Spin-rotation coupling observed in neutron interferometry

The principle of equivalence of inertial and gravitational masses is a cornerstone of general relativity, implying local indistinguishability between inertial forces and pseudo-forces. In rotating frames, wave phenomena acquire additional couplings leading to phase shifts. The Sagnac effect is a phase shift between counter-rotating waves in a rotating interferometer, proportional to the scalar product of rotation frequency and interferometer area, interpretable as a coupling between rotation and orbital angular momentum L. For matter waves, the neutron Sagnac effect was demonstrated in the late 1970s. In quantum theory, inertial properties are influenced by both inertial mass and spin. In the non-relativistic regime of Dirac’s equation in accelerated frames, the Hamiltonian includes a term −Ω·J (with Ω the rotation vector and J = L + S the total angular momentum), giving rise to spin-rotation coupling −Ω·S. Mashhoon proposed neutron interferometer experiments (insensitive to Sagnac and gravity effects) to measure spin-rotation coupling, including setups where longitudinally polarized neutrons pass through a rotating spin flipper equivalent to a rotating magnetic field. This phenomenon reveals inertial properties of intrinsic spin. Some theoretical works debated aspects of spin-rotation coupling, including doubts about its existence for fermions. Prior neutron polarimeter experiments reported measurements attributable to coupling between neutron spin and rotating magnetic fields, but those relied on rotation of the polarization vector describable by semi-classical Bloch equations and thus could be reproduced with a classical magnetic moment. In this work, a neutron interferometric experiment directly measures the relative phase between partial waves in the two paths, demonstrating the purely quantum mechanical aspect of spin-rotation coupling via direct phase measurement rather than polarization rotation. Neutron interferometry is a well-established tool to probe fundamental quantum phenomena such as 4π spinor symmetry, spin superposition, and the equivalence principle.

Prior work established the Sagnac effect for light and neutrons, linking phase shifts to rotation and orbital angular momentum. Theoretical treatments derived spin-rotation coupling in accelerated frames from Dirac/Pauli-Schrödinger equations, predicting an additional term −Ω·S in the Hamiltonian. Mashhoon and collaborators proposed neutron interferometric schemes employing rotating spin flippers (rotating magnetic fields) to observe spin-rotation coupling and emphasized its significance for the inertia of intrinsic spin. Some authors questioned its existence for fermions, while others elaborated relativistic and geometric phase treatments. Earlier neutron polarimeter experiments observed effects consistent with spin-rotation coupling but could be modeled by Bloch equations as rotations of classical-like polarization vectors, leaving ambiguity about the purely quantum phase aspect. This study addresses that gap by directly measuring phase shifts in an interferometer.

Theory: For an observer rotating relative to an inertial frame, the wavefunction transforms as ψ' = Uψ with U = exp(i Ω·J t/ħ), leading to a Hamiltonian in the rotating frame H' = UHU−1 − Ω·J. This reveals a coupling of intrinsic spin with rotation, expressible by an effective Hamiltonian H_SR = −(γ/2) Ω·S (γ: Lorentz factor). The effect can be derived by solving the Pauli-Schrödinger equation in the lab frame for a free neutron spin in a magnetic field rotating with angular velocity Ω. For a neutron traveling along +y through a uniformly rotating magnetic field B(Ω,t) = B1 (cos Ωt, 0, sin Ωt), the spinor solution contains a factor e−iΩ t/2 and a rotation operator U(α_rot(t)) = exp[−i α_rot(t)·σ/2] with α_rot(t) = (ω1 t, 0, 0), a(t) = sqrt(ω1^2 t^2 + Ω^2), and ω1 = −γ B1/(2ħ). The e−iΩ t σ_y/2 term transforms between rotating and lab frames while U(α_rot) includes Larmor precession. Experimental setup: Experiments were performed at the S18 neutron interferometry station at the Institut Laue-Langevin (ILL), Grenoble. A monochromatized neutron beam (λ = 1.9 Å) with polarization vector P parallel to +y is prepared using a magnetic prism and a guide field B0 ≈ 9 G ê_z, and a DC spin rotator (DC1) rotates |±z⟩ to |±y⟩. A perfect-crystal silicon interferometer splits the |+y⟩ polarized beam into paths I and II; the |−y⟩ beam is blocked. In path I, a rotating field generator (RFG) coil produces a magnetic field B_RFG(t) = B1 cos(Ω t) ê_x + B1 sin(Ω t) ê_z, rotating about the beam axis (y) with angular velocity Ω and amplitude B1. The RFG is a double-coil (x and z) arrangement with π/2 phase-shifted sinusoidal currents; it is water-cooled and optimized to avoid wires in the beam to preserve contrast. A phase shifter (sapphire plate) introduces a controllable relative phase χ between paths. A supermirror analyzer (CoTi array) can be inserted in the O-beam for spin analysis (transmission T ≈ 0.4 for |+z⟩). In path II, a local Helmholtz z-field B_loc acts as a Larmor accelerator to align spin orientations at the last plate by compensating path-dependent Larmor precession in the guide field. A second DC spin rotator (DC2) and a |+z⟩ spin analyzer are used during adjustment procedures. Adjustment and operation: Distances between DC1, RFG, and DC2 are tuned so that spins entering the RFG and DC2 are |+y⟩, using Larmor precession about B0 (ω0 = μ B0/ħ). With path II blocked, DC1 and RFG are set to produce successive π/2 rotations giving intensity minima at appropriate spacings; analogous tuning sets DC2. The RFG is then driven with x and z currents (π/2 phase shift) and their amplitudes are increased to achieve cyclic spin evolution inside the RFG with a(t1) = 2π over the neutron transit time t1, signalled by recurrence of the same intensity minimum. For each frequency f = Ω/(2π) from 0 to 20 kHz, B1 is adjusted (decreasing with higher f) to maintain the cyclic condition. To ensure maximal interference, spins in both paths are aligned at the exit: path I is blocked and B_loc in path II is scanned to yield a minimum, compensating guide-field precession differences (RFG locally cancels guide field z-offset). Measurement: With spin alignment established, interferograms are recorded by rotating the phase shifter and counting O-beam intensity (He-3 detector) for 20 s per point, both without and with the supermirror inserted for additional spin analysis. The relative phase of the interferograms versus the static case (Ω = 0) is extracted by sinusoidal fits. Frequencies from 0–20 kHz are tested. Contrasts are evaluated with and without supermirror to assess coherence and depolarization effects.

- Interferograms recorded for rotating field frequencies from 0 to 20 kHz exhibit continuous phase shifts increasing linearly with rotation frequency, consistent with the expected Mashhoon phase Ω t1/2 at fixed transit time t1. - The linear dependence of phase on frequency matches predictions from the Pauli-Schrödinger equation for spin-rotation coupling; deviations from perfect linearity are systematic and attributed to small misadjustments of the x and z amplitudes in the RFG. - The effect persists with a polarizing supermirror inserted (reduced counts due to T ≈ 0.4), demonstrating that the observed shift is a phase effect independent of spin-vector analysis and thus purely quantum mechanical; the slopes of linear fits with and without supermirror differ by about 8%, likely due to changes in guide-field conditions when removing the supermirror. - Interferometric contrast shows only slight reduction without supermirror, mainly due to depolarization during spin rotation in the RFG and minor field misalignments, confirming coherent spin manipulation. - The phase shift is not due to a Zeeman effect, as B1 decreases with increasing frequency to maintain a(t1) = 2π, while the phase continues to grow with frequency.

The setup could, in principle, be sensitive to the Earth's rotation (Sagnac phase), but the interferometer orientation was fixed, rendering this contribution constant and irrelevant to the measured relative phases. The observed phase shifts are not Zeeman-induced since the rotating field amplitude decreases with frequency under the cyclic evolution condition. Describing the effect in the rotating frame highlights a quantum-mechanical analog to the Sagnac effect, replacing classical orbital angular momentum with intrinsic spin and yielding a simple linear coupling. The results can be interpreted as an energy change for a rotating observer depending on the relative orientation of spin and rotation. The procedure is equivalent to a symmetric configuration with counter-rotating fields ±Ω/2 in both arms. While spin dynamics in NMR and prior neutron polarimetry can be modeled by Bloch equations (classical magnetic moment), the present neutron interferometer directly measures a phase shift in the wavefunction, a quantity with no classical analog in the Bloch picture. Thus, the measurement confirms a purely quantum mechanical consequence involving intrinsic angular momentum and supports the interpretation of an inertia associated with intrinsic spin. Spin-rotation (Mashhoon) and Sagnac effects can both serve as rotation sensors; the Mashhoon phase depends on the time spent in the interferometer (intrinsic property) rather than the interferometer area, allowing closely spaced paths (e.g., SESANS). In the present geometry with k ≈ 3.3 × 10^10 m−1 and area A ≈ 36 cm², the Mashhoon-to-Sagnac phase ratio is ~10−10, motivating high-frequency operation (kHz) to enhance detectability. Detection requires differing spin alignments between paths; no effect is seen for parallel spins or spins perpendicular to the rotation axis. The work connects to broader efforts at the interface of general relativity and quantum theory, suggesting further neutron interferometry tests (e.g., Lense–Thirring).

A neutron interferometer experiment confirmed the theoretical prediction of spin-rotation coupling between a rotating magnetic field and the neutron spin. Unlike prior polarimeter experiments, the interferometer directly measured a phase shift of the neutron wavefunction—a purely quantum effect not accounted for by semi-classical Bloch equations. The observed linear dependence of phase shift on rotation frequency validates spin-rotation coupling of neutrons as a quantum mechanical extension of the Sagnac effect. Future work can exploit the time-dependent nature of the Mashhoon phase for compact rotation sensing and explore related relativistic quantum phenomena with tailored interferometer geometries and higher frequencies.

- Systematic deviations from linear phase–frequency dependence are attributed to small misadjustments of the x and z amplitudes driving the RFG’s rotating field. - The slopes with and without supermirror differ by ~8%, likely due to unaccounted changes in the guide-field control when removing the supermirror element. - Slight contrast reduction arises from depolarization during spin rotation and minor magnetic field misalignments in the RFG. - Reduced count rates when the supermirror is inserted (T ≈ 0.4) limit statistical precision in spin-analyzed measurements. - The setup’s sensitivity to Earth’s rotation (Sagnac) is constant for a fixed orientation but could be a confounding factor in non-fixed configurations. - Observation of the Mashhoon effect requires specific spin-state configurations; parallel spins or spins perpendicular to the rotation axis would not reveal the effect.