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arXiv:cond-mat/0601628 · Physical Review A

Entanglement properties of degenerate four-wave mixing of matter waves in a periodic potential

M. K. Olsen, Matthew J. Davis

Published Jun 21, 2006• 14 citations
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Abstract

In a recent experiment, Campbell et al. [Phys. Rev. Lett. 96, 020406 (2006)] observed degenerate four-wave mixing of matter waves in a one-dimensional optical lattice. We analyze the essential quantum features of the experiment to show that entanglement is created between the quadratures of the two scattered atomic clouds and is a true many-body (rather than two-body) phenomenon. We demonstrate that well-known bipartite entanglement inequalities are significantly violated and that this is robust to a moderate level of coherent seeding. The system is thus a promising candidate for generating spatially separated macroscopically entangled atomic samples.

DOI
10.1103/physreva.73.063618

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In a recent experiment, Campbell et al. [Phys. Rev. Lett. 96, 020406 (2006)] observed degenerate four-wave mixing of matter waves in a one-dimensional optical lattice. We analyze the essential quantum features of the experiment to show that entanglement is created between the quadratures of the two scattered atomic clouds and is a true many-body (rather than two-body) phenomenon. We demonstrate that well-known bipartite entanglement inequalities are significantly violated and that this is robust to a moderate level of coherent seeding. The system is thus a promising candidate for generating spatially separated macroscopically entangled atomic samples.
The unequivocal detection of entanglement between two distinct matter-wave pulses is a significant challenge that has yet to be experimentally demonstrated. We describe a realistic scheme to generate and detect continuous-variable entanglement between two atomic matter-wave pulses produced via degenerate four-wave mixing from an initially trapped Bose-Einstein condensate loaded into a one-dimensional optical lattice. We perform a comprehensive numerical investigation for fixed condensate parameters to determine the maximum violation of separability and Einstein-Podolsky-Rosen inequalities for field quadrature entanglement, and describe and simulate an experimental scheme for measuring the necessary quadratures.
Chemical reactions in a statistical ensemble are conventionally regarded as incoherent processes driven by thermodynamics. In the quantum degenerate regime, where atoms and molecules form coherent matter waves, reactions are theoretically described by nonlinear mixing of matter-wave fields. In this scenario, we expect phase matching between reactants and products, analogous to the mixing of photonic fields in nonlinear optics. Here we report on the observation of phase coherent reaction dynamics of Bose-condensed atoms and molecules near a Feshbach resonance. Using matter-wave diffraction with optical lattices, we verify spatial coherence of both atoms and molecules and observe phase doubling when atomic waves combine into molecular waves, the matter-wave analogue of optical frequency doubling. The diffraction patterns further reveal two-atom entanglement generated during the reaction. Our observations establish phase coherence and entanglement generation as two essential features of "quantum many-body chemistry". Moreover, our work opens a pathway to control of reaction dynamics by manipulation of matter-wave phases.
Entanglement is a fundamental resource for quantum information processing [1-3], its applications including scalable quantum communication, secure quantum key distribution protocols for cryptography, and exponential speedup of quantum algorithms. It occurs naturally in manybody systems at low temperatures: This makes cold atoms good candidates to provide entangled states involving a large number of particles. For realizing and manipulating such states, an essential task is quantifying the actual amount of entanglement contained in the created state before using it for quantum information protocols. However, this task represents a major challenge, as it requires either full state tomography, scaling exponentially in the system size, or the assumption of unverified system characteristics such as its Hamiltonian or temperature. In this experiment [4], we obtain a quantification of multipartite entanglement from readily accessible measurements - that is, time-of-flight atomic density distribution - based on a recent proposal [5]. We directly observe and quantify the multipartite spatial entanglement between the sites of a periodic optical potential (optical lattice) that hosts massive bosonic particles. We characterize its behaviour when crossing the superfluid-Mott insulator transition and when varying temperature, demonstrating the robustness of our method. References [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (Cambridge, 2000). [2]R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865-942 (2009). [3]O. Göhne and G. Tóth, Entanglement detection, Physics Reports 474, 1-75 (2009). [4]M. Cramer, A. Bernard, N. Fabbri, L. Fallani, C. Fort, S. Rosi, F. Caruso, M. Inguscio, and M. B. Plenio, Entanglement of bosons in optical lattices, E-print arXiv:1302.4897 [5]M. Cramer, M. B. Plenio, and H. Wunderlich, Measuring entanglement in condensed matter systems, Phys. Rev. Lett. 106, 020401 (2011).
We show that the dispersion properties imposed by an external periodic potential ensure both energy and quasimomentum conservation such that correlated pairs of atoms can be generated by four wave mixing from a Bose-Einstein condensate moving in an optical lattice potential. In our numerical solution of the Gross-Pitaevskii equation, a condensate with initial quasimomentum k0{k}_{0}k0​ is transferred almost completely (>95(>95%)(>95 into a pair of correlated atomic components with quasimomenta k1{k}_{1}k1​ and k2{k}_{2}k2​, if the system is seeded with a smaller number of atoms with the appropriate quasimomentum k1{k}_{1}k1​.
We investigate the possibility of an experimentally feasible cascaded four-wave mixing (FWM) system [Phys. Rev. Lett. 113, 023602 (2014)] to generate tripartite entanglement. We verify that genuine tripartite entanglement is present in this system by calculating the covariances of three output beams and then considering the violations of the inequalities of the three-mode entanglement criteria, such as two-condition criterion, single-condition criterion, optimal single-condition criterion and the positivity under partial transposition (PPT) criterion. We also consider the possibilities of the bipartite entanglement of any pair of the three output beams using the Duan-Giedke-Cirac-Zoller criterion and PPT criterion. We find that the tripartite entanglement and the bipartite entanglement for the two pairs are present in the whole gain region. The entanglement characteristics under different entanglement criteria are also considered. Our results pave the way for the realization and application of multipartite entanglement based on the cascaded FWM processes.
We study atom scattering from two colliding Bose-Einstein condensates using a position sensitive, time resolved, single atom detector. In analogy to quantum optics, the process can also be thought of as spontaneous, degenerate four-wave mixing of de Broglie waves. We find a clear correlation between atoms with opposite momenta, demonstrating pair production in the scattering process. We also observe a Hanbury Brown-Twiss correlation for collinear momenta, which permits an independent measurement of the size of the pair production source and thus the size of the spatial mode. The back-to-back pairs occupy very nearly two oppositely directed spatial modes, a promising feature for future quantum optics experiments.