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Physical Review A

Proposed neutron interferometer test of some nonlinear variants of wave mechanics

Abner Shimony

Published Aug 1, 1979• 83 citations
Publisher

Abstract

A family of nonlinear variants of the Schr\"odinger equation was defined by Bialynicki-Birula and Mycielski by adding terms of the form $F({|\ensuremath{\psi}|}^{2})$ to the Hamiltonian. It is proposed that this family be tested by observing whether a phase shift occurs when an absorber is moved from one point to another along the path of one of the coherent split beams in a neutron interferometer. If $F$ is $b$ times a logarithmic function, which is the most important case, a null result with apparatus now available would impose an upper bound on $b$ of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ eV, more than two orders of magnitude smaller than the bound estimated by the above authors on the basis of the Lamb-shift measurement.

DOI
10.1103/physreva.20.394

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Proposed neutron interferometer test of some nonlinear variants of wave mechanics

Abner Shimony

Cambridge University Press eBooks•Feb 26, 1993•0 citations
A family of nonlinear variants of the Schr"odinger equation was defined by Bialynicki-Birula and Mycielski by adding terms of the form F(∣\ensuremathψ∣2)F({|\ensuremath{\psi}|}^{2})F(∣\ensuremathψ∣2) to the Hamiltonian. It is proposed that this family be tested by observing whether a phase shift occurs when an absorber is moved from one point to another along the path of one of the coherent split beams in a neutron interferometer. If FFF is bbb times a logarithmic function, which is the most important case, a null result with apparatus now available would impose an upper bound on bbb of 1.5\ifmmode\times\else\texttimes\fi{}10\ensuremath−12{10}^{\ensuremath{-}12}10\ensuremath−12 eV, more than two orders of magnitude smaller than the bound estimated by the above authors on the basis of the Lamb-shift measurement.
We analyze the free-space propagation of matter waves with a view to placing an upper limit on the strength of possible nonlinear terms in the Schr\"odinger equation. Such additional terms of the form $\ensuremath{\psi}F({|\ensuremath{\psi}|}^{2})$ were introduced by Bialynicki-Birula and Mycielski in order to counteract the spreading of wave packets, thereby allowing solutions which behave macroscopically like classical particles. For the particularly interesting case of a logarithmic nonlinearity of the form $F=\ensuremath{-}\mathit{b}\mathrm{ln}{|\ensuremath{\psi}|}^{2}$, we find that the free-space propagation of slow neutrons places a very stringent upper limit on the magnitude of b\mathit{b}b. Precise measurements of Fresnel diffraction with slow neutrons do not give any evidence for nonlinear effects and allow us to deduce an upper limit for $\mathit{b}<3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$ eV about 3 orders of magnitude smaller than the lower bound proposed by the above authors.
A slow-neutron interferometer system has been used to test a nonlinear variant of the Schr\"odinger equation $i\ensuremath{\hbar}\frac{\ensuremath{\partial}\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)}{\ensuremath{\partial}t}=[(\ensuremath{-}\frac{{\ensuremath{\hbar}}^{2}}{2m}){\ensuremath{\nabla}}^{2}+U(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)]\ensuremath{\psi}\ensuremath{-}b\mathrm{ln}({a}^{3}{|\ensuremath{\psi}|}^{2})\ensuremath{\psi}$. If this equation were correct, then, as Shimony has suggested, repositioning an attenuating plate downstream in a neutron beam would produce a phase modification. No measurable phase shift beyond experimental uncertainty was found and an upper limit of 3.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ eV for the energy constant bbb was established.
Since its advent in 1974, the neutron interferometry technique is a powerful tool for tests of quantum mechanics by matter waves. High order interference pattern could be observed and an upper limit for nonlinear terms in the Schroedinger equation has been given. The 4..pi..-symmetry of a spinor wave function was verified in 1975. Afterwards a precision measurement of this 4..pi..-symmetry effect has been performed which yields, after corrections for the new values of the fundamental constants, a periodicity factor of ..cap alpha.. = 715.87 +- 3.8 deg which does not include the expected 720 deg within its simple error limits. Alternative formulations of quantum mechanics predict a deviation of the 4..pi..-periodicity factor in the region of strong interaction. Therefore, a precision measurement of the influence of strong interactions is proposed where the spinor rotation is observed within the region of additional strong interactions caused by a nonmagnetic material of heavy nuclei inserted alternatively into the beam paths of the interferometer. It is expected to reach an accuracy concerning the additional phase shift in the range of 10/sup -5/ which corresponds to deviation factors within the region of strong interactions of about 20% from the standard formulation. Various factors limiting themore » accuracy are discussed.« less
Abstract Among the various areas of neutron optical research, neutron interferometry has provided some of the most challenging and spectacular developments. In most cases, separated coherent beams are used which are produced either by wavefront division (Young’s type) or by amplitude division (Mach− Zehnder type). These beams are subsequently coherently superposed after passing through regions of space where the neutron wavefunction is modified in phase and amplitude by various interactions: nuclear, magnetic, electromagnetic, gravitational, or by geometry and topology (as in Aharonov−Bohm-type experiments). Various types of neutron interferometers that have been tested are shown in Fig. 2.1. The first neutron interferometer was based upon wavefront division and biprism deflection, but it had several constraints for interferometric applications mainly due to the small beam separation (100 μm) (Maier-Leibnitz and Springer 1962). Nevertheless, this method gained further interest for diffraction experiments from macroscopic objects and for the detection of weak interaction effects where the long flight paths (10 m) enhance the sensitivity for the measurements of these effects (Gähler et al. 1980; Klein et al. 1981a). The perfect-crystal interferometer (Rauch et al. 1974) provides widely separated coherent beams and has become a standard method for advanced neutron optical investigations. The development of diffraction and phase grating interferometers extended the Mach-Zehnder interferometer method to very slow neutrons (loffe et al. 1985; Gruber et al. 1989). The superposition of spin-up and spin-down states in the longitudinal direction provides the basis for Larmor and Ramsey interferometry, where no lateral beam splitting is necessary. The well-known spin-echo systems (Mezei 1972) and zero-field spin-echo systems (Gähler and Golub 1987; Dubbers et al. 1989) are examples of such interferometers, which also play an important role in atom interferometry (Bord é 1989; Kasevich and Chu 1991; Riehle et al. 1991).
One of the significant successes in the field of neutron interferometry has been the experimental observation of the phase shift of a neutron de Broglie wave due to the action of the Earth's gravitational field. Past experiments have clearly demonstrated the effect and verified the quantum-mechanical equivalence of gravitational and inertial masses to a precision of about 1%. In this experiment the gravitationally induced phase shift of the neutron is measured with a statistical uncertainty of order 1 part in 1000 in two different interferometers. Nearly harmonic pairs of neutron wavelengths are used to measure and compensate for effects due to the distortion of the interferometer as it is tilted about the incident beam direction. A discrepancy between the theoretically predicted and experimentally measured values of the phase shift due to gravity is observed at the 1% level. Extensions to the theoretical description of the shape of a neutron interferogram as a function of tilt in a gravitational field are discussed and compared with experiment.