# Spin-Rotation Coupling

November 29, 2016 10:39 amThe effect of *spin–rotation coupling*, refereed to as * Sagnac–Mashhoon phase*, has been measured by our group for the first time with neutrons in a

*polarometric experiment*

^{1}. The basis of the

*Sagnac effect*lies in the influence of rotation on the phase of light passing through an interferometer. An inertial observer in the frame of an rotating interferometer of frequency , will observe that the wave vector of any radiation is . Thus a phase shift proportional to the scalar product of rotation frequency and area of the interferometer occurs. This phase shift can be extended to include the intrinsic spin of a quantum mechanical particle when replacing the orbital angular momentum, denoted as , by the total angular momentum . This results in an additional phase shift due to the spin-rotation coupling. In a paper published in 1998

^{2}Iranian born physicist Bahram Mashhoon proposed a neutron interferometric test of the Sagnac-Mashhoon phase, where a longitudinal polarized beam passes through a slowly rotating spin flipper placed in one arm, is proposed.

To describe the interaction of the spin of a free neutron in a magnetic field with angular velocity the Pauli–Schrödinger equation has to be solved for a particle propagating in − direction in a uniformly rotating magnetic field given by which gives (see here for details of the derivation). The Pauli equation in a rotating magnetic field for plane waves is analytically solvable and in this particular case given by , where generates the rotation of the initial spin state , which cab be expressed as . The square of the magnitude of the rotation vector is given by a Pythagorean equation , where we used the definition of the *Larmor frequency *. The solution from above contains the coupling term and describes the evolution of the spin states for both the rotating and the static field (). Instead of a flip it is better to choose a multiple of in which case the dynamical part becomes disengaged from the spinor function and the phase shift can be attributed purely to a * geometric phase*.

If we prepare the incident beam in our *polarimetric setup*, which is depicted above (left), to be the effect of the spin–rotation coupling can be seen when we choose to represent the initial state as a superposition of and states. The outcome of the spinor function in this set-up after a time and a rotation changes to initial state to . The corresponding change in the polarization vector (from ) is . For higher precision and feasibility it is more convenient to keep constant and scan a relative phase, just like in an interferometer experiment where phase shifter scans are performed. The observed phase shift is plotted above (right) as a function of the frequency. A linear behaviour is clearly seen, as theoretically expected. All results are within the error bounds of the expected values predicted by the *Pauli–Schrödinger equation*. The final average contrast of 95.23(84)% confirms the high reliability of our measurement.

A scheme of the original proposed measurement for spin- particles is shown aside. Polarized neutrons initially propagating in +y− direction are split into two arms of an interferometer of which one contains a static spin flipper and the other one a rotating spin flipper whose angular velocity vector is parallel to the initial neutron spin. The first proposals of the experiment suggested the use of a mechanically rotating coil, analogous to the HWP in the optical interferometer scheme, but it was pointed out later that this situation is physically equivalent to a static quadrate recoil producing a rotating magnetic field . The experimental realization of the interferometric setup is an ongoing project and foreseen to be achieved in 2018.

1. B. Demirel, S. Sponar, and Y. Hasegawa, *New. J. of Phys. A* **17**, 023065 (2015). ↩

2. B. Mashhoon, R. Neutze, M. Hannah, and G. E. Stedman, *Phys. Lett. A* **249**, 161 (1998). ↩