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Fortschritte der Physik

Simulating Decoherence of Neutron Wave Packets in Magnetic Fields

Martin Suda

Published Oct 1, 2001• 0 citations
Publisher

Abstract

Fluctuations of the magnetic field lead to a dephasing of polarized neutron wave packets and to an attenuation of the interference term of the Wigner distribution. It turns out that the Schrödinger equation can be solved exactly by using a constant space-dependent magnetic potential plus an arbitrary number of time-dependent cosine- and sine-shaped potentials with different frequencies and amplitudes. If the time oscillations are not resolved or if frequency fluctuations are taken into consideration, one can work out a procedure to take the damping characteristics of the interference term into account. An arbitrary statistical noise function can be constructed by Fourier's theorem. This potential leads to depolarization due to a loss of phase correlation. One can simulate the transition from coherence to decoherence which can be used in ongoing spin-echo and quantum state reconstruction experiments.

DOI
10.1002/1521-3978(200110)49:10/11<1103::aid-prop1103>3.0.co;2-u

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Fluctuations of the magnetic field lead to a dephasing of polarized neutron wave packets and to an attenuation of the interference term of the Wigner distribution. It turns out that the Schrödinger equation can be solved exactly by using a constant space-dependent magnetic potential plus an arbitrary number of time-dependent cosine- and sine-shaped potentials with different frequencies and amplitudes. If the time oscillations are not resolved or if frequency fluctuations are taken into consideration, one can work out a procedure to take the damping characteristics of the interference term into account. An arbitrary statistical noise function can be constructed by Fourier's theorem. This potential leads to depolarization due to a loss of phase correlation. One can simulate the transition from coherence to decoherence which can be used in ongoing spin-echo and quantum state reconstruction experiments.
If a quantum mechanical particle passes a spatially finite region of a purely time-dependent potential, coherent emission and absorption of energy take place. Due to Zeeman interaction, this situation can be realized with neutrons crossing an oscillating magnetic field. We solve the corresponding Schr\"odinger equation for fields consisting of an arbitrary number of frequencies and measure the calculated transition amplitudes for the quantized energy transfer by means of time-dependent neutron interferometry. Investigating the interferometer contrast for magnetic signals consisting of a high number of modes whose phases are randomly distributed shows that noise fields and therefore decoherence processes can be modeled in this calculation as well.
Abstract This chapter describes a rather broad selection of neutron interferometry experiments whose theoretical description requires an understanding of the mutual coherence function, and the sensitive experimental techniques utilized. Within the interferometer the coherent beams can be influenced by interactions with fields and material media in space and time causing phase shifts and wave packet delays that substantially affect the outgoing beams. Post-selection experiments involving spectral filtering and time-of-flight analysis are described. The phase-echo effect is described; it is a generalization of the spin-echo effect. Placing statistical or deterministic absorbers within the interferometer affects the outgoing coherence properties of beams quite differently. Low counting rate interference experiments show time pair-correlation effects characteristic of Poissonian statistical beams. Decoherence effects are investigated to study the transition from a quantum to a classical regime.
We analyze the coherence properties of polarized neutrons, after they have interacted with a magnetic field or a phase shifter undergoing different kinds of statistical fluctuations. We endeavor to probe the degree of disorder of the distribution of the phase shifts by means of the loss of quantum-mechanical coherence of the neutron. We find that the notion of entropy of the shifts and that of decoherence of the neutron do not necessarily agree. In some cases the neutron wave function is more coherent, even though it has interacted with a more disordered medium.
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
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.
Decoherence phenomenona in a neutron interferometer are analyzed by simulation of the effects of an environment with magnetic noise fields. Basic calculations and experiments show the validity and limitations of this model system. In particular, loss and recovery of the interference pattern with controllable noise sources in both interferometer arms are discussed in detail. In addition, the decoherence behavior at high interference order, where Schr\"odinger-cat-like states exist in the interferometer, is investigated. While at low interference order a smearing of the interference pattern is observed, at high interference order a smearing of the modulated momentum distribution occurs.