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.

Figures (10)

• Figure 1: Experimental test of motion in free space. If particles travel in a straight line, any particle that can be found simultaneously within a position interval $L$ and a momentum interval $B$ at $t=0$ will not have traveled beyond a classically expected bound of $L+(B\:t_M)/m$ at time $t_M$.
• Figure 2: Integration region in phase space for the Wigner function.
• Figure 3: Experimental Setup. The initial light with $\lambda = 810 \:\mathrm{nm}$ from the Ti:Sapphire laser was attenuated to the single photon level using ND filters and its stability was monitored by an avalanche photo diode module APD1. The input polarization used for fine-turning of amplitude and phase of the superposition state was adjusted by a half wave plate and a quarter wave plate. As a result, a superposition of the $\ket{L}$ state and the $\ket{B}$ state was prepared by the interference at the exit of the interferometer. The transverse photon distribution was measured with a detector system mounted on a movable stage connected to another avalanche photo diode module APD2 via a multi-mode fiber.
• Figure 4: Detection system to measure the transverse distribution. The photons pass through a slit of $5 \:\mathrm{\mu m}$ coupled to a multi-mode fiber by an objective lens. This system was mounted on an automated X-stage to be moved in increments of $5 \:\mathrm{\mu m}$ in a transverse direction.
• Figure 5: Preparation of $\ket{L}$ and $\ket{B}$ states in the Sagnac Interferometer. (a): Preparation of the state $\ket{L}$. The rectangular distribution just after the slit is transferred by the biconvex lens. (b): Preparation of he state $\ket{B}$. The distance between the slit and the Fourier transform lens was adjusted to be equal to its focal length. The rectangular distribution in position space was thus transformed into the rectangular distribution in momentum space.