Overview of neutron interferometry at NIST

Neutron interferometry at the National Institute of Standards and Technology is a well-established program that performs experiments in a wide range of areas including materials science, quantum information, precision measurements of coherent and incoherent scattering lengths, and dark energy/fifth force searches. Central to the continued success of this program is the further understanding and elimination of instabilities and coherence-losses whether they are from thermal, vibrational, or dynamical sources. We have spent considerable effort in fabricating new interferometer crystals which have higher maximum fringe visibilities and that can be tailored to specific experiments. We describe the current facilities and a new post-machining fabrication process of crystal annealing.


Introduction
Interferometry represents one of the most precise instruments for probing fundamental physics using neutrons. This success is despite the fact that neutron interferometry is mostly insensitive to the weak interaction, which is a major focus of fundamental physics and Beyond the Standard Model searches. Neutron interactions with nuclear, magnetic, and gravitational potentials combined with its quantum nature makes neutron interferometry a unique tool for exploring fundamental physics. However, the limited availability of high-quality neutron interferometers has prevented the technique from wider use in the neutron community. One challenge is the physical means by which to produce these types of interferometers. Another challenge is the resulting low flux because the neutron interferometer acts as a very selective momentum filter; the vast majority of neutrons pass through the device without Bragg diffracting. As such, the neutron fluence measured by the detectors is low; on the order of a few neutrons per second. The intensity limitations of neutron interferometry are being addressed [1][2][3] using a new technique to achieve a Mach-Zehnder interferometer with broad-spectrum acceptance. One issue with this new interferometric technique is that it is devoid of macroscopic spatial separation between the interferometer paths. However, its advantages may prove a e-mail: michael.huber@nist.gov invaluable for several experiments including precision measurements of gravity.
Perfect-crystal silicon interferometers remain the workhorse of the field. In this proceeding, we discuss recent advances in the fabrication and post-fabrication of silicon interferometers. In addition, we will mention the improvements to stability and noise suppression at the two neutron interferometer facilities at NIST.

Neutron Interferometry
A perfect-crystal neutron interferometer consists of a silicon ingot machined so that there are several parallel crystal blades on a common monolithic base. A schematic of an interferometer is shown in Fig. 1. Neutrons entering the interferometer Bragg diffract (Laue geometry) in the first (splitter) blade. This coherently separates the neutron into two spatially separate paths labeled I and II. Both neutron paths are Bragg diffracted in a second (mirror) blade and interfere in the final (analyzer/mixer) blade of the interferometer. Conceptually, this is analogous to a Mach-Zehnder interferometer in optics. The two beams exiting the interferometer are historically labeled the O-and H-beams. Neutrons are detected with high efficiency using cylindrical 3 He-filled proportional counters. Differences between path I and II modulate the intensity of the O-and H-beams. Typically, a sample called a phase flag is inserted to modulate the signal in a controllable way.  The effect of neutrons passing through a material is described by an index of refraction n, conceptually the same as for light optics, given by where λ is the neutron wavelength and N b is the scattering length density of the material. This gives rise to a phase shift due to an object with thickness D placed inside the interferometer of Here k = 2π/λ is the neutron wave vector. The two detectors behind the interferometer measure a signal given by The coefficients c 0 , c 1 , and c 2 are treated as fit parameters. The ratio V = c 1 /c 0 is the contrast, or fringe visibility, and is the primary indicator of the quality of an interferometer (absent environmental and experimental factors). More recently, we have begun using another method (discussed in Sect. 3.1) to quantify an interferometer's performance. In this method, the variation in the relative blade thicknesses and angular Bragg plane misalignments between the blades can be measured using refractive elements placed inside the interferometer. This can be done to the precision of a few percent of pendellösung length for relative blade thickness and at the nanoradian level for Bragg plane misalignments. The phase flag provides a phase shift of φ flag (δ) where the angle δ, defined in Fig. 1, changes the flag's effective thickness traversed by the beam paths. The phase φ 0 (t) is phase difference between the two paths of an empty interferometer and drifts with time t as the interferometer is not completely isolated from the environment.

NIST facilities
In 2011, the neutron interferometer facility at NIST was expanded to include a secondary beamline. This beamline operates at a fixed λ = 4.4Å wavelength, but with a substantial fraction, 23%, of λ/2 = 2.2Å such that it can utilize either wavelength for experiments. The facilities' overall characteristics are summarized in Shahi et al. [4,5]. At the time of Ref. [4], the facilities at NIST had 9 interferometer crystals (see Table 1) of varying quality. Since then the interferometer 'Cronus' has been on display at Purdue University and is not available for general use.
A new 2-blade interferometer 'Iris' has been machined at Riken and then post-fabricated at NIST. But most notably, the interferometer "Hera", which had low but measurable contrast, has now been greatly improved through annealing [6] and is discussed below. The secondary neutron interferometry beamline has had a number of experiments including measuring  improved phase stability of an interferometer inside a vacuum chamber [7], characterizing the diffraction pattern of high-aspect-ratio phase gratings [1,2] and sub-micron resolution tomography [8]. Current efforts are to measure pendellösung oscillations in perfect crystals.
The primary interferometry beamline that includes the vibrationally isolated Hutch structure, has been continuing its work on precision scattering length measurements, testing of fabrication techniques and various other items. The capabilities inside the Hutch have not significantly changed since 2011. However, one notable improvement has been in the stability of the interferometer Hutch facility. It was discovered during precision scattering length measurements of helium samples that the phase stability inside the Hutch had degraded over time. The Hutch was designed so that minimum personnel intervention would be required under normal operation. Sample movement, interferometer control, and counting are controlled outside of the facility. To this end, there was a multitude of cables entering the Hutch from small notches along the Hutch floor. Because of the difficulty of subtracting cabling from the Hutch, the majority of these cables were from legacy experiments and not typically utilized. These cables represented a weak but important coupling between the outside NCNR building and the interferometer. To quantify stability we used a variance defined as where φ i is the phase of the i th interferogram, φ avg is the average measured phase over a 24 hour period and N is the total number of interferograms. A standard interferogram is formed using 21 different phase flag positions stretching over 1.5 periods. For the data shown in Figs. 2-4, neutrons were counted for 60 seconds at each of the 21 phase flag angles. Statistically, this method determines each φ i to an uncertainty of ±1 degree. In Fig. 2 one can see the phase drifting and large variance of Eq. (5). This is in comparison with data taken measuring the scattering length of deuterium [9] in the earlier days of the facility and is shown in Fig. 3. Removal of these nonessential cabling/connections lead to a 10x increase in stability at the instrument which is now in line with recorded data in 2001 (see Fig. 4). One can see that the original performance has been achieved.

Fabrication and annealing
Neutron interferometers are machined from float-zone grown, dislocation-free silicon crystals. In principle PPNS 2018 germanium could also be used for interferometry and is easier to machine than silicon. However, the quality of germanium is not typically sufficient for use in neutron interferometry. The process of fabricating an interferometer is as follows: (1) start with a float-zone perfect silicon crystal ingot; (2) machine the ingot to desired geometry; (3) chemically etching the crystal surfaces; (4) testing the interferometer in a neutron beam; (5) repeating steps (3) and (4) until an acceptable V is measured. Typically, fabrication is done using a diamondembedded wheel to cut away material, leaving two or more crystal blades protruding from a common base [10]. This is done with a machining tolerance between (5-7) microns which is roughly 1/10 of the crystal's pendellösung length. The base not only adds stability but ensures that the lattice planes of each blade are aligned relative to each other. Diamond wheels and other cutting techniques leave subsurface damage and micro cracking in the surface layers that may extend up to 100 microns into the crystal blade. This damaged layer completely destroys the contrast of the interferometer. Therefore, the interferometer surfaces are chemically etched to remove any layers of machining damage. After etching the interferometer's contrast is measured using a neutron beam. The etching and testing of the interferometer is typically done iteratively until reasonable contrast is obtained [11, p. 32-34]. Etch rates vary along the length of the crystal blades with the etchant more efficiently removing silicon at the ends of crystal. The nonuniform etching rates lens the blade's surface causing thickness variations on the order of microns [10] which in turn destroys contrast and is why an iterative etching process is done. It is not guaranteed that contrast can be measured in a new interferometer as the bulk features of the crystal and other factors may still dominate.

Qualitative stress analysis
One often thinks of Bragg diffraction as a simple geometric effect and that for a neutron incident on a crystal lattice, an incident beam will be diffracted at a precise angle θ Bragg . In fact, Bragg diffraction occurs over a small angular range of θ Bragg ± D where D is the Darwin width and is typically only a few arcseconds wide. For a crystal with structure where a 3 is the volume of a unit cell. Because this angle is only a few arcsconds, neutron interferometers are machined out of single ingots and preserve a large common base in addition to the diffracting crystal blades. This ensures that the lattices of each diffracting blade observe the original ingot's uniformity and are aligned to within the Darwin width of each other. If this was not the case, neutron interference would not be observed after the final blade of the interferometer. Stresses internal to the interferometer can cause these diffracting planes to be slightly misaligned. Neutron diffraction inside a crystal lattice is highly sensitive to misalignments in the incident angle θ which can be modified using optical elements. For instance, because the index of refraction for neutrons is so close to unity, very small deflections can be achieved though fused silica prisms. Here we used a right angle circular prism [12] inserted inside the interferometer to deflect the incident neutron while maintaining the Bragg condition and allowing the study of diffraction effects and stresses in the crystal. The prism had a pitch angle of α = 6 • , which gives a total beam deflection of, to first order, where ϕ is the angle between the Bragg plane and the prism deflecting plane. Placing the prism in path II of the interferometer (see  difference . The subscript 'S' refers to the splitter or first blade of the interferometer, 'M' the mirror blade, and 'A' the analyzer or final blade. Both functions I and J are ideally written in terms of Bessel functions but allow for non-ideal behavior [13]. Details can be found Ref. [6]. The measured relative misalignments of 'Hera' were large enough to explain its historically low contrast. In order to realign the crystal blades at the 10 nrad level 'Hera' was annealed in a 208 mm ID tube furnace which is shown in Fig. 6. As seen in Fig. 7, the interferometer rested on a silicon wafer supported by ceramic crucible. Argon gas flowed continuously through the furnace. The temperature was ramped at a rate of 5 • C/min and held for 10 hours at 800 • C. The interferometer was then cooled down over a period of 17 hours.
We found that the Bragg plane misalignment gradient dropped from 10 nrad/mm to less than 5 nrad/mm after annealing, and the absolute misalignment between the mirror and analyzer blades also decreased for some  . A comparison of "Zeus" which has been the most prolific neutron interferometer at NIST with "Hera" post heat treatment. One can see that the contrast of "Hera" is now both high and uniform across its front surface. The crystal "Zeus" exhibits typical behavior where contrast is highest only for a limited area or sweet spot.
portions of the interferometer. Most importantly, we found an increase in contrast from 23% to 90% [6].
The crystals "Zeus" and "Hera" are nearly physically identical (photos of which are shown in Fig. 8) and were machined at the same facility around similar times; thus we expect structurally very little difference between the two interferometers. However, prior to heat treatment "Hera" only obtained a maximum contrast of 23%. As reported in Heacock et al. [6], to verify the prism measurements, we measured the contrast of "Hera" both before and after annealing. Figure 8 shows uniform contrast measured across the crystal surface of "Hera" post heat-treatment. For comparison, "Zeus", the most widely used interferometer at NIST because of its repeatability high 85% contrast, is also shown. Note that in practice, most interferometers have a narrow range called the sweet spot where one obtains the maximum contrast and outside of this sweet spot the contrast is significantly worse. One can see this spot in Fig. 8a for "Zeus". In addition to "Hera", the baby interferometer, "Aphrodite", has also been recently re-machined and annealed. The resulting contrast was 92% compared to <30% before.

Conclusion
We have successfully demonstrated that annealing interferometer crystals has a positive effect on the

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interferometer's performance and is the first nonchemical method to do so. By relieving internal crystal stresses through heat treatment, we hope that this lessdestructive method leads to higher-contrast interferometers with more uniform phase deviations. In addition, using prism deflection we can measure and quantify imperfections of the interferometer in a way that is more sensitive than comparing the overall contrast of the device. Our aim is that post-fabrication, improved environmental isolation, and utilization of lessons from quantum information will make neutron interferometry a more robust measurement technique and increase its usefulnesses in other applications.