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Europhysics Letters (EPL)

Observation of 4π-periodicity of the spinor using neutron resonance interferometry

Wicher H. Kraan, Сергей Валентинович Григорьев, M.Th. Rekveldt

Published Apr 1, 2004• 4 citations
Publisher

Abstract

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.

DOI
10.1209/epl/i2003-10202-4

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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.
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.
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.
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.
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.
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.