Spin-Echo System

Abstract

The neutron spin-echo technique has become a standard tool for condensed matter research [25, 64]. In most cases the semi-classical description by Lar-mor precession using the well-known Bloch equation is sufficient [65]. In this case, the rotation angle depends on the time t = l/v that the neutron spends in a perpendicular precession field of length l and, therefore, on the velocity v of the neutron. In this chapter we will show how beam polarization and its rotation within a region of a constant magnetic field results from the interference of spin-up and spin-down states. It is known that Larmor precession exists due to the coherent superposition of a spin-up and a spin-down state with slightly different momenta (see (9.10) below), ∆k = |µ|BM 2 k , (9.1) caused by the Zeeman energy splitting (see (9.11)) ∆E = ω L = 2|µ|B, (9.2) where µ = −1, 913µ N denotes the magnetic moment of the neutron, µ N = 5, 051.10 −27 J/T the nuclear magneton and B the strength of the precession field. M denotes the mass of the neutron and ω L the Larmor frequency. This Zeeman shift is orders of magnitudes smaller than the momentum width of the beam and can, therefore, be neglected in most cases. A description based on the Bloch equation can be used, which relates the precession angle ϕ = ω L t = 2|µ|Bl v (9.3) to the velocity v of the neutron and the distance l the neutron travelling inside the precession field. More generally, the Larmor precession has to be seen as an interference phenomenon in the longitudinal direction of the spin-up and the spin-down components of the initial wave packet [66-68]. This leads to a spatial separation of the wave packets and to non-classical states in neutron precession experiments [69]. A quantum optical description will show how wave packets, separated in ordinary space, become coupled in momentum space. The close