Experimental Separation of Wave and Particle Attributes of a Single Photon Based on the Quantum Cheshire Cat Concept

The paper addresses the longstanding question of which path a single quantum particle (specifically, a photon) takes in a Mach–Zehnder interferometer or double-slit setup, a central issue in wave-particle duality. While numerous theoretical and experimental studies have deepened understanding, ambiguity remains about path information at the single-particle level. The work leverages weak measurement—a minimally disturbing measurement technique—to explore this problem. Inspired by the quantum Cheshire cat concept, which suggests physical properties can be disembodied from their carriers via appropriate pre- and post-selection, the study aims to experimentally separate the wave and particle attributes of a single photon and verify the separation by extracting weak values.

Weak measurement has enabled extraction of quantum state information with reduced disturbance, avoiding collapse into eigenstates, and has been applied to phenomena such as the spin Hall effect of light, direct measurement of quantum wavefunctions, and tests of Bell inequality violations. The quantum Cheshire cat effect, motivated by “Alice in Wonderland,” proposes that intrinsic properties (e.g., spin) can be separated from particles, demonstrated in neutron and optical systems. Traditional strong and weak measurements often require auxiliary pointers, scaling complexity with system size and limiting operability. Imaginary-time evolution (ITE) provides a simpler scheme for extracting weak values, reducing experimental difficulty. A recent thought experiment proposed separation of wave and particle attributes of a quantum entity, which this work seeks to realize experimentally.

Theoretical framework: A wave-particle superposition state is prepared using a wave-particle toolbox: |ψ⟩ = cos(α)|Particle⟩ + sin(α)|Wave⟩. After beam splitter BS1, the pre-selection state is |ψ_i⟩ = (|L⟩ + |R⟩)(cos(α)|Particle⟩ + sin(α)|Wave⟩)/√2, where |L⟩ and |R⟩ denote the interferometer paths. Weak values are defined by ⟨A⟩_w = ⟨ψ_f|A|ψ_i⟩ / ⟨ψ_f|ψ_i⟩. Under the ITE-based weak measurement framework with non-unitary evolution U(H,t) = e^(−iHt) and H = A, the normalized incidence rate is N = N(U)/N0 with N0 = ⟨ψ|ψ⟩^2 and N(U) = ⟨ψ|U|ψ⟩^2. In this experiment, only the real part contributes, yielding ⟨A⟩_w = (∂N/∂t)/2. Post-selection: The target post-selection state is |ψ_f⟩ = (|L⟩|Wave⟩ + |R⟩|Particle⟩)/√2, implemented via operations U, BS2, and X. Operation U swaps wave and particle states (|Wave⟩ ↔ |Particle⟩). Operator X separates |Wave⟩ to Detector D1 and reflects |Particle⟩ to Detector D3. Validation proceeds by evolving a |Wave⟩ state from D1 backward through X and BS2 to split into (|L⟩|Wave⟩ + |R⟩|Wave⟩)/√2, then applying U on the |R⟩ path to transform |Wave⟩ to |Particle⟩, yielding (|L⟩|Wave⟩ + |R⟩|Particle⟩)/√2 at post-selection. Observables for attribute localization are defined: for particle, Π_P^L = |L⟩⟨L| ⊗ |P⟩⟨P|, Π_P^R = |R⟩⟨R| ⊗ |P⟩⟨P|, Π_P^T = Π_P^L + Π_P^R; for wave, Π_W^L = |L⟩⟨L| ⊗ |W⟩⟨W|, Π_W^R = |R⟩⟨R| ⊗ |W⟩⟨W|. Applying these with the weak value expression yields: ⟨Π_P^L⟩_w = 0, ⟨Π_P^R⟩_w = cos(α); ⟨Π_W^L⟩_w = 0, ⟨Π_W^R⟩_w = sin(α). Experimental setup: Single photons are generated via spontaneous parametric down-conversion (SPDC) in type-II β-barium borate (BBO) pumped by ultraviolet pulses (~100 mW, 400 nm). One photon serves as a trigger; the other enters the interferometer. Due to path differences, a 15 ns delay is set between trigger and core paths, with a 3 ns coincidence window. Wave-particle toolbox: A polarization beam splitter (PBS) and half-wave plate (HWP) set to α/2 prepare |ψ⟩ = cos(α)|H⟩ + sin(α)|V⟩. A beam displacer (BD) separates polarizations into up (|H_u⟩) and down (|V_u⟩) paths. Encoding: The wave and particle attribute states are prepared using sets of HWPs and quarter-wave plates (QWPs). General forms are |Wave⟩ = e^(iφ1)[cos(φ1/2)|H_u⟩ − i sin(φ1/2)|V_u⟩] and |Particle⟩ = (|H_u⟩ + e^(iφ2)|V_u⟩)/√2. As the predicted weak values depend only on α, φ1 and φ2 are set to 0 for simplicity, yielding |Wave⟩ = |H_u⟩ (HWP 45°, QWP 0°) and |Particle⟩ = (|H_u⟩ + |V_u⟩)/√2 (HWP 22.5°, QWP 45°). This prepares the input |ψ⟩ = cos(α)|Particle⟩ + sin(α)|Wave⟩. Pre-selection: Using a BD and wave plates to realize BS1, the input is split into |L⟩ and |R⟩ paths, preparing |ψ_i⟩. Weak measurement: Neutral density (ND) filters inserted into different paths simulate the weak disturbance in ITE, with transmission T = e^(−2t), t the interaction time. By varying transmission rates, N vs t is measured and fitted; the weak value is obtained from the slope (reported as minus half of the slope in practice). Post-selection: Operation U is implemented with wave plates and two BDs to exchange paths and corresponding polarization states. BS2 is realized with a BD, an HWP at 22.5°, and a PBS. In this encoding, X naturally routes |Wave⟩ to Detector D1 and reflects |Particle⟩ to Detector D3. Single-photon detectors (SPDs) D1–D3 record clicks in coincidence with the trigger. Quantum state tomography of the BS2 output assesses interference visibility and setup fidelity.

Theoretical weak values indicate spatial separation of attributes: ⟨Π_P^L⟩_w = 0 and ⟨Π_P^R⟩_w = cos(α) show particle attributes localized to the right path; ⟨Π_W^L⟩_w = 0 and ⟨Π_W^R⟩_w = sin(α) show wave attributes localized to the left path. The proportion is controlled by α; for α = π/4, ⟨Π_P^R⟩_w = ⟨Π_W^L⟩_w = 1/2, meaning half the particle attribute resides in |R⟩ and half the wave attribute resides in |L⟩. Experimentally, quantum state tomography of the BS2 output yields an average fidelity of 99.45 ± 0.26%, indicating high interference visibility and accurate state preparation. Using ND filters with varied transmissions (T = e^(−2t)), linear fits of incidence rate vs interaction time provide slopes from which weak values are extracted (minus half the slope). For α ≈ 45°, weak values for multiple observables (denoted Π_x, Π_y, Π_z, Π_w in the reported plots) are consistent with theoretical expectations, supporting successful spatial separation of wave and particle attributes.

The results address the core question of wave-particle duality by demonstrating that a single photon’s wave and particle attributes can be spatially separated within an interferometric setup via appropriate pre- and post-selection and weak measurement. Nonzero weak values in one path and zeros in the other confirm attribute localization, providing operational evidence for the quantum Cheshire cat effect applied to wave-particle properties. High-fidelity tomography validates the integrity of the state preparation and interferometric evolution. This separation opens avenues to probe foundational aspects of quantum mechanics, such as performing classic experiments (Young’s interference, photoelectric effect, diffraction) with isolated wave or particle attributes, thereby clarifying the roles each attribute plays in observed phenomena.

The study experimentally demonstrates, for the first time, the separation of the wave and particle attributes of a single photon using the quantum Cheshire cat framework and weak measurement via imaginary-time evolution. By engineering pre- and post-selected states and extracting weak values with minimal disturbance, the work confirms spatial attribute separation and provides a robust platform for investigating quantum foundations. Future research directions include exploring fundamental optical and quantum phenomena with isolated attributes (e.g., conducting interference or diffraction with only the particle attribute, studying the photoelectric effect with only the wave attribute), extending the approach to other particles and systems, and refining weak measurement techniques to access complex weak values beyond the real part.

The experiment accesses only the real part of weak values under the ITE framework, potentially limiting the scope of observable effects. The weak interaction is simulated using ND filters, an effective but approximate proxy for ideal weak couplings. Phase parameters φ1 and φ2 are fixed to zero for simplicity, reducing generality. The approach relies on precise interferometric alignment and high-quality optical components; while tomography indicates high fidelity, practical scalability and robustness in more complex settings may be challenging.