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Physical Review A

Neutron wave-interference experiments with adiabatic passage of neutron spin through resonant coils

Сергей Валентинович Григорьев, Rob Kreuger, Wicher H. Kraan, Fokko M. Mulder, M.Th. Rekveldt

Published Jun 7, 2001• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.64.013614

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Europhysics Letters (EPL)•Apr 1, 2004•4 citations
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 \ensuremathω0\ensuremathτ.{\ensuremath{\omega}}_{0}\ensuremath{\tau}.\ensuremathω0​\ensuremathτ. 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ 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.
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-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.
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.
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 effect of spin–rotation coupling is measured for the first time with neutrons. The coupling of spin with the angular velocity of a rotating spin turner can be observed as a phase shift in a neutron polarimeter set-up. After the neutron's spin is rotated by 2 π through a rotating magnetic field, different phase shifts are induced for 'up' and 'down' spin eigenstates. This phase difference results in the rotation of the neutron's spin-vector, which turns out to depend solely on the frequency of the rotation of the magnetic field. The experimental results agree well with the solutions acquired by the Pauli–Schrödinger equation.
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.