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European Journal of Physics

Free evolution of an incoherent mixture of states: a quantum mechanical approach to the van Cittert–Zernike theorem

Ignacio Fabre, Francisco Navarrete, László Sarkadi, R O Barrachina

Published Sep 22, 2017• 12 citations
Publisher

Abstract

Abstract We study the time evolution of an incoherent mixture of quantum states and demonstrate, in very simple terms, a quantum mechanical equivalent of van Cittert–Zernike theorem, which can be easily explained to Quantum Physics students with a basic knowledge of the density matrix theory. Finally, we exemplify this result by applying it to the quantitative analysis of the coherence of a beam of particles in atomic collisions.

DOI
10.1088/1361-6404/aa8e74

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Abstract We study the time evolution of an incoherent mixture of quantum states and demonstrate, in very simple terms, a quantum mechanical equivalent of van Cittert–Zernike theorem, which can be easily explained to Quantum Physics students with a basic knowledge of the density matrix theory. Finally, we exemplify this result by applying it to the quantitative analysis of the coherence of a beam of particles in atomic collisions.
Abstract This article continues and completes a recent one regarding a quantum version of van Cittert–Zernike’s theorem (Fabre et al 2018 Eur. J. Phys. 39 015401). In that paper, we considered a mixture of quantum states that only differed in their initial positions, but not on their momenta. Here, we relax this nonphysical assumption and analyse how the inclusion of both classical uncertainties in the initial position and momentum of a mixture of—otherwise identical—free quantum states which do not interact with an environment, affects its quantum coherence length. We show that the mathematical development and the final result are simple enough to make possible the incorporation of this subject into a basic course of quantum physics. Furthermore, we demonstrate that these results provide a comprehensive solution to a recent controversy in atomic collision physics.
Recent progress on quantum state engineering has enabled the preparation of quantum photonic systems comprising multiple interacting particles. Interestingly, multiphoton quantum systems can host many complex forms of interference and scattering processes that are essential to perform operations that are intractable on classical systems. Unfortunately, the quantum coherence properties of multiphoton systems degrade upon propagation leading to undesired quantum-to-classical transitions. Furthermore, the manipulation of multiphoton quantum systems requires of nonlinear interactions at the few-photon level. Here, we introduce the quantum van Cittert-Zernike theorem to describe the scattering and interference effects of propagating multiphoton systems. This fundamental theorem demonstrates that the quantum statistical fluctuations, which define the nature of diverse light sources, can be modified upon propagation in the absence of light-matter interactions. The generality of our formalism unveils the conditions under which the evolution of multiphoton systems can lead to surprising classical-to-quantum transitions. Specifically, we show that the implementation of conditional measurements may enable the all-optical preparation of multiphoton systems with attenuated quantum statistics below the shot-noise limit. Remarkably, this effect had not been discussed before and cannot be explained through the classical theory of optical coherence. As such, our work opens new paradigms within the established field of quantum coherence.
With a choice of boundary conditions for solutions of the Schrödinger equation, state vectors and density operators even for closed systems evolve asymmetrically in time. For open systems, standard quantum mechanics consequently predicts irreversibility and signatures of the extrinsic arrow of time. The result is a new framework for the treatment of decoherence, not based on a reduced dynamics or a master equation. As an application, using a general model we quantitatively match previously puzzling experimental results and can conclude that they are the measurable consequence of the indistinguishability of separate, uncontrolled interactions between systems and their environment.
A reciprocal relationship between the autocovariance of the light intensity in the source plane and in the far-field detector plane is presented in a form analogous to the classical van Cittert-Zernike theorem, but involving intensity correlation functions. A "classical" version of the reciprocity relationship is considered first, based on the assumption of circular Gaussian statistics of the complex amplitudes in the source plane. The result is consistent with the theory of Hanbury Brown-Twiss interferometry, but it is shown to be also applicable to estimation of the source size or the spatial resolution of the detector from the noise power spectrum of flat-field images. An alternative version of the van Cittert-Zernike theorem for intensity correlations is then derived for a quantized electromagnetic beam in a coherent state, which leads to Poisson statistics for the intrinsic intensity of the beam.