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Journal of Modern Optics

Transitivity of the relative localization of particles

Jacob Dunningham, A. V. Rau, Keith R. Burnett

Published Sep 14, 2004• 4 citations
Publisher

Abstract

We discuss the emergence of ‘classical’ relative positions between delocalized quantum particles when light is scattered off them. These welldefined separations are due to entanglements between the particles and can occur even though the absolute position of each particle remains undefined. This suggests that the natural spatial coordinates for such a system are those of relative position. We check that these relative localizations provide a consistent coordinate space by demonstrating their transitivity and we discuss how they may emerge in a multiparticle system.

DOI
10.1080/09500340408232661

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We discuss the emergence of ‘classical’ relative positions between delocalized quantum particles when light is scattered off them. These welldefined separations are due to entanglements between the particles and can occur even though the absolute position of each particle remains undefined. This suggests that the natural spatial coordinates for such a system are those of relative position. We check that these relative localizations provide a consistent coordinate space by demonstrating their transitivity and we discuss how they may emerge in a multiparticle system.
We consider the localization of a pair of particles in relative-position space. We show how a sequence of scattering interactions progressively entangles two particles, giving rise to a robust state of well-defined separation and thus providing a natural description of relative position. We use two thought experiments to describe the localization process. The first is an interferometer with recoiling mirrors. The second, and more general, case considers photons scattering from a pair of particles and the resulting emergence of a Young's interference pattern. The underlying framework of the localization process suggests a prominent role for entanglement and relative observables at the boundary between quantum and classical mechanics.
We describe how quasiclassical relative positions of particles emerge in an initially delocalized quantum system as scattering of a probe beam is observed. We show that in the multiparticle case this localization in position space occurs via intermediate states that are quantum superpositions of spatial configurations. These superpositions are robust to consecutive scattering events, and scattering alone does not lead to their decoherence. Instead the free evolution of the system combines with scattering to destroy these superpositions, leading the particles to adopt classical-like positions relative to one another.
In the classical world, the position of an object is well defined, but in quantum mechanics, increasing the precision of position measurement comes at the expense of losing information about the particle's momentum. This difference is well accepted, but it is still important to understand how larger objects that are made up on quantum-mechanical particles can develop their “classical” properties. Taking an approach that views position as a relative measurement, Rau et al. (p. 1081) present a thought experiment that shows how two particles, when subjected to a series of scattering interactions in which they become progressively more entangled, should localize in space with a well-defined separation. The proposal suggests that localization and the emergence of the classical world can be understood in terms of entanglement.
We examine three simple dynamical mechanisms wherein some relational degree of freedom becomes well localized between two initially uncorrelated systems. Specifically we explore the localization of relative phase between two optical modes, spatial interference between two independently prepared BEC.’s and the localization of relative position between two massive particles as they scatter light. Emphasis is placed on elements common to all three and a simple heuristic approach to the dynamics is adopted yielding much analytic insight. The results have application to several topics — interference experiments, the emergence of classicality, symmetry breaking and defining frames of reference in quantum mechanical terms.
This thesis presents a wide-ranging study of localising relational degrees of freedom. Three physical systems are studied in depth, each built upon a simple measurement-based process. For each physical system - light from independent sources leaking onto a beam splitter monitored by photodetectors, spatially interfering Bose-Einstein condensates extracted from separate preparation procedures, and delocalised massive particles or mirrors scattering light which is detected in the far-field - the thesis investigates the key features of the underlying process of localisation, and explores the properties of the induced post-measurement states of the system. A range of analytical and numerical methods are used. Many new results are presented - for example, cases of mixed initial states are considered in addition to the more commonly considered pure states. Up to now there has been little attempt to develop in detail the themes common to studies concerned with specific physical examples. This thesis addresses this, and sets out a "modus operandi" that can be applied widely. A variety of specific applications are considered - both in the context of controlled laboratory experiments, and with a view to understanding processes occurring in nature. The thesis is introduced and summarised in Chapter 1. Chapter 2 studies localising relative optical phase in the canonical interference process. This discussion is extended and applied in Chapter 3. Chapter 4 discusses localising relative atomic phase in interference experiments with Bose-Einstein condensates. The emergence of relative positions between particles scattering light is explored in Chapter 5. The thesis concludes with an Outlook, Chapter 6.
The concept of position and its measurement are fundamental to modern navigation technology as well as to theoretical physics. Nevertheless, a consistent notion of position has not been found in modern physics because of apparent inconsistencies between quantum mechanics and relativity. In quantum mechanics the probability distribution for a particle's position is determined by its wave function. A wave function strictly localized in space describes a particle within that locality, and nowhere else. But the quantum equations of motion predict the function will disperse, after any length of time at all, to occupy all space. Therefore the theoretical prediction is that, shortly after detection in a laboratory, a particle may be found anywhere in the universe, including regions requiring faster-than-light travel to reach. This contradicts the theory of relativity, which precludes a body traveling faster than the speed of light. The conflict arises from unexamined assumptions regarding the time of the measurement event and the time of the system under measurement. The prevailing view in physics is that time is only a parameter, having a definite value, describing dynamic changes in physical systems. This view endures even though quantum mechanics recognizes position as a quantum variable with inherent uncertainty, and relativity indicates that time should be placed on the same footing as position. But if time is recognized as a quantum variable with inherent uncertainty, the conceptual problems with position are resolved. The particle has finite probability of being in the past (or future) of the observer, while the quantum mechanical operator describing the position measurement contains a term which extrapolates from the intrinsic time of the particle to the time of the observer. Being potentially well in the past of the measurement event, the particle has time to travel to remote regions without exceeding the speed of light, and the conflict with relativity is resolved. A relativistic quantum theory can therefore be constructed incorporating a well-defined concept of position.