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Journal of the Optical Society of America B

Continuous variable tripartite entanglement and steering using a third-order nonlinear optical interaction

Tarek Peña-Armendáriz, Roberto Ramírez-Alarcón, L. Rosales-Zárate

Published Dec 1, 2020• 9 citations
Publisher

Abstract

Nonlinear optical devices are very useful for generating entanglement. One proposal for the generation of genuine tripartite entanglement [ Phys. Rev. Lett. 120 , 043601 ( 2018 ) PRLTAO 0031-9007 10.1103/PhysRevLett.120.043601 ] is to use a third-order nonlinear optical interaction. Here, we investigate a generalization of this, where we consider the quantum nature of all modes; moreover, we also investigate quantum steering, which is a stronger and asymmetric non-local correlation. We show that the final state presents not only tripartite continuous variable entanglement but also tripartite steering.

DOI
10.1364/josab.403813

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Nonlinear optical devices are very useful for generating entanglement. One proposal for the generation of genuine tripartite entanglement [ Phys. Rev. Lett. 120 , 043601 ( 2018 ) PRLTAO 0031-9007 10.1103/PhysRevLett.120.043601 ] is to use a third-order nonlinear optical interaction. Here, we investigate a generalization of this, where we consider the quantum nature of all modes; moreover, we also investigate quantum steering, which is a stronger and asymmetric non-local correlation. We show that the final state presents not only tripartite continuous variable entanglement but also tripartite steering.
Utilizing a tritter with variable parameter TTT and induced by vacuum--one-photon superpositions $|0\ensuremath{\rangle}+\ensuremath{\alpha}|1\ensuremath{\rangle}$ with $\ensuremath{\alpha}=|\ensuremath{\alpha}|{e}^{i\ensuremath{\phi}}$, we propose a scheme to prepare a class of three-qubit pure states. These states take the form of ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}={c}_{0}|000\ensuremath{\rangle}+{c}_{1}|100\ensuremath{\rangle}+{c}_{2}|010\ensuremath{\rangle}+{c}_{3}|001\ensuremath{\rangle}$. The coefficients (c0,({c}_{0},(c0​, c1,{c}_{1},c1​, c2,andc3){c}_{2}, \text{and} {c}_{3})c2​,andc3​) can be manipulated through interaction parameters $(|\ensuremath{\alpha}|,$ $\ensuremath{\phi}, \text{and} T)$. In line with Xie and Eberly's work [Phys. Rev. Lett. 127, 040403 (2021)], we investigate the genuine tripartite entanglement for ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}$ by using the measure of concurrence fill. Drawing on Hao et al.'s research [Phys. Rev. Lett. 128, 120402 (2022)], we examine tripartite steering for ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}$ under certain measurements based on the uncertainty relations criterion. We identify nine potential configurations exhibiting varying steerability across different parameter spaces. It is important to highlight that, while the state ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}$ exhibits entanglement, steering remains unattainable in a substantial portion of the parameter space.
Entanglement is a property of composite physical systems distinct from any classical description and therefore has fundamental significance in quantum theory. Although a great deal of progress has been made in understanding entanglement, there is still much unknown about this phenomenon and interest in the nature of entanglement alone continues to motivate research in this area. Moreover, a better understanding of entanglement inevitably leads to a deeper knowledge of physical processes in quantum systems. This thesis explores continuous-variable entanglement in many-body nonlinear bosonic systems with the overall aim of broadening our knowledge of entanglement, and in turn, the quantum optical systems under consideration. The secondary aim of this thesis is to further the development of entanglement as a resource for use in quantum information, communication or computing. After introducing some of the terminology and basics of entanglement theory (Chapter 1), we then provide a brief review of the main theoretical tools employed throughout this thesis (Chapter 2). We begin our investigations of entanglement by considering an evanescently coupled pair of nonlinear crystals contained in a pumped optical cavity (Chapter 3). This two-mode system is known as the intracavity Kerr nonlinear coupler and it is the output beams emerging from the cavity that provide a source of continuous-variable entangled states. Within this framework, we examine the concept of steering, a term introduced by Schr{\o}dinger to generalize the effect described by the Einstein-Podolsky-Rosen paradox. In particular, we investigate bipartite asymmetric steering. This is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories, depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. We predict that asymmetric steering should arise in experiments using the intracavity nonlinear coupler, at least in the case where Alice and Bob can only make Gaussian measurements. We continue our investigations of nonlinear quantum optical systems and entanglement by studying an experimentally feasible intra-cavity coupled down-conversion scheme that makes use of concurrent nonlinearities (Chapter 4). From a theoretical perspective, we demonstrate that this scheme gives rise to a set of continuous-variable multipartite entangled output beams. Specifically, we verify that genuine quadripartite entanglement is present in this system by calculating the output fluctuation spectra and then considering violations of multipartite entanglement inequalities. The entanglement characteristics both above and below the oscillation threshold of the cavity are considered. Finally, we build on these studies of quadripartite entanglement and examine a modified intra-cavity concurrent down-conversion scheme (Chapter 5). We analyze the feasibility and potential of this scheme for generating continuous-variable cluster states. Cluster states have been proposed as a resource state for continuous-variable one-way quantum computing. We demonstrate genuine quadripartite entanglement and investigate the degree of entanglement present. We find that above the oscillation threshold, the basic cluster state geometry under consideration suffers from phase diffusion. We alleviate this problem by incorporating a small injected signal into our analysis. We also investigate squeezed joint operators, which must approach zero in the limit of large squeezing if the cluster state defining relation is to be satisfied. While the squeezed joint operators approach zero in the undepleted regime, we find that this is not the case when we consider the full interaction Hamiltonian and the presence of a cavity. In fact, we find that the decay of these operators is minimal in a cavity and even depletion alone inhibits cluster state formation.
It is theoretically shown that tripartite entanglement with different wavelengths can be generated by cascaded nonlinear interaction in an optical parametric oscillator cavity with parametric down conversion and sum-frequency generation. A sufficient inseparability criterion for continuous-variable tripartite entanglement proposed by van Loock and Furusawa was used to evaluate the degree of the quadrature-phase amplitude correlations between the three modes. The dependences of correlation on the cavity parameters and pump intensity are discussed.
The generation and manipulation of multipartite entanglement and EPR steering in macroscopic systems not only play a fundamental role in exploring the nature of quantum mechanics, but are also at the core of current developments of various nascent quantum technologies. Here we report a theoretical method using squeezing-phase-controlled quantum noise flows to selectively generate and manipulate quantum entanglement and asymmetric EPR steering in a nonlinear χ(2)χ^{(2)}χ(2) whispering-gallery-mode (WGM) optomechanical resonator. We show that by pumping the χ(2)χ^{(2)}χ(2) nonlinear medium with two-photon optical fields and broadband squeezed lights, a pair of counterpropagating squeezed optical modes could be introduced to the WGM resonator, each coupled with an independent squeezed vacuum reservoir. This configuration could enable squeezing-phase-controlled light-reservoir interaction for each squeezed optical mode, providing a flexible tool for tailoring asymmetric optical noise flows in the counterpropagating modes. Based on this unique feature, it is found that with the injection of asymmetric noise flows, the generation of various types of bipartite and tripartite entanglement become phase-dependent and thus they can be produced in an asymmetric way. More excitingly, it is also found that by further properly adjusting the squeezing parameters, the overall asymmetry of EPR steering can also be stepwise driven from no-way regime, one-way regime to two-way regime. These findings, holding promise for preparing rich types of entangled quantum resources with asymmetric features, may have potential applications in the area of secure quantum information processing such as quantum secure direct communication and one-way quantum computing.
We investigate the quantum entanglement of the three modes associated with the three-photon states obtained by triple-photon generation in a phase-matched third-order nonlinear optical interaction. Although the second-order processes have been extensively dealt with, there is no direct analogy between the second and third-order mechanisms. We show, for example, the absence of quantum entanglement between the quadratures of the three modes in the case of spontaneous parametric triple-photon generation. However, we show robust, seeding-dependent, genuine triple-photon entanglement in the fully seeded case.
Asymmetric quantum steering can achieve one-way control, which is different from multipartite quantum steering and can accomplish some work that quantum entanglement cannot complete. The cascaded nonlinear process of spontaneous parametric down-conversion cascaded with double sum-frequency generations can be realized in one optical superlattice by quasi-phase-matching technique. Here the asymmetric quantum steering is investigated in this cascaded nonlinear process and the asymmetry of quantum steering is confirmed in a wide range of the parameters both below and without the oscillation threshold, respectively. The generation of asymmetric quantum steering in present scheme has potential applications in quantum key distribution and continuous variable teleportation.