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AIP conference proceedings

Measurement Induced Localization of Relative Parameters

Hugo V. Cable

Published Jan 1, 2004• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.1834392

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This Week in Science

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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.
We present a comprehensive study, using both analytical and numerical methods, of measurement-induced localization of relational degrees of freedom. Looking first at the interference of two optical modes, we find that the localization of the relative phase can be as good for mixed states---in particular, for two initially Poissonian or thermal states---as for the well-known case of two Fock states. In a realistic setup the localization for mixed states is robust and experimentally accessible, and we discuss applications to superselection rules. For an ideal setup we show how a relational Schr\"odinger cat state emerges and investigate circumstances under which such a state is destroyed. In our second example we consider the localization of relative atomic phase between two Bose Einstein condensates, looking particularly at the build up of spatial interference patterns, an area which has attracted much attention since the work of Javanainen and Yoo. We show that the relative phase localizes much faster than was intimated in previous studies focusing on the emerging interference pattern itself. Finally, we explore the localization of relative spatial parameters discussed in recent work by Rau, Dunningham, and Burnett. We retain their models of indistinguishable scattering but make different assumptions. In particular we consider the case of a real distant observer monitoring light scattering off two particles, who records events only from a narrow field of view. The localization is only partial regardless of the number of observations. This paper contributes to the wider debate on relationism in quantum mechanics, which treats fundamental concepts---reference frames and conservation laws---from a fully quantum and operational perspective.
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.
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.
One interpretation of how the classical world emerges from an underlying quantum reality involves the build-up of certain robust entanglements between particles due to scattering events [Science Vol.301 p.1081]. This is an appealing view because it unifies two apparently disparate theories. It says that the uniquely quantum effect of entanglement is associated with classical behaviour. This is distinct from other interpretations that says classicality arises when quantum correlations are lost or neglected in measurements. To date the weakness of this interpretation has been the lack of a clear experimental signature that allows it to be tested. Here we provide a simple experimentally accessible scheme that enables just that. We also discuss a Bayesian technique that could, in principle, allow experiments to confirm the theory to any desired degree of accuracy and we present precision requirements that are achievable with current experiments. Finally, we extend the scheme from its initial one dimensional proof of principle to the more real world scenario of three dimensional localisation.
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.