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arXiv:2011.11076 · Physical Review A

Spatial entanglement in interacting Bose-Einstein condensates

Natalia Sánchez-Kuntz, Stefan Floerchinger

Published Apr 21, 2021• 7 citations
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Abstract

The entanglement between spatial regions in an interacting Bose-Einstein condensate is investigated using a quantum field theoretic formalism. Regions that are small compared to the healing length are governed by a nonrelativistic quantum field theory in the vacuum limit, and we show that the latter has vanishing entanglement. In the opposite limit of a region that is large compared to the healing length, the entanglement entropy is like in the vacuum of a relativistic theory where the velocity of light is replaced with the velocity of sound and where the inverse healing length provides a natural ultraviolet regularization scale. Besides the von Neumann entanglement entropy, we also calculate R\'enyi entanglement entropies for a one-dimensional quasicondensate.

DOI
10.1103/physreva.103.043327

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The entanglement between spatial regions in an interacting Bose-Einstein condensate is investigated using a quantum field theoretic formalism. Regions that are small compared to the healing length are governed by a nonrelativistic quantum field theory in the vacuum limit, and we show that the latter has vanishing entanglement. In the opposite limit of a region that is large compared to the healing length, the entanglement entropy is like in the vacuum of a relativistic theory where the velocity of light is replaced with the velocity of sound and where the inverse healing length provides a natural ultraviolet regularization scale. Besides the von Neumann entanglement entropy, we also calculate R'enyi entanglement entropies for a one-dimensional quasicondensate.
Consider an arbitrary local quantum field theory with a gap or an arbitrary gapless free theory. We consider states in such a theory, that describe two entangled particles localized in disjoint regions of space. We show that in such a state, to leading order, Rényi entropies of spatial regions, containing only one of the particles are same as their quantum mechanical counterparts, after subtraction of vacuum contribution. Subleading corrections depend on overlap of wave functions. These results suggest that Von Neumann entropy of a spatial region, after subtraction of vacuum contribution, can serve as a measure of entanglement of indistinguishable particles in pure states.
Entanglement is defined between subsystems of a quantum system, and at fixed time two regions of space can be viewed as two subsystems of a relativistic quantum field. The entropy of entanglement between such subsystems is ill-defined unless an ultraviolet cutoff is introduced, but it still diverges in the continuum limit. This behavior is generic for arbitrary finite-energy states, hence a conceptual tension with the finite entanglement entropy typical of nonrelativistic quantum systems. We introduce a novel approach to explain the transition from infinite to finite entanglement, based on coarse graining the spatial resolution of the detectors measuring the field state. We show that states with a finite number of particles become localized, allowing an identification between a region of space and the nonrelativistic degrees of freedom of the particles therein contained, and that the renormalized entropy of finite-energy states reduces to the entanglement entropy of nonrelativistic quantum mechanics.
Quasi-particle excitation in a Bose-Einstein condensate leads to quantum entanglement between real bosonic atoms in the system. By using spectral expansion method, the eigenvalues and eigenstates of Bogoliubov-de Gennes equation are numerically calculated in a quasi-one-dimensional infinite square well potential. For the low-energy collective excitations of the quasi-particles, we explore the dependence of quantum entanglement entropy of the Bose-Einstein condensate on scattering length. Our results show that the entanglement entropy increases slowly with the increase of the scattering length, and such an increasing trend can be well described by a power function. These results are analogous to those in a one-dimensional uniform BEC, where the entanglement entropy of the Bogoliubov ground state is approximately proportional to the square root of the scattering length. This work provides a viable way for investigating many-particle entanglement in a quasi-one-dimensional trapped Bose-Einstein condensate where the quantum entanglement is closely related to the interaction strength between particles.
We compute the entanglement entropy associated to the Hawking emission of a (1+11+11+1)-dimensional acoustic black hole in a Bose-Einstein condensate. We use the brick wall model proposed by 't Hooft, adapted to the momentum space, in order to tackle the case when high frequency dispersion is taken in account. As expected, we find that in the hydrodynamic limit the entropy only depends on the size of the box in the near-horizon region, as for gravitational (1+11+11+1)-dimensional black holes. When dispersion effects are considered, we find a correction that depends on the square of the size of the near-horizon region measured in units of healing length, very similar to the universal correction to the entropy found in the case of spin-1/21/21/2 Heisenberg XX chains.
We study the entanglement entropy and spectrum between components in binary Bose-Einstein condensates in ddd spatial dimensions. We employ effective field theory to show that the entanglement spectrum exhibits an anomalous square-root dispersion relation in the presence of intercomponent tunneling (Rabi coupling) and a gapped dispersion relation in its absence. These spectral features are associated with the emergence of long-range interactions in terms of the superfluid velocity and the particle density in the entanglement Hamiltonian. Our results demonstrate that unusual long-range interactions can be emulated in a subsystem of multicomponent BECs that have only short-range interactions. We also find that for a finite Rabi coupling the entanglement entropy exhibits a volume-law scaling with subleading logarithmic corrections originating from the Nambu-Goldstone mode and the symmetry restoration for a finite volume.
We study the entanglement entropy in a relativistic quantum field theory for regions which are not included in a single spatial hyperplane. This geometric configuration cannot be treated with the Euclidean time method and the replica trick. Instead, we use a real time method to calculate the entropy for a massive free Dirac field in two dimensions in some approximations. We find some specifically relativistic features of the entropy. First, there is a large enhancement of entanglement due to boosts. As a result, the mutual information between relatively boosted regions does not vanish in the limit of zero volume and large relative boost. We also find extensivity of the information in a deeply Lorentzian regime with large violations of the triangle inequalities for the distances. This last effect is relevant to an interpretation of the amount of entropy enclosed in the Hawking radiation emitted by a black hole.