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Photonics Letters of Poland

Quantum steering in an asymmetric chain of nonlinear oscillators

Joanna K. Kalaga, W. Leoński, Radosław Szczȩśniak

Published Sep 30, 2017• 5 citations
Publisher

Abstract

We discuss here a possibility of generation of steerable states in asymmetric chains comprising three Kerr-like nonlinear oscillators. We show that steering between modes can be generated in the system and it strongly depends on the asymmetry of internal couplings in our model. We can lead to the appearance of new steering effects, which were not present in symmetric models already studied in the literature. Full Text: PDF ReferencesE. Schrödinger, "Discussion of Probability Relations between Separated Systems", Math. Proc. Camb. Phil. Soc. 31, 555 (1935). CrossRef M.D. Reid, "Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification", Phys. Rev. A 40, 913 (1989). CrossRef E.G. Cavalcanti, M.D. Reid, "Uncertainty relations for the realization of macroscopic quantum superpositions and EPR paradoxes", Journal of Modern Optics 54, 2373 (2007). CrossRef S.P. Walborn, A. Salles, R.M. Gomes, F. Toscano, P.H. Souto Ribeiro, "Revealing Hidden Einstein-Podolsky-Rosen Nonlocality", Phys. Rev. Lett. 106, 130402 (2011). CrossRef H.M. Wiseman, S.J. Jones, A.C. Doherty, "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox", Phys. Rev. Lett. 98, 140402 (2007). CrossRef S.J. Jones, H.M. Wiseman, A.C. Doherty, "Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering", Phys. Rev. A 76, 052116 (2007). CrossRef J.K. Kalaga, W. Leoński, "Quantum steering borders in three-qubit systems", Quantum Inf Process 16, 175 (2017). CrossRef Q. He, Z. Ficek, "Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system", Phys. Rev. A 89, 022332 (2014). CrossRef S. Kiesewetter, Q.Y. He, P.D. Drummond, M.D. Reid, "Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics", Phys. Rev. A 90, 043805 (2014). CrossRef K. Bartkiewicz, A. Cernoch, K. Lemr, A. Miranowicz, F. Nori, "Experimental temporal quantum steering", Scientific Reports 6, 38076 (2016). CrossRef A. Barasiński, B. Brzostowski, R. Matysiak, P. Sobczak, D. Woźniak, In: R. Wyrzykowski, J. Dongarra, K. Karczewski, J. Wasniewski editor, Parallel Processing and Applied Mathematics (PPAM 2013), Lecture Notes in Computer Science, vol 8385. Springer, Berlin, Heidelberg (2014). CrossRef A. Drzewiński, J. Sznajd, "On the real-space renormalization-group study of some 2D quantum spin systems", Physica A 170, 415 (1991). CrossRef G.J. Milburn, C.A. Holmes, "Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity", Phys. Rev. A 44, 4704 (1991). CrossRef W. Leoński, "Quantum and classical dynamics for a pulsed nonlinear oscillator", Physica A 233, 365 (1996). CrossRef A. Kowalewska-Kudłaszyk, J.K. Kalaga, W. Leoński, "Long-time fidelity and chaos for a kicked nonlinear oscillator system", Physics Letters A 373, 1334 (2009). CrossRef J.K. Kalaga, W. Leoński, "Two proposals of quantum chaos indicators related to the mean number of photons: pulsed Kerr-like oscillator case", Proc. SPIE 10142, 101421L (2016). CrossRef A. Barasiński, W. Leoński, T. Sowiński, "Ground-state entanglement of spin-1 bosons undergoing superexchange interactions in optical superlattices", J. Opt. Soc. Am. B 31, 1845 (2014). CrossRef A. Barasiński, W. Leoński, "Symmetry restoring and ancilla-driven entanglement for ultra-cold spin-1 atoms in a three-site ring", Quantum Inf Process 16, 6 (2017). CrossRef D. Woźniak, A. Drzewiński, G. Kamieniarz, "Relaxation Dynamics in the Spin-1 Heisenberg Antiferromagnetic Chain after a Quantum Quench of the Uniaxial Anisotropy", Acta Physica Polonica A 130, 1395 (2016). CrossRef R. Szczęśniak, D. Szczęśniak, E.A. Drzazga, "Superconducting state in the atomic metallic hydrogen just above the pressure of the molecular dissociation", Solid State Communications 152, 2023 (2012). CrossRef A. P. Durajski, R. Szczęśniak, M.W. Jarosik, "Properties of the superconducting state in compressed sulphur", Phase Transitions 85, 727 (2012). CrossRef R. Szczęśniak, A. P. Durajski, "The thermodynamic properties of the high-pressure superconducting state in the hydrogen-rich compounds", Solid State Sciences 25, 45 (2013). CrossRef X. Wang, A. Miranowicz, H.R. Li, F. Nori, "Multiple-output microwave single-photon source using superconducting circuits with longitudinal and transverse couplings", Phys. Rev. A 94, 053858, (2016). CrossRef Y.X. Liu, X.W. Xu, A. Miranowicz, F. Nori, "From blockade to transparency: Controllable photon transmission through a circuit-QED system", Phys. Rev. A 89, 043818 (2014). CrossRef M.K. Olsen, "Spreading of entanglement and steering along small Bose-Hubbard chains", Phys. Rev. A 92, 033627 (2015). CrossRef E.G. Cavalcanti, Q.Y. He, M.D. Reid, H.M. Wiseman, "Unified criteria for multipartite quantum nonlocality", Phys. Rev. A 84, 032115 (2011). CrossRef

DOI
10.4302/plp.v9i3.759

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We discuss here a possibility of generation of steerable states in asymmetric chains comprising three Kerr-like nonlinear oscillators. We show that steering between modes can be generated in the system and it strongly depends on the asymmetry of internal couplings in our model. We can lead to the appearance of new steering effects, which were not present in symmetric models already studied in the literature. Full Text: PDF ReferencesE. Schrödinger, "Discussion of Probability Relations between Separated Systems", Math. Proc. Camb. Phil. Soc. 31, 555 (1935). CrossRef M.D. Reid, "Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification", Phys. Rev. A 40, 913 (1989). CrossRef E.G. Cavalcanti, M.D. Reid, "Uncertainty relations for the realization of macroscopic quantum superpositions and EPR paradoxes", Journal of Modern Optics 54, 2373 (2007). CrossRef S.P. Walborn, A. Salles, R.M. Gomes, F. Toscano, P.H. Souto Ribeiro, "Revealing Hidden Einstein-Podolsky-Rosen Nonlocality", Phys. Rev. Lett. 106, 130402 (2011). CrossRef H.M. Wiseman, S.J. Jones, A.C. Doherty, "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox", Phys. Rev. Lett. 98, 140402 (2007). CrossRef S.J. Jones, H.M. Wiseman, A.C. Doherty, "Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering", Phys. Rev. A 76, 052116 (2007). CrossRef J.K. Kalaga, W. Leoński, "Quantum steering borders in three-qubit systems", Quantum Inf Process 16, 175 (2017). CrossRef Q. He, Z. Ficek, "Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system", Phys. Rev. A 89, 022332 (2014). CrossRef S. Kiesewetter, Q.Y. He, P.D. Drummond, M.D. Reid, "Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics", Phys. Rev. A 90, 043805 (2014). CrossRef K. Bartkiewicz, A. Cernoch, K. Lemr, A. Miranowicz, F. Nori, "Experimental temporal quantum steering", Scientific Reports 6, 38076 (2016). CrossRef A. Barasiński, B. Brzostowski, R. Matysiak, P. Sobczak, D. Woźniak, In: R. Wyrzykowski, J. Dongarra, K. Karczewski, J. Wasniewski editor, Parallel Processing and Applied Mathematics (PPAM 2013), Lecture Notes in Computer Science, vol 8385. Springer, Berlin, Heidelberg (2014). CrossRef A. Drzewiński, J. Sznajd, "On the real-space renormalization-group study of some 2D quantum spin systems", Physica A 170, 415 (1991). CrossRef G.J. Milburn, C.A. Holmes, "Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity", Phys. Rev. A 44, 4704 (1991). CrossRef W. Leoński, "Quantum and classical dynamics for a pulsed nonlinear oscillator", Physica A 233, 365 (1996). CrossRef A. Kowalewska-Kudłaszyk, J.K. Kalaga, W. Leoński, "Long-time fidelity and chaos for a kicked nonlinear oscillator system", Physics Letters A 373, 1334 (2009). CrossRef J.K. Kalaga, W. Leoński, "Two proposals of quantum chaos indicators related to the mean number of photons: pulsed Kerr-like oscillator case", Proc. SPIE 10142, 101421L (2016). CrossRef A. Barasiński, W. Leoński, T. Sowiński, "Ground-state entanglement of spin-1 bosons undergoing superexchange interactions in optical superlattices", J. Opt. Soc. Am. B 31, 1845 (2014). CrossRef A. Barasiński, W. Leoński, "Symmetry restoring and ancilla-driven entanglement for ultra-cold spin-1 atoms in a three-site ring", Quantum Inf Process 16, 6 (2017). CrossRef D. Woźniak, A. Drzewiński, G. Kamieniarz, "Relaxation Dynamics in the Spin-1 Heisenberg Antiferromagnetic Chain after a Quantum Quench of the Uniaxial Anisotropy", Acta Physica Polonica A 130, 1395 (2016). CrossRef R. Szczęśniak, D. Szczęśniak, E.A. Drzazga, "Superconducting state in the atomic metallic hydrogen just above the pressure of the molecular dissociation", Solid State Communications 152, 2023 (2012). CrossRef A. P. Durajski, R. Szczęśniak, M.W. Jarosik, "Properties of the superconducting state in compressed sulphur", Phase Transitions 85, 727 (2012). CrossRef R. Szczęśniak, A. P. Durajski, "The thermodynamic properties of the high-pressure superconducting state in the hydrogen-rich compounds", Solid State Sciences 25, 45 (2013). CrossRef X. Wang, A. Miranowicz, H.R. Li, F. Nori, "Multiple-output microwave single-photon source using superconducting circuits with longitudinal and transverse couplings", Phys. Rev. A 94, 053858, (2016). CrossRef Y.X. Liu, X.W. Xu, A. Miranowicz, F. Nori, "From blockade to transparency: Controllable photon transmission through a circuit-QED system", Phys. Rev. A 89, 043818 (2014). CrossRef M.K. Olsen, "Spreading of entanglement and steering along small Bose-Hubbard chains", Phys. Rev. A 92, 033627 (2015). CrossRef E.G. Cavalcanti, Q.Y. He, M.D. Reid, H.M. Wiseman, "Unified criteria for multipartite quantum nonlocality", Phys. Rev. A 84, 032115 (2011). CrossRef
We study multipartite entanglement, the generation of Einstein-Podolsky-Rosen (EPR) states, and quantum steering in a three-mode optomechanical system composed of an atomic ensemble located inside a single-mode cavity with a movable mirror. The cavity mode is driven by a short laser pulse, has a nonlinear parametric-type interaction with the mirror and a linear beam-splitter-type interaction with the atomic ensemble. There is no direct interaction of the mirror with the atomic ensemble. A threshold effect for the dynamics of the system is found, above which the system works as an amplifier and below which as an attenuator of the output fields. The threshold is determined by the ratio of the coupling strengths of the cavity mode to the mirror and to the atomic ensemble. It is shown that above the threshold, the system effectively behaves as a two-mode system in which a perfect bipartite EPR state can be generated, while it is impossible below the threshold. Furthermore, a fully inseparable tripartite entanglement and even further a genuine tripartite entanglement can be produced above and below the threshold. In addition, we consider quantum steering and examine the monogamy relations that quantify the amount of bipartite steering that can be shared between different modes. It is found that the mirror is more capable for steering of entanglement than the cavity mode. The two-way steering is found between the mirror and the atomic ensemble despite the fact that they are not directly coupled to each other, while it is impossible between the output of cavity mode and the ensemble which are directly coupled to each other.
Einstein-Podolsky-Rosen (EPR) steering, that is, an intermediate form of quantum correlations with asymmetry in between entanglement and Bell nonlocality, plays an essential role in quantum information processing, such as one-sided device-independent quantum cryptography tasks. Here we theoretically study the direct generation of EPR steering in ${\ensuremath{\chi}}^{(2)}$ nonlinear optical processes with cascaded crystals in a self-phase-locked optical parametric oscillator. Based on a standard linearization method, we analyze different conditions for generating EPR steering in Gaussian states with vanishing or nonzero mean fields, and also show bistable regions for different types of EPR steering. In particular, such system can intrinsically generate asymmetry one-way EPR steering, without introducing any postoperation. Our results suggest that the self-phase-locked optical parametric oscillator is a good candidate to generate different types of EPR steering at will by varying the cavity detunings, photon loss, or input pump power.
We describe how to generate an Einstein-Podolsky-Rosen (EPR) paradox between a mesoscopic mechanical oscillator and an optical pulse. We find two types of paradox, defined by whether it is the oscillator or the pulse that shows the effect Schr\"odinger called ``steering''. Only the oscillator paradox addresses the question of mesoscopic local reality for a massive system. In that case, EPR's ``elements of reality'' are defined for the oscillator, and it is these elements of reality that are falsified (if quantum mechanics is complete). For this sort of paradox, we show that a thermal barrier exists, meaning that a threshold level of pulse-oscillator interaction is required for a given thermal occupation n0{n}_{0}n0​ of the oscillator. We find there is no equivalent thermal barrier for the entanglement of the pulse with the oscillator or for the EPR paradox that addresses the local reality of the optical system. Finally, we examine the possibility of an EPR paradox between two entangled oscillators. Our work highlights the asymmetrical effect of thermal noise on quantum nonlocality.
In the last few years, several criteria to identify Einstein-Podolsky-Rosen steering have been proposed and experimentally implemented. On the operational side, however, the evaluation of the steerability degree of a given state has shown to be a difficult task and only a few results are known. In this work, we propose a measure of steering that is based on the maximal violation of well-established steering inequalities. Applying this approach to two-qubit states, we managed to derive simple closed formulas for steering in the two- and three-measurement scenarios. We also provide closed formulas for quantifiers of Bell nonlocality in the respective scenarios. Finally, we show that our measures of steering verify the entanglement-steering-nonlocality hierarchy and reproduce results reported so far.
We describe how to generate an Einstein-Podolsky-Rosen (EPR) paradox between a mesoscopic mechanical oscillator and an optical pulse. We find two types of paradox, defined by whether it is the oscillator or the pulse that shows the effect Schrodinger called steering. Only the oscillator paradox addresses the question of mesoscopic local for a massive system. In that case, EPR's of reality are defined for the oscillator, and it is these elements of that are falsified (if quantum mechanics is complete). For this sort of paradox, we show that a thermal barrier exists, meaning that a threshold level of pulse-oscillator interaction is required for a given thermal occupation n_0 of the oscillator. We find there is no equivalent thermal barrier for the entanglement of the pulse with the oscillator, nor for the EPR paradox that addresses the local of the optical system. Finally, we examine the possibility of an EPR paradox between two entangled oscillators. Our work highlights the asymmetrical effect of thermal noise on quantum nonlocality.
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum correlation and its intrinsic asymmetry makes it distinct from entanglement and Bell nonlocality. We propose here a scheme for realizing one-way Gaussian steering of two electromagnetic fields mediated by a mechanical oscillator. We reveal that the steady-state one-way steering of the intracavity and output fields is obtainable with different cavity losses or strong mechanical damping. The conditions for achieving this asymmetric steering are found, and it shows that the steering is robust against thermal mechanical fluctuations. The present scheme can realize hybrid microwave-optical asymmetric steering by optoelectromechanics. In addition, our results are generic and can also be applied to other three-mode parametrically coupled bosonic systems.