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arXiv:1503.03999 · Journal of Physics B Atomic Molecular and Optical Physics

Entanglement-like properties in spin–orbit coupled ultra cold atom and violation of Bell-like inequality

Rahul Kumar, Sankalpa Ghosh

Published Jul 5, 2018• 3 citations
arXivHTMLPDF

Abstract

Abstract We show that the general quantum state of synthetically spin–orbit coupled ultra cold bosonic atoms, whose condensate was experimentally created recently by Lin et al (2011 Nature 471 , 83), shows entanglement between motional degrees of freedom (momentum) and internal degrees of freedom (hyperfine spin). We demonstrate the violation of Bell-like inequality (CHSH) for such states that provide a unique opportunity to verify fundamental principles like quantum contextuality for commutating observables which are not spatially separated. We analyze in detail the Rabi oscillation executed by such an atom-laser system and how this influences quantities like entanglement entropy, the violation of Bell-like inequality, etc. We also discuss the implications of our result in testing quantum contextuality and Bell’s Inequality violation by macroscopic quantum object like Bose–Einstein condensate of ultra cold atoms.

DOI
10.1088/1361-6455/aad149

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Abstract We show that the general quantum state of synthetically spin–orbit coupled ultra cold bosonic atoms, whose condensate was experimentally created recently by Lin et al (2011 Nature 471 , 83), shows entanglement between motional degrees of freedom (momentum) and internal degrees of freedom (hyperfine spin). We demonstrate the violation of Bell-like inequality (CHSH) for such states that provide a unique opportunity to verify fundamental principles like quantum contextuality for commutating observables which are not spatially separated. We analyze in detail the Rabi oscillation executed by such an atom-laser system and how this influences quantities like entanglement entropy, the violation of Bell-like inequality, etc. We also discuss the implications of our result in testing quantum contextuality and Bell’s Inequality violation by macroscopic quantum object like Bose–Einstein condensate of ultra cold atoms.
Bell's theorem revealed the fundamental incompatibility between the predictions of quantum mechanics and local realism. Bell inequality violations have since demonstrated quantum nonlocality using photons and internal states of massive particles, but never using their motional states. Here we report the first Bell inequality violation in the motional states of massive particles. Using momentum-entangled pairs of metastable helium atoms manipulated by matter-wave interferometry, we measure a Clauser-Horne-Shimony-Holt (CHSH) Bell parameter of S=2.52±0.17S = 2.52 \pm 0.17S=2.52±0.17, violating the CHSH-Bell inequality (S≤2S \le 2S≤2). Our work completes a long-standing objective in quantum atom optics by extending Bell tests from internal quantum variables to the external degrees of freedom of massive particles, opening a new regime for exploring quantum nonlocality in matter waves and for investigating the interplay between quantum mechanics and gravity.
In cold atom, the coupling between "spin" (atomic internal hyperfine states) and "orbit" (atomic center-of-mass motion) can be induced by Raman transition, where different hyperfine states are coupled by a pair of Raman lasers. In recent years, this synthetic spin-orbit coupling has received tremendous attention, as it leads to a variety of novel quantum phenomena in precisely controllable cold atom systems. In this thesis, we first present a comprehensive analysis of one-, two- and many- particle physics of harmonically trapped atoms with spin-orbit coupling, followed by the study of "novel spin-orbit coupling" in two different systems: (1) cold atom spinor mixtures and (2) cold atoms in an optical cavity. In the first system, we consider a spinor mixture consisting of two species of cold atoms, where the spin-orbit coupling can be transmitted from one species to the other, and we discuss novel topological properties and the supersolid stripe phase in this mixture. In the second system, we consider the coupling among three parts: the cavity photon field, the atomic internal hyperfine states, and the atomic external center-of-mass motion, and we discuss how this coupling affects familiar quantum optical phenomena, such as Rabi oscillation and Dicke superradiance phase transition. Our novel systems contribute new and practical platforms for the research field of synthetic spin-orbit coupling in cold atoms.
We consider a many-body system of pseudo-spin-1∕21∕21∕2 bosons with spin-orbit interactions, which couple the momentum and the internal pseudo-spin degree of freedom created by spatially varying laser fields. The corresponding single-particle spectrum is generally anisotropic and contains two degenerate minima at finite momenta. At low temperatures, the many-body system condenses into these minima generating a type of entangled Bose-Einstein condensate. We show that in the presence of weak density-density interactions the many-body ground state is characterized by a twofold degeneracy. The corresponding many-body wave function describes a condensate of ``left-'' and ``right-moving'' bosons. By fine-tuning the parameters of the laser field, one can obtain a bosonic version of the spin-orbit coupled Rashba model. In this symmetric case, the degeneracy of the ground state is very large, which may lead to phases with nontrivial topological properties. We argue that the predicted type of Bose-Einstein condensates can be observed experimentally via time-of-flight imaging, which will show characteristic multipeak structures in momentum distribution.
We propose and theoretically simulate an experiment for demonstrating a motional-state Bell inequality violation for pairs of momentum-entangled atoms produced in Bose-Einstein condensate collisions. The proposal is based on realizing an atom-optics analog of the Rarity-Tapster optical scheme: it uses laser-induced Bragg pulses to implement two-particle interferometry on the underlying Bell state for two pairs of atomic scattering modes with equal but opposite momenta. The collision dynamics and the sequence of Bragg pulses are simulated using the stochastic Bogoliubov approach in the positive-PPP representation. We predict values of the Clauser-Horne-Shimony-Holt (CHSH) parameter up to $S\ensuremath{\simeq}2.5$ for experimentally realistic parameter regimes, showing a strong violation of the CSHS-Bell inequality bounded classically by $S\ensuremath{\le}2$.
Entanglement is at the core of quantum information processing and may prove essential for quantum speed-up. Inspired by both theoretical and experimental studies of spin-momentum coupling in systems of ultra-cold atoms, we investigate the entanglement between the spin and momentum degrees of freedom of an optically trapped BEC of 87Rb atoms. We consider entanglement that arises due to the coupling of these degrees of freedom induced by Raman and radio-frequency fields and examine its dependence on the coupling parameters by evaluating von Neumann entropy as well as concurrence as measures of the entanglement attained. Our calculations reveal that under proper experimental conditions significant spin-momentum entanglement can be obtained, with von Neumann entropy of 80% of the maximum attainable value. Our analysis sheds some light on the prospects of using BECs for quantum information applications.
In analogy with Bell’s inequality for two-qubit quantum states we propose an inequality criterion for the non-separability of the spin-orbit degrees of freedom of a laser beam. A definition of separable and non-separable spin-orbit modes is used in consonance with the one presented in Phys. Rev. Lett. 99, 160401 (2007). As the usual Bell’s inequality can be violated for entangled two-qubit quantum states, we show both theoretically and experimentally that the proposed spin-orbit inequality criterion can be violated for non-separable modes. The inequality is discussed both in the classical and quantum domains.