnetic resonance experiments with both the Rabi and Ramsey methods, neutron resonance spin echo and a neutron interferometer experiment.

DOI
10.1119/1.17459

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We present a quantum mechanical treatment of particle beam magnetic resonance and related experiments, in terms of static plane waves, applicable to experiments with oscillating fields. The results yield an alternate view of Larmor precession, showing it as a beating between different waves, and help to shed new light on several aspects of the problems discussed. We discuss magnetic resonance experiments with both the Rabi and Ramsey methods, neutron resonance spin echo and a neutron interferometer experiment.
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Abstract The behavior of a magnetic dipole in magnetic fields is of interest in many systems including magnetic resonance imaging. The explanation presented in this chapter combines quantum physics and electricity and magnetism. The chapter begins with an explanation of magnetic dipole Larmor precession in static magnetic fields. The discussion then shifts to time-dependent magnetic fields and how they lead to transitions between the dipole’s energy states. The calculations are carried out both by using the time dependence of energy eigenstates and by solving the differential equations describing the quantum state vector amplitudes. The results are expressed in terms of the Rabi frequency and the Rabi formula. For the special cases of π and π/2 pulses of the oscillating magnetic field, the results show how those pulses implement quantum gates for the dipole system.
Neutron-multiwave-interference phenomena based on Ramsey's resonance method of ``separated oscillating fields'' are studied. A neutron passes through N successive resonant coils $(\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0}=2{\ensuremath{\mu}}_{n}{B}_{0}),$ which flip the neutron spin with a probability $\ensuremath{\rho}$ smaller than 1. These coils are separated by path lengths L over which a homogeneous field B1{B}_{1}B1​ is present. Because the spin-flip probability $\ensuremath{\rho}$ is smaller than 1, the number of waves for a neutron is doubled after each flipper, so as to produce 2N{2}^{N}2N neutron waves at the end of the setup. The phase difference between any pair of waves is a multiple of a ``phase quantum'' determined by the line integral of the field difference ${B}_{1}\ensuremath{-}{B}_{0}$ over the length L. Highly regular patterns of the quantum-mechanical probability R in ${(B}_{1},\ensuremath{\rho})$ space appear due to pair interference between individual waves. Possible applications of this phenomenon, such as a direct measurement of n-particle correlation function, are pointed out.
The observation of the 4π symmetry of spinor wave functions was one of the first and highly debated neutron interference experiments. Polarized neutrons have been used to verify the spin-superposition principle on a macroscopic level when the spin in one beam path is inverted with respect to the other one. This results in a spin polarization perpendicular to both initial spins. When a Rabi flipper is used the final spin polarization rotates in this perpendicular plane with the Larmor frequency of the flipper field. Thus, it produces a “frozen” Larmor spin rotation in free space. In this case the spin-flip is associated with an energy (photon) transfer which causes this effect. A magnetic Josephson effect-like situation exists when spin-flippers with slightly different resonance frequencies are used in both beam paths. When such a spin reversal is done in both beam paths a magnetic Josephson effect analog has been realized which shows a remarkably high energy resolution. Multiphoton exchange can be achieved when off-resonance fields are used.
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.
In continuation of previous work explicitly demonstrating the basic quantum-mechanical principles of the static and the time-dependent superposition of spinors, a new double-resonance coil experiment is described, where a spin-flip process associated with an exchange of energy quanta happens in both beams of a perfect-crystal neutron interferometer. It is shown that under the given circumstances of neutron self-interference coherence is preserved in spite of the energy transfer to every neutron, which may stimulate renewed epistemological discussions about the interpretation of quantum mechanics. Furthermore quantum beating on a macroscopic time scale has been observed by introducing an exceedingly small energy difference of 8.6\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}17}$ eV between the two interfering beams. According to the statistical accuracy of the data thereby an energy sensitivity of 2.7\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}19}$ eV has been achieved.