# Frequency Dependence of the Entanglement Entropy Production in a System of Coupled Driven Nonlinear Oscillators

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## Abstract

**:**

## 1. Introduction

## 2. System of Interest

## 3. Entanglement Entropy and Its Frequency Response

## 4. Entropy Production during the Quantum Evolution

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information: 10th Anniversary Edition; Cambridge University Press: Cambridge, UK, 2011; pp. 571–580. [Google Scholar]
- Raimond, J.M.; Brune, M.; Haroche, S. Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys.
**2001**, 73, 565. [Google Scholar] [CrossRef] - Berrada, K.; Abdel-Khalek, S.; Eleuch, H.; Hassouni, Y. Beam splitting and entanglement generation: Excited coherent states. Quant. Inf. Process.
**2013**, 12, 69–82. [Google Scholar] [CrossRef] - Mohamed, A.B.; Eleuch, H. Non-classical effects in cavity QED containing a nonlinear optical medium and a quantum well: Entanglement and non-Gaussanity. Eur. Phys. J. D
**2015**, 69, 191. [Google Scholar] [CrossRef] - Mohamed, A.B.A.; Eleuch, H. Quantum correlation control for two semiconductor microcavities connected by an optical fiber. Phys. Scr.
**2017**, 92, 065101. [Google Scholar] [CrossRef] - Berrada, K.; Eleuch, H.; Hassouni, Y. Asymptotic dynamics of quantum discord in open quantum systems. J. Phys. B
**2011**, 44, 145503. [Google Scholar] [CrossRef] - Sete, E.A.; Eleuch, H.; Das, S. Semiconductor cavity QED with squeezed light: Nonlinear regime. Phys. Rev. A
**2011**, 84, 053817. [Google Scholar] [CrossRef] - Vacanti, G.; Paternostro, M.; Palma, G.M.; Kim, M.S.; Vedral, V. Non-classicality of optomechanical devices in experimentally realistic operating regimes. Phys. Rev. A
**2013**, 88, 1023–1027. [Google Scholar] [CrossRef] - Rips, S.; Kiffner, M.; Wilsonrae, I.; Hartmann, M.J. Steady-state negative wigner functions of nonlinear nanomechanical oscillators. New J. Phys.
**2012**, 4, 023042. [Google Scholar] [CrossRef] - Aspelmeyer, M.; Kippenberg, T.J.; Marquardt, F. Cavity optomechanics. Rev. Mod. Phys.
**2014**, 86, 1391–1452. [Google Scholar] [CrossRef] - Poot, M.; Zant, H.S.J.V.D. Mechanical systems in the quantum regime. Phys. Rep.
**2012**, 511, 273–335. [Google Scholar] [CrossRef][Green Version] - Kaltenbaek, R.; Kiesel, N.; Romero-Isart, O.; Schwab, K.C.; Johann, U.; Aspelmeyer, M. Macroscopic quantum resonators (MAQRO): 2015 Update. Exp. Astron.
**2015**, 34, 123–164. [Google Scholar] [CrossRef] - Yukihiro, T.; Hiroya, T.; Dykman, M.I. Driven nonlinear nanomechanical resonators as digital signal detectors. Sci. Rep.
**2018**, 8, 11284. [Google Scholar] - Peano, V.; Thorwart, M. Macroscopic quantum effects in a strongly driven nanomechanical resonator. Phys. Rev. B
**2005**, 70, 155–163. [Google Scholar] [CrossRef] - Imboden, M.; Williams, O.; Mohanty, P. Nonlinear dissipation in diamond nanoelectromechanical resonators. Appl. Phys. Lett.
**2013**, 102, 103502. [Google Scholar] [CrossRef] - Rips, S.; Wilsonrae, I.; Hartmann, M.J. Nonlinear nanomechanical resonators for quantum optoelectromechanics. Phys. Rev. A
**2014**, 89, 58–61. [Google Scholar] [CrossRef] - Almog, R.; Zaitsev, S.; Shtempluck, O.; Buks, E. Noise squeezing in a nanomechanical duffing resonator. Phys. Rev. Lett.
**2007**, 98, 078103. [Google Scholar] [CrossRef] [PubMed] - Ong, F.R.; Boissonneault, M.; Mallet, F.; Palacioslaloy, A.; Dewes, A.; Doherty, A.C.; Blais, A.; Bertet, P.; Vion, D.; Esteve, D. Circuit QED with a nonlinear resonator: Ac-stark shift and dephasing. Phys. Rev. Lett.
**2011**, 106, 167002. [Google Scholar] [CrossRef] [PubMed] - Murch, K.W.; Vijay, R.; Barth, I.; Naaman, O.; Aumentado, J.; Friedland, L.; Siddiqi, I. Quantum fluctuations in the chirped pendulum. Nat. Phys.
**2011**, 7, 105–108. [Google Scholar] [CrossRef] - Yang, C.P.; Chu, S.I.; Han, S. Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity qed. Phys. Rev. A
**2003**, 67, 042311. [Google Scholar] [CrossRef] - Stannigel, K.; Komar, P.; Habraken, S.J.M.; Bennett, S.D.; Lukin, M.D.; Zoller, P.; Rabl, P. Optomechanical quantum information processing with photons and phonons. Phys. Rev. Lett.
**2012**, 109, 013603. [Google Scholar] [CrossRef] - Cleland, A.N.; Geller, M.R. Superconducting qubit storage and entanglement with nanomechanical resonators. Phys. Rev. Lett.
**2004**, 93, 070501. [Google Scholar] [CrossRef] [PubMed] - Ikeda, D.S.; Fillaux, F. Incoherent elastic-neutron-scattering study of the vibrational dynamics and spin-related symmetry of protons in the crystal. Phys. Rev. B
**1999**, 59, 4134. [Google Scholar] [CrossRef] - Audenaert, K.; Eisert, J.; Werner, R.F. Entanglement properties of the harmonic chain. Phys. Rev. A
**2002**, 66, 042327. [Google Scholar] [CrossRef] - Fillaux, D.F. Quantum entanglement and nonlocal proton transfer dynamics in dimers of formic acid and analogues. Chem. Phys. Lett.
**2005**, 408, 302. [Google Scholar] [CrossRef] - Chung, N.N.; Chew, L.Y. Two-step approach to the dynamics of coupled anharmonic oscillators. Phys. Rev. A
**2009**, 80, 012103. [Google Scholar] [CrossRef][Green Version] - Qiao, G.J.; Gao, H.X.; Liu, H.D.; Yi, X.X. Quantum synchronization of two mechanical oscillators in coupled optomechanical systems with Kerr nonlinearity. Sci. Rep.
**2018**, 8, 15614. [Google Scholar] [CrossRef] [PubMed] - Chakraborty, S.; Sarma, A.K. Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics. Phys. Rev. A
**2018**, 97, 022336. [Google Scholar] [CrossRef][Green Version] - Ralph, T.C.; Gilchris, A.; Milburn, G.J.; Munro, W.J.; Glancy, S. Quantum computation with optical coherent states. Phys. Rev. A
**2003**, 68, 042319. [Google Scholar] [CrossRef][Green Version] - Jaffe, C.; Brumer, P. Classical Liouville mechanics and intramolecular relaxation dynamics. J. Phys. Chem.
**1984**, 88, 4829–4839. [Google Scholar] [CrossRef] - Caldeira, A.O.; Leggett, A.J. Path integral approach to quantum brownian motion. Physica A
**1983**, 121, 587–616. [Google Scholar] [CrossRef] - Rabinovich, M.I.; Trubetskov, D.I. Oscillations and Waves: In Linear and Nonlinear Systems; Kluver Academic Publisher: Dordrecht, The Netherlands, 1989; pp. 286–297. [Google Scholar]
- Lefebvrebrion, H.; Field, R. The Spectra and Dynamics of Diatomic Molecules Revised and Enlarged Edition; Elsevier: San Diego, CA, USA, 2004; pp. 27–47. [Google Scholar]
- Bracewell, R. The Fourier Transform and Its Applications; McGraw-Hill: New York, NY, USA, 1985; pp. 112–113. [Google Scholar]
- Louisell, W.H. Quantum Statistical Properties of Radiation; Wiley: New York, NY, USA, 1973; pp. 331–468. [Google Scholar]
- Duan, L.-M.; Guo, G.-C. Preserving coherence in quantum computation by pairing the quantum bits. Phys. Rev. Lett.
**1997**, 79, 1953. [Google Scholar] [CrossRef] - Eleuch, H.; Guérin, S.; Jauslin, H. Effects of an environment on a cavity-quantum-electrodynamics system controlled by bichromatic adiabatic passage. Phys. Rev. A
**2012**, 85, 13830. [Google Scholar] [CrossRef] - Eleuch, H.; Rotter, I. Resonances in open quantum systems. Phys. Rev. A
**2017**, 95, 022117. [Google Scholar] [CrossRef][Green Version] - Eleuch, H.; Rotter, I. Open quantum systems and Dicke superradiance. Eur. Phys. J. D
**2014**, 68, 74. [Google Scholar] [CrossRef] - Schlosshauer, M. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys.
**2004**, 76, 1267. [Google Scholar] [CrossRef] - Berrada, K.; El Baz, M.; Eleuch, H.; Hassouni, Y. A comparative study of negativity and concurrence based on spin coherent states. Int. J. Mod. Phys. C
**2010**, 21, 291. [Google Scholar] [CrossRef] - Bogoliubov, N.N. A New method in the theory of superconductivity. I. J. Exp. Theor. Phys.
**1958**, 34, 58. [Google Scholar] [CrossRef] - Nambu, Y.; Jona-Lasinio, G. Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I. Phys. Rev.
**1961**, 122, 345. [Google Scholar] [CrossRef] - Nambu, Y.; Jona-Lasinio, G. Dynamical model of elementary particles based on an analogy with superconductivity. II. Phys. Rev.
**1961**, 124, 246. [Google Scholar] [CrossRef] - Arraut, I. The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem. Symmetry
**2019**, 11, 803. [Google Scholar] [CrossRef] - Nambu, Y. From Yukawa’s Pion to spontaneous symmetry breaking. J. Phys. Soc. Jpn.
**2007**, 76, 111002. [Google Scholar] [CrossRef] - Arraut, I. The Nambu-Goldstone theorem in non-relativistic systems. Int. J. Mod. Phys.
**2017**, A32, 1750127. [Google Scholar] [CrossRef] - Arraut, I. The origin of the mass of the Nambu-Goldstone bosons. Int. J. Mod. Phys.
**2018**, A33, 1850041. [Google Scholar] [CrossRef] - Xu, Y.; Li, Y.; Li, J.; Feng, J.; Zhang, H. The Phase Transition in a Bistable Duffing System Driven by Lèvy Noise. J. Stat. Phys.
**2015**, 158, 120–131. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Time evolution of the entropy S for $\omega =0$. (

**b**) Average of the entropy over time $\overline{S}$ versus the driving frequency $\omega $ (the increment of $\omega $ is $0.02$ for $\omega \in [0.5,1.8]$ and $0.1$ for the other intervals). When $\omega =0$, the external driving is zero and the system is isolated.

**Figure 2.**(

**a**) Time evolution of the entropy S for $\omega =0$ (black line), $\omega =1$ (red line), $\omega =1.28$ (blue line), and $\omega =1.5$ (magenta line). Fourier spectra of the correlation function $C\left(t\right)$ for (

**b**) $\omega =0$, (

**c**) $\omega =1$, (

**d**) $\omega =1.28$, and (

**e**) $\omega =1.5$.

**Figure 3.**(

**a**–

**d**) Probability distribution $|\langle {q}_{1},{q}_{2}\left(\right)open="|"\; close="\rangle ">\Psi \left(t\right)$ for (

**a**) $\omega =0$, (

**b**) $\omega =1$, (

**c**) $\omega =1.28$ and (

**d**) $\omega =1.5$. (

**e**) Average of the uncertainty over time $\overline{\Delta}$ versus the driving frequency $\omega $ (the increment of $\omega $ is $0.02$ for $\omega \in [0.5,1.8]$ and $0.1$ for the other intervals).

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**MDPI and ACS Style**

Zhang, S.-H.; Yan, Z.-Y.
Frequency Dependence of the Entanglement Entropy Production in a System of Coupled Driven Nonlinear Oscillators. *Entropy* **2019**, *21*, 889.
https://doi.org/10.3390/e21090889

**AMA Style**

Zhang S-H, Yan Z-Y.
Frequency Dependence of the Entanglement Entropy Production in a System of Coupled Driven Nonlinear Oscillators. *Entropy*. 2019; 21(9):889.
https://doi.org/10.3390/e21090889

**Chicago/Turabian Style**

Zhang, Shi-Hui, and Zhan-Yuan Yan.
2019. "Frequency Dependence of the Entanglement Entropy Production in a System of Coupled Driven Nonlinear Oscillators" *Entropy* 21, no. 9: 889.
https://doi.org/10.3390/e21090889