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Oxford University Press eBooks

Spinor Symmetry and Spin Superposition

Helmut Rauch, Samuel A. Werner

Published Jan 15, 2015• 0 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780198712510.003.0005

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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.
The coherent superposition of particle quantum states resulting from different spatial paths shows well-known interference effects. Another type of interference is Larmor precession, which can be described as neutron wave interference in spin space, i.e. interference on the same spatial path. Here we show several new manifestations of interference phenomena of the latter type. The interference is produced by coherent population of different I z = ±½ spin states in a polarized neutron beam between two resonant radio-frequency (RF) spin flippers. Their spin flip probability, and thus the occupation numbers of the spin states, can be tuned between 0 and 1 by changing the RF amplitude. This leads to interference effects in the polarization since the energy exchanged between the RF field and spin yields different phase evolution in the spatial parts of the neutron states. Depending on the preparation of the initial neutron states, five values for the time-independent phase difference between the final neutron states can be observed.
The coherent superposition of oppositely polarized neutron beams of equal amplitude results in a final beam polarization perpendicular to the polarization of both initial beams. This polarization can be rotated by purely scalar interaction applied to the beams before superposition, which is equivalent to an additional Larmor precession applied to the beam after superposition. We have directly observed these effects in an experiment performed using the perfect-crystal neutron interferometer at the high-flux reactor at Grenoble. This paper gives the experimental results and discusses their theoretical foundation.
A polarised neutron beam is passed through a gradient resonance flipper. By the amplitude of their RF field, such flippers can be set at flip probability ρ = 0, 1, or ½. At ½, the neutron wave splits into a flipped and a non-flipped part with different precession. We measure the polarisation after a spin-echo (SE) setup with each precession arm made up of 2 such flippers. Offset from SE is made by varying the static fields in one flipper while the other flippers stay unchanged. This shows up as a periodic behaviour of the polarisation. With incoming polarisation parallel to the static field in the flippers–set at ρ = ½–this period is twice the period measured for both ρ = 0 and 1 and with polarisation perpendicular to the static field. In the latter experiments both components of the spinor are affected, whereas in the former experiment we create spin states in which only one spinor component in one state is affected. Hence, this experiment demonstrates explicitly the 4π-periodicity of the spinor.
Neutron resonance spin-echo (NRSE) phenomena are investigated experimentally and theoretically. In our experiments, spin precession produced in a classical manner and by neutron resonance is combined as two arms of a spin-echo machine. A magnetic-field scan in the classical SE coil reveals a spin-echo signal of the precession produced by the NRSE arm. The neutron spin-flip probability $\ensuremath{\rho}$ in the resonance coils turns out to be a key parameter of the NRSE arm. The limiting cases $\ensuremath{\rho}=0$ and $\ensuremath{\rho}=1$ lead, respectively, to Larmor precession with phase ${\ensuremath{\varphi}}_{1}$ in the static magnetic fields of the NR flippers or to NRSE precession with ${\ensuremath{\varphi}}_{2}.$ The case $0<\ensuremath{\rho}<1$ produces quantum interference resulting in additional echoes with phases ${\ensuremath{\varphi}}_{3}=({\ensuremath{\varphi}}_{1}+{\ensuremath{\varphi}}_{2})/2$ and ${\ensuremath{\varphi}}_{4}=\ifmmode\pm\else\textpm\fi{}({\ensuremath{\varphi}}_{1}\ensuremath{-}{\ensuremath{\varphi}}_{2})/2.$ The amplitude of each pattern depends on the spin-flip probability $\ensuremath{\rho}$ and the initial polarization. These experiments demonstrate explicitly the quantum-mechanical principle of linear spin state superposition of neutron particle waves, and the interference as a result of that.
If a spin-½ particle is placed in a constant uniform magnetic field for a time t during which, classically, the Larmor precession angle would be θ, then according to quantum mechanics the components of the spin part of the wave function undergo phase changes of ±θ/2. Hence, corresponding to a classical precession of 2π radians, there is a sign change of the spinor, whatever the initial polarization may be. Bernstein pointed out that this sign change can be observed by coherently splitting a fermion beam, subjecting one part to the magnetic field, and coherently recombining the parts. Evidently, the same interference effect can be achieved by subjecting the two parts of the beam to fields which would classically cause precessions of π radians in opposite senses. He noted advantages in studying neutral rather than charged particles, and outlined a proposal for an interferometry experiment with neutrons. The recent invention and application of a remarkable single-crystal neutron interferometer greatly increases the feasibility of observing the sign change. In the experiment of Ref. 3 a neutron beam is split, and the two parts follow paths of different elevations. The interference of the recombined parts exhibits the effect of gravitational potential difference between the two elevations. In the experiment which we propose, the paths could be kept at the same elevation, but phase differences would be produced by appropriate magnetic fields.
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.