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arXiv:quant-ph/0607069

Spatial Entanglement of a Free Bosonic Field

Libby Heaney, Janet Anders, Vlatko Vedral

Published Jul 11, 2006• 7 citations
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Abstract

In this paper we discuss the entanglement properties of a thermal non-relativistic free bosonic field. We demonstrate how to formally construct spatial modes in order to use a continuous variable separability criterion and show that the construction of the modes plays a significant role in the way the entanglement manifests itself. For instance the presence of entanglement depends on the shape of the modes and their separation. The temperature of the gas is another crucial factor and for one choice of modes we show that entanglement can be found at arbitrarily high temperatures as long as we can zoom in on suitably small regions in space. Moreover, we show that the entanglement here is useful as it can be extracted to a pair of localised systems.

DOI
10.48550/arxiv.quant-ph/0607069

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In this paper we discuss the entanglement properties of a thermal non-relativistic free bosonic field. We demonstrate how to formally construct spatial modes in order to use a continuous variable separability criterion and show that the construction of the modes plays a significant role in the way the entanglement manifests itself. For instance the presence of entanglement depends on the shape of the modes and their separation. The temperature of the gas is another crucial factor and for one choice of modes we show that entanglement can be found at arbitrarily high temperatures as long as we can zoom in on suitably small regions in space. Moreover, we show that the entanglement here is useful as it can be extracted to a pair of localised systems.
We present a general argument showing that the temperature as well as other thermodynamical state variables can qualify as entanglement witnesses (EWs) for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple free non-interacting bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We then discuss the relationship between the occurrence of Bose–Einstein condensation and our conditions for the presence of entanglement and compare the respective critical temperatures. We close with a short discussion of the idea of seeing entanglement as a macroscopic property in thermodynamical systems and its possible relation to phase transitions in general.
Abstract. We present a general argument showing that the temperature as well as other thermodynamical state variables qualify as entanglement witnesses for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple non-interacting Bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We close with a short discussion of the relationship between the occurrence of Bose-Einstein condensation and our conditions for the presence of entanglement. PACS numbers: Submitted to: New Journal of Physics Keywords: macroscopic entanglement, entanglement witnesses, bosonic gas, BEC, continuous variable systemsDetecting entanglement with a thermometer 2 1.
We outline an interferometric scheme for the detection of bimode and multimode spatial entanglement of finite-temperature interacting Bose gases. Whether entanglement is present in the gas depends on the existence of the single-particle reduced density matrix between different regions of space. We apply the scheme to the problem of a harmonically trapped repulsive boson pair and show that while entanglement is rapidly decreasing with temperature, a significant amount remains for all interaction strengths at zero temperature. Thus, by tuning the interaction parameter, the distribution of entanglement between many spatial modes can be modified.
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We investigate spatial entanglement---particle-number entanglement between regions of space---in an ideal bosonic gas. We quantify the amount of spatial entanglement around the transition temperature for condensation (TC)({T}_{C})(TC​) by probing the gas with two localized two-level systems. We show that spatial entanglement in the gas is directly related to filling of the ground-state energy level and therefore to the off-diagonal long-range order of the system and the onset of condensation.