Characterization of non-classical particle propagation using superpositions of position and momentum

Abstract: The uncertainty principle suggests a quantitative trade-off between the control of position and the control of momentum in particle propagation. However, a superposition of two states with very different uncertainty trade-offs introduces an interference term that seems to combine precise statements about position and about momentum, allowing us to study how quantum mechanics describes the propagation of individual particles in free space. Here, we present a detailed experimental study of photons prepared in a superposition of position and momentum generated in a Sagnac interferometer. The transverse distribution of photons was obtained with three different measurement settings at the output port of the interferometer, corresponding to the initial position distribution, the initial momentum distribution, and an intermediate propagation time at which the contributions of initial position and momentum uncertainties are approximately equal to each other. We show that the interference effect localizes the photons in narrow intervals of position and momentum, resulting in a quantitative violation of Newton's first law as the interference pattern spreads out at the intermediate position. The data obtained can be used to demonstrate the negativity of the Wigner function in regions outside the position and momentum intervals in which the position and momentum contributions are confined.

• The paper demonstrates that superpositions of localized position and momentum produce interference that violates classical Newtonian propagation laws.
• The methodology employs a Sagnac interferometer with precise optical configurations to separately measure position, momentum, and intermediate propagation effects.
• Results reveal significant defect probabilities and Wigner negativity, challenging classical trajectories and supporting quantum contextuality.

Characterization of Non-Classical Particle Propagation via Superpositions of Position and Momentum

Introduction and Motivation

The statistical interplay between position and momentum in quantum dynamics, crucially constrained by the uncertainty principle, remains central to foundational questions in quantum mechanics. Notably, the creation of superpositions of states highly localized in either position ($\ket{L}$) or momentum ($\ket{B}$) enables quantum interference which can directly test and quantify violations of classical propagation rules such as Newton’s first law. This paper presents a rigorous experimental investigation of such superpositions in the context of single-photon propagation, probing the resulting statistical structures in both coordinate and momentum space, as well as at intermediate propagation times. Particularly, the experiment measures quantitative violations of classical path expectations, and employs Wigner function analysis to relate these violations to negative probability regions in phase space.

Theoretical Framework: The Particle Propagation Paradox

The experiment tests a fundamental inequality implied by Newtonian propagation: any particle confined within a region $L$ in position and $B$ in momentum at $t=0$ must, by straight-line evolution, be found within an expanded region $M=L+Bt_M/m$ at time $t_M$. Translating this to probabilities, the classical bound is $P(M) \geq P(L) + P(B) - 1$, since overlap between intervals may be nonzero, but is bounded by the populations.

Quantum mechanically, for a superposition state $\ket{\psi}$ composed of $\ket{L}$ and $\ket{B}$0, interference terms appear which can boost the individual probabilities $\ket{B}$1 and $\ket{B}$2 beyond what a positive joint distribution permits, while $\ket{B}$3 remains limited by the orthogonality and spreading of the underlying wavefunctions. This leads to a possible positive “defect” probability $\ket{B}$4, signaling direct violation of Newton’s first law for single particles.

Figure 1: Schematic of the test for Newtonian propagation; only particles within prescribed intervals of $\ket{B}$5 and $\ket{B}$6 at $\ket{B}$7 are classically allowed within $\ket{B}$8 at $\ket{B}$9.

Crucially, this paradox maps onto the theory of quasi-probabilities: in Wigner phase space, any such violation requires negativity of the Wigner function outside the $L$0-$L$1 ‘cross’ region, as positive Wigner functions cannot violate the classical marginal bounds.

Figure 2: Visualization of phase-space integration regions for the Wigner function corresponding to various measurement domains.

Experimental Methodology

To realize and analyze these interference effects, the experiment employed a Sagnac interferometer to generate well-controlled superpositions of $L$2 (position-localized) and $L$3 (momentum-localized) photon states. Input light from an attenuated Ti:sapphire laser at $L$4 was manipulated in polarization and path length to tune overlap and phase between the desired components, and the resulting quantum states were probed at distinct propagation distances.

Figure 3: Experimental setup highlighting the Sagnac interferometer design and polarization controls for superposition state preparation.

The detection apparatus involved scanning single-photon counting modules behind narrow slits and lenses to measure spatial distributions directly, implement Fourier-space (momentum) detection, and access the intermediate location corresponding to maximal expected inequality violation.

Figure 4: Detailed schematic of the detection system for spatial probability distributions.

The separate preparation of $L$5 and $L$6 within the interferometer was achieved through lens configurations effecting direct real-space imaging and Fourier transformation using carefully aligned optical elements.

Results

Characterization at $L$7: Position and Momentum Distributions

Recorded data for both position and momentum at $L$8 exhibit two prominent features: sharp rectangular peaks reflecting population within $L$9 and $B$0, and broad sinc-like diffraction tails from their complementary Fourier-limited spreads. Notably, the intervals relevant to violation calculation are set at $B$1, with the overlap $B$2. Precise background subtraction and normalization yielded $B$3 and $B$4.

Figure 5: Experimental probability densities, showing central peaks at the prescribed $B$5 and $B$6 intervals and low-probability sidelobes from sinc-type diffraction.

Decomposition into contributions from $B$7, $B$8, and their interference term via quasi-probabilities provided $B$9, $t=0$0, and interference weight $t=0$1. Crucially, the interference term significantly enhances joint probability within $t=0$2 and $t=0$3 intervals, the direct source of classical inequality violation.

Figure 6: Fits to the diffraction tails in position and momentum, isolating sinc contributions and refining estimates for overlap and quasi-probability weights.

Intermediate Propagation: Distributions at $t=0$4

At the propagation time $t=0$5, the measured position distribution was characterized by high-visibility interference fringes over an envelope, confirming coherent superposition rather than statistical mixing. The probability within the classical region $t=0$6 was found to be $t=0$7, yielding a clear defect probability $t=0$8, well above statistical uncertainty and fully inconsistent with Newtonian evolution for any potential hidden variable theory.

Figure 7: Measured position distribution at $t=0$9, showing interference structure and the expanded classical interval $M=L+Bt_M/m$0.

Envelope fitting and fringe analysis confirm that only a substantially reduced fraction of the interference term is present within $M=L+Bt_M/m$1, validating the hypothesis that Wigner negativity (localization beyond classical bounds) is responsible for the propagation anomaly.

Figure 8: Quantitative estimation of signal within the interference envelope, used to apportion contributions from position, momentum, and interference terms.

Wigner Function Negativity and Quantum Propagation

Analysis of the data within the Wigner function framework demonstrates that the observed probabilities require significant negativity in $M=L+Bt_M/m$2, the quasi-probability integrated beyond the $M=L+Bt_M/m$3-$M=L+Bt_M/m$4 cross. Explicitly, $M=L+Bt_M/m$5, more than enough to support the observed $M=L+Bt_M/m$6.

The observed constructive interference within $M=L+Bt_M/m$7 and $M=L+Bt_M/m$8 does not translate into a correspondingly high $M=L+Bt_M/m$9, revealing a fundamental asymmetry enforced by phase-space negativity. This effect cannot be reconciled with any positive probability distribution over phase space or with deterministic particle trajectories, as in Bohmian mechanics.

Implications and Outlook

The findings have direct consequences for foundational discussions of quantum mechanics:

• Experimental quantification of Wigner negativity: The results decisively confirm that negative quasi-probabilities are not just theoretical artifacts but observable properties manifested in single-particle propagation.
• Refutation of realist trajectory interpretations: Classical or hidden variable models, including Bohmian mechanics, cannot reproduce the defect probability without violating the linearity or positivity constraints of the density matrix.
• Measurement contextuality and causal structure: The data indicate that identification of a photon at a certain position, momentum, or time does not imply an underlying trajectory, but reflects different sub-ensembles of the superposed quantum state.

Practically, this work suggests that exploiting non-classical interference between canonical observables could become a resource for quantum information processing, quantum measurement theory, and metrology. The methodology provides a pathway for further studies of quantum backflow, entanglement in single-particle phase space, and the dynamics of quasi-probability distributions.

Conclusion

Through a sophisticated combination of interferometric state preparation, coordinate and momentum-resolved measurement, and quasi-probability analysis, this study provides robust experimental evidence for the violation of Newtonian propagation in quantum systems via non-classical superpositions. The results demonstrate the necessity of Wigner function negativity for such violations and reinforce the incompatibility of deterministic ontological trajectories with quantum interference phenomena. Future work may explore generalizations to higher dimensions, other forms of quasi-probability, and the incorporation of these findings into applied quantum technologies.