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arXiv:quant-ph/0110114 · Physical Review A

Natural entanglement in Bose-Einstein condensates

Christoph Simon

Published Nov 25, 2002• 63 citations
arXivHTMLPDF

Abstract

It is shown that every zero-temperature Bose-Einstein condensate is a highly entangled state, as a consequence of the fact that the particles in a condensate are distributed over space in a coherent way. In particular, any two regions within a condensate of finite particle number are entangled. This entanglement does not depend on the distance between the two regions. Criteria for the presence of entanglement are derived in the context of interference experiments. For separable states there is a trade-off between fluctuations in particle number and interference visibility.

DOI
10.1103/physreva.66.052323

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In this paper we study the entanglement properties of free nonrelativistic Bose gases. At zero temperature, we calculate the bipartite block entanglement entropy of the system and find that it diverges logarithmically with the particle number in the subsystem. For finite temperatures, we study the mutual information between the two blocks. We first analytically study an infinite-range hopping model, then numerically study a set of long-range hopping models in one dimension that exhibit Bose-Einstein condensation. In both cases we find that a Bose-Einstein condensate, if present, makes a divergent contribution to the mutual information which is proportional to the logarithm of the number of particles in the condensate in the subsystem. The prefactor of the logarithmic divergent term is model dependent.
We propose three criteria for identifying continuous variable entanglement between two many-particle systems with no restrictions on the quantum state of the local oscillators used in the measurements. Mistakenly asserting a coherent state for the local oscillator can lead to incorrectly identifying the presence of entanglement. We demonstrate this in simulations with 100 particles and also find that large number fluctuations do not prevent the observation of entanglement. Our results are important for quantum information experiments with realistic Bose-Einstein condensates or in optics with arbitrary photon states.
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A numerical experiment based on a particle number-conserving quantum field theory is performed for two initially independent Bose–Einstein condensates that are coherently coupled at two temperatures. The present model illustrates ab initio that the initial phase of each of the two condensates does not remain random at the Boltzmann equilibrium, but is distributed around integer multiple values of 2π from the interference and thermalization of forward and backward propagating matter waves. The thermalization inside the atomic vapors can be understood as an intrinsic measurement process that defines a temperature for the two condensates and projects the quantum states to an average wave field with zero (relative) phases. Following this approach, focus is put on the original thought experiment of Anderson on whether a Josephson current between two initially separated Bose–Einstein condensates occurs in a deterministic way or not, depending on the initial phase distribution. Quanta 2024; 13: 28–37.