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

1. Problem Statement

The paper addresses a fundamental paradox in quantum mechanics regarding the propagation of individual particles in free space. Classical mechanics (Newton's First Law) dictates that a particle with a specific position and momentum at time $t=0$ must follow a straight-line trajectory. Consequently, the probability of finding a particle in a specific position interval $M$ at a later time $t_M$ should be bounded by the joint probability of it having been in a position interval $L$ and a momentum interval $B$ at $t=0$.

This leads to the inequality:
$$P(M) \geq P(L) + P(B) - 1$$
where $P(L)$, $P(B)$, and $P(M)$ are probabilities of finding the particle in intervals $L$, $B$, and $M$ respectively.

The authors investigate the violation of this inequality using a quantum superposition of a state localized in position ($|L\rangle$) and a state localized in momentum ($|B\rangle$). Such a violation implies that quantum particles do not follow classical straight-line trajectories and that the Wigner function (a quasi-probability distribution) must take negative values, signifying a breakdown of classical concepts of motion.

2. Methodology

Experimental Setup:

• Source: A Ti:Sapphire laser ($\lambda = 810$ nm) attenuated to the single-photon level.
• Interferometer: A Sagnac interferometer was used to generate a coherent superposition of the position state $|L\rangle$ and the momentum state $|B\rangle$.
  • Path 1 (Counter-clockwise): Prepared the $|L\rangle$ state by imaging a rectangular slit (width $L$) directly onto the output plane.
  • Path 2 (Clockwise): Prepared the $|B\rangle$ state by performing a Fourier transform of a rectangular slit (width $d$) using a lens, creating a rectangular distribution in momentum space.
  • Superposition: The two paths were recombined at the output PBS. Polarization control (HWP/QWP) allowed for the tuning of the relative amplitude and phase between the two states.
• Detection: A movable detection system consisting of a 5 $\mu$m slit coupled to a multi-mode fiber and an Avalanche Photodiode (APD) scanned the transverse spatial distribution.
• Measurement Points: Three distinct measurement settings were employed:
  1. $t=0$: Direct measurement of the initial position distribution.
  2. $t \to \infty$ (Momentum): Measurement of the momentum distribution via a Fourier transform lens ($f=100$ mm).
  3. $t=t_M$: Measurement at an intermediate propagation distance ($z=100$ mm) where the contributions of position and momentum uncertainties are approximately equal, maximizing the inequality violation.

Theoretical Framework:
The authors utilized the concept of quasi-probabilities to decompose the observed statistics. They modeled the density operator $\hat{\rho}$ as a sum of three components:
$$\hat{\rho} = w_L |L\rangle\langle L| + w_B |B\rangle\langle B| + w_{inter} \left( \frac{|L\rangle\langle B|}{2\langle B|L\rangle} + \text{h.c.} \right)$$
where $w_L$, $w_B$, and $w_{inter}$ are the quasi-probabilities associated with the position state, momentum state, and the interference term, respectively.

3. Key Contributions

1. Experimental Realization of the Paradox: The team successfully generated a high-quality superposition of position and momentum states and experimentally demonstrated the violation of the particle propagation inequality ($P_{defect} > 0$).
2. Decomposition of Interference: Unlike previous studies that focused solely on the violation, this work quantitatively isolates the specific contributions of the position state, momentum state, and the interference term to the total probability distributions.
3. Link to Wigner Function Negativity: The authors explicitly map the observed inequality violation to the negativity of the Wigner function. They demonstrate that the violation arises because the interference term creates a "joint control" of position and momentum that cannot be represented by a positive joint probability distribution.
4. Quantitative Analysis of Non-Locality: The study provides evidence that photons detected in different measurement contexts (position, momentum, intermediate) cannot be assigned to the same classical trajectory, challenging hidden-variable theories like Bohmian mechanics.

4. Results

• Inequality Violation:

  • Measured probabilities: $P_{exp}(L) = 0.5249 \pm 0.0007$ and $P_{exp}(B) = 0.6445 \pm 0.0007$.
  • Measured intermediate probability: $P_{exp}(M) = 0.1092 \pm 0.0002$.
  • Defect Probability: $P_{defect} = P(L) + P(B) - 1 - P(M) = 0.0602$. This positive value confirms the violation of Newton's first law for these quantum states.

• Quasi-Probability Decomposition:
  By analyzing the diffraction patterns and fitting sinc functions, the authors determined the weights of the components:

  • $w_L \approx 0.355$ (Position state contribution)
  • $w_B \approx 0.493$ (Momentum state contribution)
  • $w_{inter} \approx 0.152$ (Interference term contribution)

• Role of Interference:

  • The interference term contributes positively (approx. 15.2%) to both $P(L)$ and $P(B)$, effectively "localizing" photons in both intervals simultaneously.
  • However, at the intermediate time $t_M$, the interference pattern spreads out, contributing only $\approx 5.4\%$ to $P(M)$.
  • This discrepancy (high contribution to $L$ and $B$, low contribution to $M$) is the source of the inequality violation.

• Wigner Function Negativity:

  • Theoretical limits on the joint probability $W_{LB}$ (integral of Wigner function over $L$ and $B$) are bounded by $LB/\pi\hbar \approx 0.074$.
  • The required joint probability from the interference term ($w_{inter} = 0.152$) exceeds this bound.
  • To satisfy normalization, the Wigner function must be negative in the regions outside the intervals $L$ and $B$. The authors calculate a required negativity of $W_{out} \leq -0.095$, which is directly responsible for the observed violation.

5. Significance

This work provides rigorous experimental evidence that quantum interference fundamentally alters particle propagation in free space, preventing the assignment of classical trajectories.

• Failure of Classical Intuition: The results confirm that the classical assumption—that a particle's future position is strictly determined by its initial position and momentum—is invalid in quantum mechanics.
• Nature of Quantum Reality: The authors argue against the idea that photons follow hidden trajectories (as in Bohmian mechanics). Instead, they suggest that the "particle" is delocalized until detected, and the measurement context (position vs. momentum vs. intermediate) defines the causal connection.
• Wigner Function Validation: The study offers a direct quantitative link between the violation of a classical inequality and the mathematical necessity of negative values in the Wigner function, reinforcing the interpretation of the Wigner function as a true indicator of non-classicality.
• Methodological Advance: The ability to decompose the density matrix into distinct quasi-probability contributions ($w_L, w_B, w_{inter}$) using only standard spatial measurements offers a powerful tool for characterizing complex quantum states and their non-classical propagation dynamics.

In conclusion, the paper demonstrates that the "particle propagation paradox" is not merely a theoretical curiosity but a measurable phenomenon where quantum interference creates a statistical reality that defies classical straight-line motion, necessitating a description via negative quasi-probabilities.