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Journal of the Optical Society of America B

Oscillating between classically entangled state and separable state: an analogy between classical and quantum optics

Shahram Dehdashti, Firat Yasar

Published May 26, 2020• 0 citations
Publisher

Abstract

In this paper, we study the possible realization of a classical system with quantum characteristics on the level of classical optics. Indeed, following Arrizon et al. [ J. Opt. Soc. Am. A 32 , 1140 ( 2015 ) JOAOD6 0740-3232 10.1364/JOSAA.32.001140 ], we first use quantum optics formalism to consider the propagation of two coherent states in a Kerr medium where the interaction between the two states is described by the cross-Kerr interaction. We then draw an analogy between the dynamical process of this structure and that of a Gaussian wave propagating in a quadratic gradient index medium. We demonstrate that by using this structure, we can generate a state that oscillates between a classically entangled state and a separable one.

DOI
10.1364/josab.386100

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The emerging field of entanglement or nonseparability in classical optics is reviewed, and its similarities with and differences from quantum entanglement clearly pointed out through a recapitulation of Hilbert spaces in general, the special restrictions on Hilbert spaces imposed in quantum mechanics and the role of Hilbert spaces in classical polarization optics. The production of Bell-like states in classical polarization optics is discussed, and new theorems are proved to discriminate between separable and nonseparable states in classical wave optics where no discreteness is involved. The influence of the Pancharatnam phase on a classical Bell-like state is deived. Finally, to what extent classical polarization optics can be used to simulate quantum information processing tasks is also discussed. This should be of great practical importance because coherence and entanglement are robust in classical optics but not in quantum systems.

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In this paper, we study the possible realization of a classical system with quantum characteristics on the level of classical optics. Indeed, following Arrizon et al. [ J. Opt. Soc. Am. A 32 , 1140 ( 2015 ) JOAOD6 0740-3232 10.1364/JOSAA.32.001140 ], we first use quantum optics formalism to consider the propagation of two coherent states in a Kerr medium where the interaction between the two states is described by the cross-Kerr interaction. We then draw an analogy between the dynamical process of this structure and that of a Gaussian wave propagating in a quadratic gradient index medium. We demonstrate that by using this structure, we can generate a state that oscillates between a classically entangled state and a separable one.
We consider a model of Kerr-like nonlinear coupler linearly excited by a external classical field involving coherent part and white noise and solved a set of coupled stochastic integro-differential equations, deriving exactly analytical formulae for the probability amplitudes of n-photon Fock states. As a consequence, this nonlinear coupler deliberated behaves as a two-qubit system. The effects of dissipation on entanglement of formation parameterised by concurrence are also analyzed and compare these results with that obtained previously in the literature. As a interesting result, initially for a period of time the entanglement is enlarged when the chaotic parameter increases. Full Text: PDF References A. Einstein, B. Podolski, N. Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?", Phys. Rev. 47, 777 (1935). CrossRef A. Miranowicz, W. Leoński, "Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers", J. Phys. B 39, 1683 (2006). CrossRef T.D. Nguyen, W. Leoński, V. Cao Long, "Computer Simulation of Two-Mode Nonlinear Quantum Scissors", Comput. Method. Sci. Tech. 19, 175 (2013). CrossRef N. Korolkova, J. Peřina, "Quantum statistics and dynamics of Kerr nonlinear couplers", Opt. Commun. 136, 135 (1997). CrossRef A. Kowalewska-Kudłaszyk, W. Leoński, "Finite-dimensional states and entanglement generation for a nonlinear coupler", Phys. Rev. A 73, 042318 (2006). CrossRef A. Kowalewska-Kudłaszyk, W. Leoński, T.D. Nguyen, V. Cao Long, "Kicked nonlinear quantum scissors and entanglement generation", Phys. Scr. T 160, 014023 (2014). CrossRef K. Doan Quoc, V. Cao Long, W. Leoński, "Electromagnetically induced transparency for Λ-like systems with a structured continuum and broad-band coupling laser", Phys. Scr. T 147, 014008 (2012). CrossRef K. Doan Quoc, V. Cao Long, W. Leoński, "A broad-band laser-driven double Fano system—photoelectron spectra", Phys. Scr. 86, 045301 (2012). CrossRef K. Doan Quoc, V. Cao Long, L. Chu Van, P. Huynh Vinh, "Electromagnetically induced transparency for Λ-like systems with degenerate autoionizing levels and a broadband coupling laser", Optica Applicata 46, 93 (2016). CrossRef W. Leoński, A. Miranowicz, "Kerr nonlinear coupler and entanglement", J. Opt. B 6, S37 (2004). CrossRef K. Doan Quoc, V. Cao Long, to be Published. W. Leoński, "Finite-dimensional coherent-state generation and quantum-optical nonlinear oscillator models", Phys. Rev. A 55, 3874 (1997). CrossRef
Abstract Quantum Mechanics and classical optics feature similar phenomena such as superposition, interference and even entanglement. Hence, techniques from optics can be used in quantum mechanics and vice versa. In this article I address the question: What can we learn from formulating optics in the language of quantum mechanics? It is argued that the solutions of the wave equations for the electromagnetic field form a tensor product of Hilbert spaces corresponding to the degrees of freedom of classical light. Therefore, it comprises non-separable solutions reminiscent of entanglement. Moreover, the two spatial degrees of freedom each carry non-commuting position and momentum variables forming a Heisenberg algebra like quantum particles moving in a two dimensional space. An analogy between the dynamics of a quantum harmonic oscillator and paraxial light propagating through a converging lens is drawn. This article presents a modern formulation of optics in the language of state vectors and operators (based on Dirac’s notation) along the lines of an earlier contribution [1], specifying and explaining its results.
We found the analytical expression for cross-Kerr nonlinear coefficient in a four-level Y-type atomic system. The analytical model is applied to 85Rb atoms and shown that under electromagnetically induced transparency, cross-Kerr nonlinear coefficient is enhanced by several orders of magnitude. At the same time, the amplitude and the sign of cross-Kerr nonlinear coefficient are controlled with respect to the intensity and the frequency of the coupling laser field. The analytical model can be useful to explain the experimental results and to study related effects in nonlinear optics. Full Text: PDF ReferencesC. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media", Phys. Rev. Lett. 90, 197902 (2003). CrossRef C. Zhu, G. Huang, "Giant Kerr Nonlinearity, Controlled Entangled Photons and Polarization Phase Gates in Coupled Quantum-Well Structures", Opt. Express 19, 23364 (2011). CrossRef C. Hang, G. Huang, "Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet", Opt. Express 18(3), 2952 (2010). CrossRef M. Fleischhauer, I. Mamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media", Rev. Mod. Phys. 77, 633 (2005). CrossRef H. Schmidt, And A. Imamogdlu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency", Opt. Lett., 21(23), 1936 (1996). CrossRef H. Kang And Y. Zhu, Phys. "Observation of Large Kerr Nonlinearity at Low Light Intensities", Rev. Lett., 91, 093601 (2003). CrossRef J. Kou, R. G. Wan, Z. H. Kang, H. H. Wang, L. Jiang, X. J. Zhang, Y. Jiang, and J. Y. Gao, "EIT-assisted large cross-Kerr nonlinearity in a four-level inverted-Y atomic system", J. Opt. Soc. Am. B. 27(10), 2035 (2010). CrossRef X. Yang, S. Li, C. Zhang, and H. Wang, "Enhanced cross-Kerr nonlinearity via electromagnetically induced transparency in a four-level tripod atomic system", J. Opt. Soc. Am. B. 26(7), 1423 (2009). CrossRef C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media", Phys. Rev. Lett. 90, 197902 (2003). CrossRef H. Sun, Y. Niu, S. Jin and S. Gong, "Phase control of cross-phase modulation with electromagnetically induced transparency", J. Phys. B: At. Mol. Opt. Phys. 40, 3037 (2007). CrossRef L.V. Doai, P.V. Trong, D.X. Khoa, and N.H. Bang, "Electromagnetically induced transparency in five-level cascade scheme of 85Rb atoms: An analytical approach", Optik, 125, 3666 (2014). CrossRef D. X. Khoa, P. V. Trong, L. V. Doai and N. H. Bang, "Electromagnetically induced transparency in a five-level cascade system under Doppler broadening: an analytical approach", Phys, Scr. 91, 035401 (2016). CrossRef D.X. Khoa, L.C. Trung, P.V. Thuan, L.V. Doai and N.H. Bang, "Measurement of dispersive profile of a multiwindow electromagnetically induced transparency spectrum in a Doppler-broadened atomic medium", J. Opt. Soc. Am. B 34 (6), 1255 (2017). CrossRef D. X. Khoa, L. V. Doai, D. H. Son, and N. H. Bang, "Enhancement of self-Kerr nonlinearity via electromagnetically induced transparency in a five-level cascade system: an analytical approach", J. Opt. Soc. Am. B., 31, 1330 (2014). CrossRef L.V. Doai, N.L.T. An, D.X. Khoa, V.N. Sau and N.H. Bang, "Manipulating giant cross-Kerr nonlinearity at multiple frequencies in an atomic gaseous medium", J. Opt. Soc. Am. B 36, 2856 (2019). CrossRef D. X. Khoa, L. V. Doai, L. N. M. Anh, L. C. Trung, P. V. Thuan, N. T. Dung, and N. H. Bang, "Optical bistability in a five-level cascade EIT medium: an analytical approach", J. Opt. Soc. Am. B, Vol. 33, 735 (2016). CrossRef N. T. Anh, L. V. Doai, and N. H. Bang, "Manipulating multi-frequency light in a five-level cascade-type atomic medium associated with giant self-Kerr nonlinearity", J. Opt. Soc. Am. B 35, 1233 (2018). CrossRef N. T. Anh, L. V. Doai, D. H. Son, and N. H. Bang, "Manipulating multi-frequency light in a five-level cascade EIT medium under Doppler broadening", Optik 171, 721 (2018). CrossRef D.A. Steck, Rb85 D Line Data: http://Steck.Us/Alkalidata/rubidium85numbers.pdf CrossRef
In this paper, we follow the model presented in our previous work [J. Opt. Soc. Am. B30, 1109 (2013)], in which the interaction between a Λ-type three-level atom and a quantized two-mode radiation field in a cavity in the presence of nonlinearities was studied. After giving a brief review of the procedure for obtaining the state vector of the atom-field system, we investigated some further interesting and important physical features of the whole system state, which are of particular interest to the quantum optics field of research, i.e., the number-phase entropic uncertainty relation (based on the two-mode Pegg–Barnett formalism) and a few of the nonclassicality signs that consist of sub-Poissonian statistics, Cauchy–Schwartz inequality, and two types of squeezing phenomena. During our presentation, the effects of intensity-dependent coupling, deformed Kerr medium, and the detuning parameters on the depth and domain of each of the mentioned nonclassical criteria of the considered quantum system are described in detail. The results show that each of the mentioned nonclassicality aspects can be obtained by appropriately choosing the related parameters.
The emerging field of entanglement or nonseparability in classical optics is reviewed, and its similarities with and differences from quantum entanglement clearly pointed out through a recapitulation of Hilbert spaces in general, the special restrictions on Hilbert spaces imposed in quantum mechanics and the role of Hilbert spaces in classical polarization optics. The production of Bell-like states in classical polarization optics is discussed, and new theorems are proved to discriminate between separable and nonseparable states in classical wave optics where no discreteness is involved. The influence of the Pancharatnam phase on a classical Bell-like state is deived. Finally, to what extent classical polarization optics can be used to simulate quantum information processing tasks is also discussed. This should be of great practical importance because coherence and entanglement are robust in classical optics but not in quantum systems.