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arXiv:quant-ph/0411167 · Physical Review A

Measurement-induced localization of relative degrees of freedom

Hugo V. Cable, P. L. Knight, Terry Rudolph

Published Apr 22, 2005• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.71.042107

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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.
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 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.
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.
Additional variables (also often called ``hidden variables'') are sometimes added to standard quantum mechanics in order to remove its indeterminism or ``incompleteness'' and to make the measurement process look more classical. Here we discuss a case in which an additional variable arises almost spontaneously from the quantum formalism: the emergence of a relative phase between two highly populated Fock-state Bose-Einstein condensates. The model simulated here involves the interference of two Bose condensates, one with all up spins and the other with down spins, along a zzz axis. With the clouds overlapping, we consider the results of measuring spins in a transverse plane (the general direction is studied in an appendix). The determination of the previously ``hidden'' phase becomes progressively more definite as additional measurements are made. We also provide an analysis of a recent and closely related experiment.
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
Quantum measurement can profoundly impact the dynamics of many-body systems, for instance, inducing phase transitions from volume-law chaotic phases to area-law localized phases. While chaotic phases emerge naturally in generic many-body systems, the experimental realization of a fully localized phase driven by measurement remains challenging, primarily owing to the stringent requirement of near-perfect isolation of the system from environmental dissipation. Here we experimentally observe measurement-induced localization phenomena in an interacting 13C^{13}\mathrm{C}13C nuclear spin system in diamond. In contrast to previous circuit-model-based experiments, our well-isolated system evolves under a physical many-body Hamiltonian with measurements intervening at intervals. By implementing either projective measurements on a single spin or global weak measurements on the entire system, we observe substantially prolonged spin autocorrelation times as the measurement frequency increases. Our results, combined with numerical simulations, provide evidence that quantum measurement can drive a many-body system into a localized regime, and lay the groundwork for future experimental investigations into measurement-induced many-body dynamics and related entanglement phenomena.