In a neutron interferometer ¹ a monochromatic beam of massive particles is split by amplitude division (Mach-Zehnder type) via Bragg reflection, in triple Laue (LLL) configuration. At the first plate of a silicon perfect crystal interferometer (silicon has essentially zero absorption for thermal neutrons), which is cut from a single rod.  The coherently superposed sub-beams (partial wave functions) passes through different regions of space.

(left) Mach-Zehnder-interferometer scheme (right) triple Laue (LLL) neutron interferometer together with monochromator crystal.

Behind the first plate (beam splitter) the neutron’s wave function is found in a coherent superposition state of the transmitted and a reflected sub-beams, which can be written in terms of , where and   are the transmission and reflection amplitudes (  for neglectable absorption) with  . Next  an additional slab (phase flag), inducing an adjustable phase shift is inserted, which yields , where , with the thickness of the phase shifter plate  , the neutron wavelength , the coherent scattering length   and the particle density   in the phase shifter plate. This can be rewritten as , with  . By rotating the plate,   can be varied systematically. Leaving the interferometer at the third plate, where the sub-beams are recombines, the neutron’s final wave function on forward direction (O-direction) is given by . Thus the intensity in the O-detector is given by , with . In similar manner we get in deviated (diffracted) direction (H-beam) , with . From particle conservation it follows immediately that . This behavior of the two intensities and is illustrated in the animation below.

A brief historical digression

In 1964, when advances in semiconductor technology had allowed the production of large monolithic perfect crystal silicon ingots, U. Bonse and M. Hart invented a single-crystal interferometer for X-rays based on the effects of dynamical diffraction in perfect crystals . This type of interferometer was then applied to neutrons, resulting in the first interference fringes sighted in 1974 by Rauch, Treimer and Bonse at the rather small (250kW) TRIGA reactor the Atominstitut (ATI) in Vienna. A picture of the first obtained interferogram is given below on the right hand side. The obtained interference demonstrates in impressive manner the wave-like nature of neutrons. Over the years numerous remarkable experiments on the fundamentals of quantum mechanics, have been carried out using neutron interferometry. Just to mention a few, there is the verification of the 4-spinor symmetry, followed by investigations of the influence of gravitation of the earth on the neutron’s wavefunction, and  experiments on spin superposition, as well as topological phases. A detailed summary is given in the book Neutron Interferometry: Lessons in Experimental Quantum Mechanics, by Helmut Rauch and Samuel A. Werner ².

Dynamical theory of diffraction

The neutron wave function inside the crystal is deduced by solving the stationary Schrödinger equation for the time-independent crystal potential , where the potential is given by the sum over all nuclear scattering centres located at .

(a) far off a Bragg condition (or homogeneous non-crystalline media) only the phase of the wavefunction is affected, whereas when the Bragg condition is fulfilled (b) a transmitted and refracted beam appears.

1. H. Rauch, W. Treimer, and U. Bonse, Phys. Lett. 47A, 369-371 (1974).  ↩

2. H. Rauch and S.A. Werner, Neutron Interferometry, Clarendon Press Oxford (2000).  ↩