Three-path quantum Cheshire cat observed in neutron interferometry

Since the introduction of quantum mechanics, its theoretical framework has suggested counter-intuitive and paradoxical phenomena such as entanglement, Schrödinger’s cat, and wave–particle duality. Their study deepens understanding and enables technologies, while interpretations of quantum mechanics, though operationally equivalent, differ in assumptions. One such effect concerns the apparent spatial separation of a particle and its properties in an interferometer—the quantum Cheshire Cat (qCC)—proposed by Aharonov et al., where weak, path-local interactions combined with pre- and post-selection suggest that different properties are localized in different paths. The first neutron experiment (two-path interferometer) indicated separation of particle and spin via distinct reactions to absorption (particle) and magnetic fields (spin). Subsequent work proposed improvements (simultaneous weak measurements, second-order effects, additional degrees of freedom) and raised critiques regarding quantumness and interaction strengths. Pan proposed a generalized qCC with arbitrarily many degrees of freedom. Here, the authors report a three-path neutron interferometer experiment that adds the neutron’s energy as a third property, aiming to clarify the origin of the qCC by showing how the effect emerges from the inner product between weakly perturbed intermediate states and post-selected states and from cross-terms between path amplitudes.

The paper situates its work within prior demonstrations and analyses of the quantum Cheshire Cat: the initial neutron-based observation separating particle and spin, optical analogues, theoretical debates on whether the effect is genuinely quantum, and methodological suggestions including simultaneous weak measurements and accounting for higher-order responses. A generalized N-path, multi-property framework by Pan is referenced and used for comparison in the discussion. The authors also note critiques regarding weak interaction strengths and alternative classical interpretations.

- Experimental platform: Neutron interferometry station S18 at the Institut Laue-Langevin (ILL), Grenoble. Monochromatized thermal neutrons (λ0 = 1.92 Å, δλ/λ0 ≈ 2%, E ≈ 25 meV). Spin polarized to +z via magnetic prisms. - Interferometer: Silicon perfect-crystal three-path interferometer producing three spatially separated sub-beams (paths I, II, III) recombined into O and H output beams. Thermal and environmental stabilization limits phase drift to ~1°/hour. - Detection and analysis: O-beam analyzed by a polarizing supermirror (up-spin selection) and recorded by a 3He counter. Two phase shifters (PS1, PS2) control relative phases χ1, χ2. - State preparation (pre-selection): Creates three pairwise orthogonal sub-states across path, spin, and energy. Implement DC spin flip in path I (↓), RF spin flip in path III (↓ with energy change), and leave path II as ↑. RF flipper frequency f = 60 kHz imparts energy shift ΔE = hf = 0.25 neV, producing two effective energy levels treated as orthogonal (time-averaged detection). Pre-selected tripartite entangled state: |i⟩ = (1/√3)(|I,↓,E0⟩ + |II,↑,E0⟩ + |III,↓,E0⟩). - Post-selection: Implemented by PS1, PS2 and supermirror up-spin projection; |f(χ1,χ2)⟩ ∝ e^{iχ1}|I⟩ + e^{iχ2}|II⟩ + |III⟩ (no energy selection). Given this pre-/post-selection, only the amplitude through path II contributes without further perturbation; path II is the reference beam. - Weak interactions (one at a time, in one path): • Weak DC spin rotation around x with angle αrot = π/9 = 20°. • Weak RF spin rotation (coupled spin-energy flip) with αrot = 20°. • Weak absorption via Indium foil (0.125 mm), absorption coefficient A = 0.10(1). Each weak interaction can be applied in any path, yielding 3×3 cases. Interactions are weak to ensure small state disturbance. - Theoretical modeling: Path-selective unitary rotations Uj^DC/RF(αrot) ≈ 1 − (1−cos(αrot/2))Πj − i sin(αrot/2) σx Πj; weak absorber operator ÂAbs(A) = 1 − (1−√(1−A)) Πj. Time-averaged intensities derived for DC and RF cases include terms linear in αrot (from cross-terms) and quadratic mean-intensity shifts, while absorption yields linear attenuation in the reference path. - Interferograms (IFGs): Record I(χ) = I0 + B sin(ωχ + φ). Three sets: • Empty interferometer to characterize maximum achievable contrast (≈50–57%). • Preparational IFGs (with DC/RF flips only) to verify orthogonality (low contrasts ≤4%). • Weak-interaction IFGs (with additional weak interaction) for all 9 path–interaction combinations. - Measurement protocol: Alternating on/off scheme for weak interactions to ensure stable phase comparison, except absorber IFGs measured sequentially with thermal stabilization. - Extraction of weak values: Model weak-interaction IFGs as sum of preparational oscillation plus a “signal” oscillation; extract signal amplitude B_signal and phase via fits and error propagation. Calibrate using empty-interferometer contrast C_empty and αrot to obtain moduli of weak values for Πj^x (spin-x in path j) and Πj^E (energy transition in path j). For absorption, path weak values obtained directly from mean intensity change: A_eff = 1 − I0,weak/I0,prep. - Adjustments and calibration: RF resonance at 60 kHz (local guide field ≈20 G), global guide field ≈10 G with local compensation/amplification for DC/RF. Coil currents tuned via polarimetric and interferometric contrast-minimization scans to achieve efficient flips. Systematic effects (field inhomogeneities, mutual induction between adjacent coils, eddy currents in absorber) assessed and bounded.

- Demonstration of a three-path quantum Cheshire Cat with neutrons: different properties exhibit conspicuous reactions only when weakly perturbed in specific paths. - Interferograms: Weak-interaction IFGs show strong, first-order (in αrot) oscillatory signals in diagonal cases (interaction matched to its associated path), with off-diagonal cases showing only small or negligible reactions, consistent with theoretical predictions of sensitive/robust/indifferent regimes. - Weak values: Extracted moduli form approximately an identity matrix across properties vs. paths, indicating: • Spin x-component effectively in path I. • Particle (path occupancy relevant to absorption) effectively in path II. • Energy degree of freedom (via coupled RF spin–energy manipulation) effectively in path III. Deviations are generally within statistical/systematic uncertainties; the largest deviation occurs for weak RF in path III (measured modulus ≈0.75 vs. expected 1), attributed to coil interactions. - Mean intensity changes (Table 3): • DC rotation: relative I (I, II, III) = 1.02(1), 0.99(1), 1.01(1) (≈+3% in non-reference paths; ≈−3% in reference path expected). • Absorber: 0.99(1), 0.92(1), 0.99(1) (≈10% drop in path II as predicted; others ≈unchanged). • RF rotation: 1.02(1), 0.99(1), 1.05(1) (increase in non-reference paths; small decrease in reference path). - Preparational contrasts ≤4% confirm high orthogonality of pre-selected sub-states; empty interferometer contrasts ≈50–57% indicate coherence limits of the device. - The observations align with a model where the inner product between weakly rotated intermediate states and the post-selected state generates cross-terms between amplitudes from the reference and a selected non-reference path, producing the conspicuous first-order signals responsible for the qCC signatures.

The authors analyze the qCC emergence by comparing with Pan’s generalized N-path framework. In both the generalized and experimental three-path cases, the post-selection admits only the reference-path amplitude unless a weak, path-local unitary generates a component parallel to the post-selected state. This creates a cross-term between the reference amplitude and the perturbed non-reference path amplitude, producing an intensity oscillation proportional to sin(χ1 − χj) with magnitude linear in the interaction strength for small α. They classify responses as: sensitive (linear-in-α, when a rotation in a non-reference path generates a component parallel to post-selection), robust (only quadratic mean-intensity change when rotating the reference path), and indifferent (no response when the rotation cannot generate a post-selected component). The absorber acts on the particle ‘property’ and reveals localization in the reference path. The data corroborate that the conspicuous reactions stem from interference cross-terms, not necessarily implying a literal physical separation of properties; instead, an effective separation emerges conditioned on pre- and post-selection. Considering completeness across all possible output channels, expectation values are recovered by averaging weak values weighted by post-selection probabilities, resolving apparent contradictions between zero expectation values and nonzero weak values in a given port. The authors thus argue the observed qCC can be understood as an interference-induced, effective delocalization of properties rather than requiring realist separability, while acknowledging that their data do not definitively exclude a realist interpretation.

A three-path quantum Cheshire Cat is realized with thermal neutrons, showing that neutron spin (x-component), particle (path occupancy relevant to absorption), and energy respond as if located in different interferometer paths under weak, path-local interactions and specific pre-/post-selection. The conspicuous responses are explained by inner products generating cross-terms between path amplitudes. Despite no energy discrimination in post-selection, the expected qCC signatures appear in the weak limit. This establishes a clear, interference-based mechanism behind the effect. Potential future directions include implementing energy-selective post-selection, decoupling energy changes from spin flips (e.g., RF plus compensating DC to isolate energy), improving statistical precision to resolve higher-order intensity changes, and mitigating coil coupling to refine weak-value extraction.

- Post-selection not energy selective; energy-sensitive signatures inferred via coupled RF spin–energy operations and time-averaged detection (energy-beat interferences not directly observed). - Time-integrating detection precludes observation of microsecond-scale energy interference; energy levels treated as orthogonal effective modes. - Systematic effects: field inhomogeneities lowering spin-manipulation efficiency; mutual induction between closely spaced RF coils (notably in path III) causing unintended manipulations; eddy currents in the Indium absorber; limited thermal/phase stability and monochromaticity. - Largest systematic uncertainty in weak RF rotation in path III, consistent with the observed deviation of its weak value from unity. - Statistical precision insufficient to quantitatively confirm all predicted second-order mean-intensity changes; higher statistics desired.