Spin-rotation coupling observed in neutron interferometry

www.nature.com/npjqi ARTICLE OPEN Spin-rotation coupling observed in neutron interferometry Armin Danner Yuji Hasegawa 1✉ , Bülent Demirel1, Wenzel Kersten 1,3 ✉ 1 , Hartmut Lemmel 1,2 , Richard Wagner 1 , Stephan Sponar 1 and Einstein’s theory of general relativity and quantum theory form the two major pillars of modern physics. However, certain inertial properties of a particle’s intrinsic spin are inconspicuous while the inertial properties of mass are well known. Here, by performing a neutron interferometric experiment, we observe phase shifts arising as a consequence of the spin’s coupling with the angular velocity of a rotating magnetic field. This coupling is a purely quantum mechanical extension of the Sagnac effect. The resulting phase shifts linearly depend on the frequency of the rotation of the magnetic field. Our results agree with the predictions derived from the Pauli–Schrödinger equation. 1234567890():,; npj Quantum Information (2020)6:23 ; https://doi.org/10.1038/s41534-020-0254-8 INTRODUCTION The principle of equivalence of inertial and gravitational masses is a corner stone of Einstein’s theory of general relativity.1 It follows from this principle that one cannot locally distinguish between inertial forces and pseudo-forces. Examples of pseudo-forces are the gravitational force, as experienced in the presence of a massive object, or Coriolis and centrifugal forces, which originate from circular motion of an observer in a non-inertial frame of reference. In terms of wave phenomena and in a rotating frame, the respective phase shifts are described by additional couplings compared to an inertial frame. The Sagnac effect2 refers to the observed phase shift induced between two counter-rotating light waves in a rotating interferometer. The phase shift is proportional to the scalar product of the rotation frequency and the area of the installed interferometer. This can also ! be written in the ^ 0 as a coupling  ! Hamiltonian H Ω  L between the rotation ! ! vector Ω and the orbital angular momentum L of the light wave around the center of rotation. The Sagnac effect for the de Broglie waves of neutrons was first demonstrated experimentally in the late 1970s.3 The orbital angular momentum in the coupling term of the Sagnac effect contains the mass which is usually the quantity associated with inertia. In quantum theory the inertial properties of a particle are influenced not only by its inertial mass, but also by its spin. When solving Dirac’s equation in accelerated frames of reference in the non-relativistic the Hamiltonian of a particle includes the ! ! 4 regime, ! term  Ω  J , where Ω is the rotation vector of the frame and ! ! ! J ¼ L þ S is the ! total angular momentum of the particle with the contribution ! ! S of the spin angular momentum. The additional term  Ω  S is called spin-rotation coupling. To measure the spin-rotation coupling, Mashhoon first published a proposal by S. A. Werner for an experiment involving a rotating neutron interferometer5 (in an arrangement insensitive to the Sagnac and gravity effects). In the further course, Mashhoon et al. suggested interferometer setups where longitudinally polarized neutrons pass through a rotating spin flipper6 which is in turn equivalent to a rotating magnetic field.7 The authors of Mashhoon et al.6 stated that “the phenomenon of spin-rotation coupling is of basic interest since it reveals the inertial properties of intrinsic spin.” For further theoretical contributions about spin- rotation coupling consider.8–13 One of them12 even doubted the existence of spin-rotation coupling for fermions. Recently, we reported on neutron polarimeter experiments14,15 whose measurement results can be attributed to the coupling of the neutron’s spin with the rotation of a magnetic field. However, the results of these experiments rely on the rotation of the polarization vector which can also be described with the semiclassical Bloch equations. Therefore, these previous results could in principle be reproduced with a classical magnetic moment. In this letter, we present the results of the neutron interferometric experiment as suggested by Mashhoon and Kaiser.7 The relative phase between the partial wave functions of paths I and II in the interferometer is directly measured. By applying a direct measurement of the relative phase, instead of measuring the rotation of the polarization vector as in a polarimeter experiment, the purely quantum mechanical aspect of the spin-rotation coupling is demonstrated. This aspect is discussed in more detail in a later section. Neutron interferometry16–18 is a technique to observe the interference effect of matter waves passing through a perfect silicon-crystal interferometer. It is an established, powerful tool to investigate fundamental quantum mechanical concepts with massive particles. Using neutron interferometry the 4π spinor symmetry of fermions,19,20 the spin-superposition law21,22 and the equivalence principle23,24 have been demonstrated. RESULTS Theory Let us consider an observer rotating relative to an inertial observer as discussed in Mashhoon.5 The wave function ψ0 ð! r ; tÞ, with respect to the rotating frame of reference, is given by the wave 0 ^ function ψð! r ; tÞ in the inertial frame ! !as ψ ¼ Uψ.!The unitary ^ ^ operator U is given by U ¼ expði Ω  J t=_Þ, with J being the total angular momentum, consisting of orbital and spin angular momentum. If the wave function ψ satisfies the Schrödinger ^ ¼ i_ ∂ψ=∂t, the wave function ψ0 represents a equation Hψ ^ 0 ψ0 ¼ i_ ∂ψ0 =∂t0 with solution of the ! Schrödinger equation H ^0 ¼ U ^H ^U ^y  ! H Ω  J . A detailed comparison of the latter equations4,10 reveals the existence of a new effect associated with the 1 Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria. 2Institut Laue-Langevin, 38000 Grenoble, France. 3Department of Applied Physics, Hokkaido University, Kita-ku, Sapporo 060-8628, Japan. ✉email: ; Published in partnership with The University of New South Wales A. Danner et al. 2 coupling of intrinsic spin with!rotation which is expressed by the ^ 0 ¼ γ ! Hamiltonian δH Ω  S with the Lorentz factor γ. As SR suggested by Mashhoon, the effect can indeed be derived as done before (e.g. Weinfurter and Badurek25) by solving the Pauli–Schrödinger equation in the lab frame for the interaction of the spin of a free neutron in a magnetic field with angular velocity Ω. For a neutron propagating in +y-direction through an uniformly rotating magnetic field, which is expressed as ! B ðΩ; tÞ ¼ B1 ðcosðΩtÞ; 0; sinðΩtÞÞT , a solution is given by   1 iky  i_k2 t  (1) e 2m ξðtÞ; ψðy; tÞ ¼ pffiffiffiffiffiffi e 2π 1234567890():,; where ξðtÞ generates the rotation of the initial spin state in the rotating frame ξ rot ð0Þ and is given as ! i i! ^ ^ ! (2) Uð α rot Þξ rot ð0Þ ξðtÞ ¼ e_ΩSy t e2 α rot  σ rot ξ rot ð0Þ ¼ UðΩÞ with the vector ! σ rot comprising the Pauli matrices. The operator i e_ΩSy t is the (...truncated)