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Physical Review A

Neutron-multiwave-interference experiments with many resonance coils

Сергей Валентинович Григорьев, Yu. O. Chetverikov, A. V. Syromyatnikov, Wicher H. Kraan, M.Th. Rekveldt

Published Sep 12, 2003• 5 citations
Publisher

Abstract

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 ${B}_{1}$ 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 ${2}^{N}$ 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.

DOI
10.1103/physreva.68.033603

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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 ${B}_{1}$ 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 ${2}^{N}$ 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 B1\ensuremath−B0{B}_{1}\ensuremath{-}{B}_{0}B1​\ensuremath−B0​ over the length L. Highly regular patterns of the quantum-mechanical probability R in (B1,\ensuremathρ){(B}_{1},\ensuremath{\rho})(B1​,\ensuremathρ) 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.
We present an analytical and numerical analysis of neutron multilevel interference phenomena generated when a neutron passes through a series of NNN resonant coils operated at the successive conditions $\ensuremath{\hbar}({\ensuremath{\omega}}_{0}+n\ensuremath{\Delta}\ensuremath{\omega})=2{\ensuremath{\mu}}_{n}({B}_{0}+n\ensuremath{\Delta}B)$ with $n=0,1,\dots{},N\ensuremath{-}1$. Each coil produces spin flip with probability $\ensuremath{\rho}$ between 0 and 1; thus the number of waves for the neutron is doubled after each coil, finally giving 2N{2}^{N}2N interfering neutron waves. The phase difference between any pair is a multiple of a time dependent ``phase quantum'' $\ensuremath{\Delta}\ensuremath{\Phi}(t)$. The analysis predicts for each number NNN a highly regular pattern for the quantum mechanical probability to find the neutron spin in one specific state as a function of $\ensuremath{\rho}$ and $\ensuremath{\Delta}\ensuremath{\Phi}$. These patterns evolve in time and show revivals after a time TTT determined by the step $\ensuremath{\Delta}\ensuremath{\omega}$ according to $T=2\ensuremath{\pi}∕\ensuremath{\Delta}\ensuremath{\omega}$. For some adjustments of the system an analytical solution is obtained. Application of multilevel interference in high-resolution neutron modulated-intensity-by-zero-effort--type spectrometers is discussed.
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.
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.
The implications of complementarity on two-path neutron interferences and on separated-oscillatory-field resonances are discussed. The studies are extensions of those by Furry and Ramsey [Phys. Rev. 118, 623 (1960)] on two-path electron interferences which showed that an apparatus used to determine the electron path introduces uncertainties in the scalar and vector potentials which in turn disturb the phase of the electron wave function so much through the Aharonov-Bohm effects [Phys. Rev. 115, 485 (1959)] that the interference fringes disappear. A similar result is derived here for the neutron, but with the phase uncertainties coming from the magnetic moment's motion through an electric field as discussed by Anandan [Phys. Rev. Lett. 48, 1660 (1982)], and Aharonov and Casher [Phys. Rev. Lett. 53, 319 (1984)]. A corresponding result is also obtained for separated-oscillatory-fields resonances, which can be interpreted as an interference between two different paths in spin space. An interesting difference between the separated-path and separated-oscillatory-field experiments is that the latter may be interpreted classically.
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.