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arXiv:1303.7101 · Physical Review A

Multislit interferometry and commuting functions of position and momentum

Johannes C. G. Biniok, Paul Busch

Published Jun 21, 2013• 4 citations
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Abstract

In a recent, modified double-pinhole diffraction experiment the existence of an interference pattern was established indirectly along with a near-perfect imaging of the double pinhole. Our theoretical analysis shows that the experiment constitutes a preparation of a quantum state that is, to a good approximation, a joint eigenstate of commuting functions of position and momentum. Gaining information about the momentum distribution by means of the particular experimental setup is thus possible with negligible impact on the position distribution. Furthermore, we construct explicitly a class of states simultaneously localized on periodic sets in position and momentum space, which are therefore eigenstates of the observables being measured jointly (to a good approximation) in multislit interferometry. Finally, we show that with an appropriate change of settings the experiment demonstrates the mutual disturbance of position and momentum measurements.

DOI
10.1103/physreva.87.062116

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In a recent, modified double-pinhole diffraction experiment the existence of an interference pattern was established indirectly along with a near-perfect imaging of the double pinhole. Our theoretical analysis shows that the experiment constitutes a preparation of a quantum state that is, to a good approximation, a joint eigenstate of commuting functions of position and momentum. Gaining information about the momentum distribution by means of the particular experimental setup is thus possible with negligible impact on the position distribution. Furthermore, we construct explicitly a class of states simultaneously localized on periodic sets in position and momentum space, which are therefore eigenstates of the observables being measured jointly (to a good approximation) in multislit interferometry. Finally, we show that with an appropriate change of settings the experiment demonstrates the mutual disturbance of position and momentum measurements.
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The present investigation is about the quantum mechanics of multislit interference experiments. One of the cornerstones of our understanding of quantum mechanics is provided by the analysis of this type of experiments. Beginning with Bohr and Einstein, the discussion revolved mostly around single-slit diffraction and double-slit interference, probing the complementarity of path determination and the appearance of interference fringes, and contrasting the familiar with the quantum. Here, we are concerned with developing a systematic understanding and description of multislit interference experiments, i.e. setups in which a plurality of slits is illuminated. We provide a characterisation of a number of relevant observables, discussing those that are compatible and may be measured jointly, and also incompatible observables which cannot be measured jointly but instead display quantum uncertainty. \nWe begin with a discussion of a particular modification of the classic double- slit interference experiment which highlights the realisation of specific position and momentum observables which are jointly measurable. Although there are technical results regarding the coexistence of specific position and momentum observables, it may be surprising that ubiquitous experimental setups provide a preparation of such quantum states. We proceed with a discussion of the particular character of the incompatibility of certain measurements by building on an initial heuristic argument provided by Aharonov, Pendleton and Petersen. We prove, extend and discuss a formulation of un- certainty suitable for the context of multislit experiments. We conclude with a comparison of this formulation of uncertainty with an alternative uncertainty formulation developed by Uffink and Hilgevoord. Although these two uncertainty formulations are very different technically, we demonstrate that the same tradeoff is expressed independently.
We present a treatment of the double-slit interference of matter-waves represented by Gaussian wavepackets. The interference pattern is modelled with Green's function propagator which emphasizes the coordinate correlations and phases. We explore the connection between phases and position–momentum correlations in the intensity, visibility and predictability of the wavepacket interference. This formulation will indicate some aspects that can be useful for theoretical and experimental treatment of particle, atom or molecule interferometry and can be discussed in introductory quantum mechanics courses.
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.
A quantitative measure of quantum coherence was recently introduced in the context of quantum information theory. This measure has also been propounded as a good quantifier of the wave nature of quantum objects. However, actually measuring coherence in an experiment is still considered a challenge. A procedure for measuring coherence in a multislit interference is proposed here. It can be used for experimentally testing duality relations for interference experiments involving more than two slits.