Experimental evidence for the two-path description of neutron spin echo

We describe an experiment that strongly supports a two-path interferometric model in which the spin-up and spin-down components of each neutron propagate coherently along spatially separated parallel paths in a typical neutron spin-echo small-angle scattering (SESANS) experiment. Specifically, we show that the usual semi-classical, single-path treatment of Larmor precession of a polarized neutron in an external magnetic field predicts a damping as a function of the spin-echo length of the SESANS signal obtained with a periodic phase grating when the transverse width of the neutron wave packet is finite. However, no such damping is observed experimentally, implying either that the Larmor model is incorrect or that the transverse extent of the wave packet is very large. In contrast, we demonstrate theoretically that a quantum-mechanical interferometric model in which the two mode-entangled (i.e., intraparticle entangled) spin states of a single neutron are separated in space when they interact with the grating accurately predicts the measured SESANS signal, which is independent of the wave packet width.

Figure 1

Simplified diagrams of a typical SESANS setup using rf flippers (pink parallelograms). (a) The two-path quantum model (see Sec. 3) in which the two rf flippers before the sample separate the initial neutron wave packet (purple) polarized in the $\widehat{y}$ direction by the polarizer (P) into two mode-entangled (i.e., intraparticle entangled) wave packets (orange and blue) that are spatially separated by the spin-echo length $\xi$. After the neutron has interacted with the phrase grating (sample), the last two rf flippers spatially recombine the two scattered wave packets (green). The final $\pi /2$ flipper, polarization analyzer (A), and detector then measure the contribution of the overlapped states to the neutron polarization along the $\widehat{x}$ direction. (b) The single-path Larmor precession model (see Sec. 2) wherein the wave nature of the neutron is treated separately from its spin: the polarization of the scattered beam arises from the difference in Larmor precession phase before (${\mathrm{\Phi }}_1$) and after (${\mathrm{\Phi }}_2$) the sample. The scattering angle, wave packet separation, and intrinsic coherence width are all greatly exaggerated for clarity.

Figure 2

(a) SESANS echo polarization $P$ normalized by the empty-beam polarization $P_0$ calculated in the single-path model for a $2\mu \mathrm{m}$ period silicon grating illuminated by a neutron beam incident along the normal to the plane of the grating. The calculation uses the parameters of the grating and the Larmor instrument described in the text and assumes that each neutron is an infinite plane wave. Notice that peak amplitudes return to unity for all orders. (b) SESANS polarization predicted for the same grating using the semi-classical model of SESANS including a finite neutron intrinsic coherence width of $60\mu \mathrm{m}$. The effect of a finite coherence width is to increasingly reduce the peak amplitudes as spin-echo length increases and to slightly modify the sloping background. The sloping background in (a,b) is a result of using a pulsed neutron time-of-flight source with multiple neutron wavelengths.

Figure 3

Diffraction of an entangled neutron state from a phase grating of period $p=a+b$ with $a$ the width of the grating walls and $b-a$ the width of the channels. The light (orange) and dark (blue) paths represent the spin-up and spin-down wave packet components, respectively. The transverse wave packet width and path separation are represented by $\mathrm{\Delta }$ and $\xi$, respectively. The grating is positioned at $z=0$. Each of the grey boxes represents a combination of two rf flippers as shown in detail in Fig. 1.

Figure 4

(a) Plot of the measured normalized spin-echo polarization versus spin-echo length at an rf frequency of 2 MHz and grating angle of 90 degrees such that the grating channels were along the $\widehat{x}$ direction in Fig. 1. (b) The same plot for an rf frequency of 3 MHz. The background functions (BG), which appear because the spin-echo length $\xi$ varies quadratically with the neutron wavelength $\lambda$ and the phase imparted on the neutron from the silicon grating is also proportional to $\lambda$, are fitted from the local minima of the echo polarization. The resolution curves are the expected maxima of the echo polarization peaks when the theoretical echo polarization is convolved with the Gaussian instrumental resolution function [see Eq. (20)].

Figure 5

Plot of normalized spin-echo polarization versus spin-echo length measured at an rf frequency of 2 and 3 MHz and grating angles of 8 and 5 degrees. The gray dashed lines are the background (BG) fits as described in Sec. 4 and in the caption of Fig. 4. Notice that the amplitude of the peaks return to unity (purple dashed line).