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arXiv:2207.07270 · Physical Review A

Controlling and measuring a superposition of position and momentum

Takafumi Ono, Nigam Samantaray, John Rarity

Published Jul 25, 2023• 0 citations
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Abstract

We present an observation of quantum interference between position and momentum with photons. We prepared a superposition of position and momentum states by using slits, lenses, and an interferometer. The precise control and measuring of propagation time of the photon results in a clear quantum interference of $85%$ visibility. The generated quantum state is then used to verify the violation of particle propagation inequality proposed by Hofmann [Phys. Rev. A 96, 020101(R) (2017)], resulting in $5.9%$ below the statistically predicted lower bound. We believe that this result is a major step towards further developments in quantum theory, such as fundamental modifications to the interpretation of quantum mechanics.

DOI
10.1103/physreva.108.012215

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We present an observation of quantum interference between position and momentum with photons. We prepared a superposition of position and momentum states by using slits, lenses, and an interferometer. The precise control and measuring of propagation time of the photon results in a clear quantum interference of 8585%85 visibility. The generated quantum state is then used to verify the violation of particle propagation inequality proposed by Hofmann [Phys. Rev. A 96, 020101(R) (2017)], resulting in 5.95.9%5.9 below the statistically predicted lower bound. We believe that this result is a major step towards further developments in quantum theory, such as fundamental modifications to the interpretation of quantum mechanics.
As shown recently [H. F. Hofmann, Phys. Rev. A 96, 020101(R) (2017)], it is possible to demonstrate that quantum particles do not move along straight lines in free space by increasing the probability of finding the particles within narrow intervals of position and momentum beyond the ``either/or'' limit of 0.5 using constructive quantum interference between a component localized in position and a component localized in momentum. The probability of finding the particle in a corresponding spatial interval at a later time then violates the lower bound of the particle propagation inequality which is based on the validity of Newton's first law. In this paper, the problem of localizing the two state components in their respective target intervals is addressed by introducing a set of three coefficients that describe the localization of arbitrary wave functions quantitatively. This characterization is applied to a superposition of Gaussians, obtaining a violation of the particle propagation inequality by more than 5% if the width of the Gaussian wave function is optimized along with the size of the position and momentum intervals. It is shown that the violation of the particle propagation inequality originates from the fundamental way in which quantum interferences relate initial position and momentum to the future positions of a particle, indicating that the violation is a fundamental feature of causality in the quantum limit.
Optimal simultaneous control of position and momentum can be achieved by maximizing the probabilities of finding their experimentally observed values within two well-defined intervals. The assumption that particles move along straight lines in free space can then be tested by deriving a lower limit for the probability of finding the particle in a corresponding spatial interval at any intermediate time ttt. Here, it is shown that this lower limit can be violated by quantum superpositions of states confined within the respective position and momentum intervals. These violations of the particle propagation inequality show that quantum mechanics changes the laws of motion at a fundamental level, providing a different perspective on causality relations and time evolution in quantum mechanics.
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.
Since the wavefunction of a photon only describes the probability of photon detection in time and space, it is impossible to derive uniquely defined trajectories describing the path taken by the photon between emission and detection. However, it is possible to test whether a particular set of trajectories is consistent with the statistics observed at different times for photons in the same initial state. Recently, I have shown that quantum interference effects between position and momentum can result in a violation of inequalities associated with motion along straight lines. Here, I present a more detailed analysis on the origin of the effect and its relation with other experimentally observable aspects of quantum statistics such as weak measurements and quantum tomography. It is shown that the interference pattern between a quantum state component of well-defined position and a quantum state component of well-defined momentum describes a modified causality relation between the positions detected at different times. The phase of the interference pattern is identified with the classical action of particle motion and the relation between uncertainty and causality is considered. The specific case of single photon wavefunctions is used to explain the possibilities and limitations of control at the ultimate quantum limit.
In the experiments considered here, we measure the y-component of momentum for a particle passing through a system of slits. The source-slit system is the preparation apparatus that determines the state vector. Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. Then, writing the state vector in momentum space provides a straightforward calculation for the probability amplitude and its corresponding probability function. Interference effects, if any, are inherent in the probability function We determine the statistical distribution of scattered particles for four different slit systems. The results are in agreement with the well-known interference patterns obtained in classical wave optics.
A quantum experiment consists of a preparation procedure followed by the measurement of a specified observable. In the experiments considered here, the y -component of momentum is measured after a particle passes through a system of slits. The source–slit system is the preparation apparatus that determines the initial state |ψ⟩. Recognizing that a system of slits is a position-measuring device allows us to determine |ψ⟩. Then, a straightforward calculation gives P ( p y ) = |⟨ p y |ψ⟩| 2 the probability that a particle will pass through a system of slits with momentum p y = p sin θ. Interference effects are inherent in this probability function. We perform the calculation for four different slit systems. The method is suitable for an upper level course in quantum mechanics.