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arXiv:quant-ph/0702067 · Physical Review A

Spatial entanglement from off-diagonal long-range order in a Bose-Einstein condensate

Libby Heaney, Janet Anders, Dagomir Kaszlikowski, Vlatko Vedral

Published Nov 7, 2007• 29 citations
arXivHTMLPDF

Abstract

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 $({T}_{C})$ 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.

DOI
10.1103/physreva.76.053605

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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.
We determine the behaviour of entanglement between regions of space in a Bose gas of fixed particle number around the critical temperature condensation. Long-range correlations develop in the Bose-Einstein condensate (BEC) phase transition and the aim here is to find out whether spatial coherence alone implies entanglement. We use a purity measure of entanglement to derive an entanglement witness that detects entanglement between two regions of space in the BEC. It is shown that spatial coherence between the two regions is necessary for entanglement with coherence and entanglement becoming equivalent only when the regions occupy the entire confining volume of the gas. The probabilities for Bosons to occupy the regions is the only other parameter that influences the amount of entanglement. We calculate explicitly the amount of entanglement between two regions for a cigar-shaped harmonic potential and find that it increases with decreasing temperature. A second entanglement witness is derived for entanglement between three regions of space, where again spatial coherence between each pair of regions is precursor to entanglement. It is shown that the state with the maximum amount of entanglement occurs when the probabilities for the Bosons to occupy the three regions are equal.
Based on the paper "Fourier formula for quantum partition functions", arXiv:2106.10032 [math-ph], we show that in an infinite system of identical bosons interacting via a positive-type pair potential there is off-diagonal long-range order if and only if a nonzero fraction of the particles form infinite permutation cycles. In particular, there is Bose-Einstein condensation if and only if the diverging cycle lengths increase at least as fast with NNN, the number of particles, as N2/dN^{2/d}N2/d in d≥3d\geq 3d≥3 dimensions. This extends a similar result known for the ideal Bose gas.
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.
We introduce an entanglement witness that identifies off-diagonal long-range order (ODLRO) -- a distinctive form of entanglement -- in systems containing both fermionic and bosonic particles. By analyzing the particle-hole reduced density matrices of each subsystem, the approach detects ODLRO independently in both fermionic and bosonic sectors and identifies when long-range order develops across the entire mixed-particle system. The witness also quantifies the magnitude of ODLRO within each particle type, revealing how fermionic and bosonic correlations combine to form the total entanglement of the system, including a bosonic condensation of particle-hole pairs driven by many-body correlations rather than particle statistics. Using the Lipkin-Meshkov-Glick spin model, we show how the transition from ODLRO localized to one particle type to ODLRO shared by both particle types captures the onset of collective entanglement in a mixed-particle environment, providing new insight into systems where fermionic and bosonic correlations coexist.
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 have experimentally investigated the formation of off-diagonal long-range order in a gas of ultracold atoms. A magnetically trapped atomic cloud prepared in a highly nonequilibrium state thermalizes and thereby crosses the Bose-Einstein condensation phase transition. The evolution of phase coherence between different regions of the sample is constantly monitored and information on the spatial first-order correlation function is obtained. We observe the growth of the spatial coherence and the formation of long-range order in real time and compare it to the growth of the atomic density. Moreover, we study the evolution of the momentum distribution during the nonequilibrium formation of the condensate.