# Quasi-Probability Husimi-Distribution Information and Squeezing in a Qubit System Interacting with a Two-Mode Parametric Amplifier Cavity

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## Abstract

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## 1. Introduction

## 2. Physical Model

## 3. Husimi Distribution (HD)

#### 3.1. Husimi Function

#### 3.2. Wehrl Entropy

## 4. Squeezing Phenomenon

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Mohamed, A.-B.A. Quantum correlation of correlated two qubits interacting with a thermal field. Phys. Scr.
**2012**, 85, 055013. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G. Quantum cryptography: Public key distribution and coin tossing. Theor. Comput. Sci.
**2014**, 560, 7. [Google Scholar] [CrossRef] - Pljonkin, A.P. Vulnerability of the Synchronization Process in the Quantum Key Distribution System. Int. J. Cloud Appl. Comput.
**2019**, 9, 50. [Google Scholar] [CrossRef][Green Version] - Pljonkin, A.; Rumyantsev, K.; Singh, P.K. Synchronization in Quantum Key Distribution Systems. Cryptography
**2017**, 1, 18. [Google Scholar] [CrossRef][Green Version] - Puigibert, M.l.G.; Askarani, M.F.; Davidson, J.H.; Verma, V.B.; Shaw, M.D.; Nam, S.W.; Lutz, T.; Amaral, G.C.; Oblak, D.; Tittel, W. Nonclassical Effects Based on Husimi Distributions in Two Open Cavities Linked by an Optical Waveguide. Phys. Rev. Res. A
**2020**, 2, 013039. [Google Scholar] [CrossRef][Green Version] - Sete, E.A.; Eleuch, H. High-efficiency quantum state transfer and quantum memory using a mechanical oscillator. Phys. Rev. A
**2015**, 91, 032309. [Google Scholar] [CrossRef][Green Version] - Baumgratz, T.; Cramer, M.; Plenio, M.B. Quantifying Coherence. Phys. Rev. Lett.
**2014**, 113, 140401. [Google Scholar] [CrossRef][Green Version] - Bennett, C.H.; DiVincenzo, D.P. Quantum information and computation. Nature
**2000**, 404, 247. [Google Scholar] [CrossRef] - Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- Shannon, C.E.; Weaver, W. Mathematical Theory of Communication; Urbana University Press: Chicago, IL, USA, 1949. [Google Scholar]
- Obada, A.-S.F.; Hessian, H.A.; Mohamed, A.-B.A. Effects of a phase-damping cavity on entanglement and purity loss in two-qubit system. Opt. Commun.
**2008**, 281, 5189. [Google Scholar] [CrossRef] - Husimi, K. Some properties of the Husimi function. Proc. Phys. Math. Soc. Jpn.
**1940**, 22, 264. [Google Scholar] - Yazdanpanah, N.; Tavassoly, M.K.; Juárez-Amaro, R.; Moya-Cessa, H.M. Reconstruction of quasiprobability distribution functions of the cavity field considering field and atomic decays. Opt. Commun.
**2017**, 400, 69. [Google Scholar] [CrossRef][Green Version] - Miller, C.A.; Hilsenbeck, J.; Risken, H. Asymptotic approximations for the Q function in the Jaynes-Cummings model. Phys. Rev. A
**1992**, 46, 4323. [Google Scholar] [CrossRef] - Mohamed, A.-B.A.; Eleuch, H. Coherence and information dynamics of a Λ-type three-level atom interacting with a damped cavity field. Eur. Phys. J. Plus
**2017**, 132, 75. [Google Scholar] [CrossRef] - Hessian, H.A.; Mohamed, A.-B.A. Quasi-Probability Distribution Functions for a Single Trapped Ion Interacting with a Mixed Laser Field. Laser Phys.
**2008**, 18, 1217–1223. [Google Scholar] [CrossRef] - Furusawa, A.; Sørensen, J.L.; Braunstein, S.L.; Fuchs, C.A.; Kimble, H.J.; Polzik, E.S. Unconditional Quantum Teleportation. Science
**1998**, 282, 706. [Google Scholar] [CrossRef][Green Version] - Sorensen, A.; Molmer, K. Spin-Spin Interaction and Spin Squeezing in an Optical Lattice. Phys. Lett. A
**1999**, 83, 2274. [Google Scholar] [CrossRef][Green Version] - Flouris, K.; Jimenez, M.M.; Debus, J.-D.; Herrmann, H.J. Landau levels in wrinkled and rippled graphene sheets. Phys. Rev. B
**2018**, 98, 155419. [Google Scholar] [CrossRef][Green Version] - Jabri, H.; Eleuch, H. Quantum fluctuations inside a microcavity with a pair of quantum wells: Linear regime. JOSA B
**2018**, 35, 2317. [Google Scholar] [CrossRef] - Qasymeh, M.; Eleuch, H. Quantum microwave-to-optical conversion in electrically driven multilayer graphene. Opt. Express
**2019**, 27, 5945. [Google Scholar] [CrossRef] - Fang, M.F.; Zhou, P.; Swain, S. Entropy squeezing of an atom with a k-photon in the Jaynes-Cummings model. J. Mod. Opt
**2000**, 47, 1043. [Google Scholar] [CrossRef] - Wineland, D.J.; Bollinger, J.J.; Itano, W.M.; Heinzen, D.J. Squeezed atomic states and projection noise in spectroscopy. Phys. Rev. A
**1994**, 50, 67. [Google Scholar] [CrossRef] - Hillery, M. Quantum cryptography with squeezed states. Phys. Rev. A
**2000**, 61, 022309. [Google Scholar] [CrossRef][Green Version] - Sete, E.A.; Eleuch, H.; Das, S. Semiconductor cavity QED with squeezed light: Nonlinear regime. Phys. Rev. A
**2011**, 84, 053817. [Google Scholar] [CrossRef] - Korbicz, J.K.; Gühne, O.; Lewenstein, M.; Häffner, H.; Roos, C.; Blatt, R. Generalized spin-squeezing inequalities in N-qubit systems: Theory and experiment. Phys. Rev. A
**2006**, 74, 052319. [Google Scholar] [CrossRef][Green Version] - Mohamed, A.-B.A.; Eleuch, H.; Obada, A.-S.F. Quantum effects in two-qubit systems interacting with two-mode fields: Dissipation and dipole-dipole interplay effects. Results Phys.
**2020**, 17, 103019. [Google Scholar] [CrossRef] - Yu, M.; Fang, M. Steady and optimal entropy squeezing of a two-level atom with quantum-jump-based feedback and classical driving in a dissipative cavity. Quantum Inf. Process.
**2016**, 15, 4175. [Google Scholar] [CrossRef] - Liu, X.-J.; Luo, A.; Peng, Z.-H.; Jia, C.-X.; Jiang, C.-L.; Zhou, B.-J. Generation and Preparation of the Sustained Optimal Entropy Squeezing State of a Motional Atom Inside Vacuum Cavity. Int. J. Theor. Phys.
**2018**, 57, 138. [Google Scholar] [CrossRef] - Obada, A.-S.F.; Abdalla, M.S.; Khalil, E.M. Statistical properties of two-mode parametric amplifier interacting with a single atom. Physica A
**2004**, 336, 433. [Google Scholar] [CrossRef] - Obada, A.-S.F.; Ahmed, M.M.A.; Faramawy, F.K.; Khalil, E.M. Entropy and entanglement of the nonlinear Jaynes-Cummings model. Chin. J. Phys.
**2004**, 42, 79. [Google Scholar] - Golkar, S.; Tavassoly, M.K. Atomic motion and dipole–dipole effects on the stability of atom-atom entanglement in Markovian/non-Markovian reservoir. Mod. Phys. Lett.
**2019**, 34, 1950077. [Google Scholar] [CrossRef] - Mohamed, A.-B.A. Non-local correlations via Wigner Yanase skew information in two SC-qubit having mutual interaction under phase decoherence. Eur. Phys. J. D
**2017**, 71, 261. [Google Scholar] [CrossRef] - Eberly, J.H.; Narozhny, N.B.; Mondragon, J.J.S. Periodic Spontaneous Collapse and Revival in a Simple Quantum Model. Phys. Rev. Lett.
**1980**, 44, 1323. [Google Scholar] [CrossRef] - Brune, M.; Raimond, J.M.; Goy, P.; Davidovich, L.; Haroche, S. Realization of a two-photon maser oscillator. Phys. Rev. Lett.
**1987**, 59, 1899. [Google Scholar] [CrossRef] - Louisell, W.H. Coupled Mode and Parametric Electronics; Wiley: New York, NY, USA, 1960. [Google Scholar]
- Kumar, S.; Mehta, C.L. Theory of the interaction of a single-mode radiation field with N two-level atoms. II. Time evolution of the statistics of the system. Phys. Rev. A
**1981**, 24, 1460. [Google Scholar] [CrossRef] - Kuang, L.M.; Tong, Z.Y.; Oyang, Z.W.; Zeng, H.S. Decoherence in two Bose-Einstein condensates. Phys. Rev. A
**2000**, 61, 013608. [Google Scholar] [CrossRef][Green Version] - 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] - Redwan, A.; Abdel-Aty, A.-H.; Zidan, N.; El-Shahat, T. Dynamics of the entanglement and teleportation of thermal state of a spin chain with multiple interactions. Chaos
**2019**, 29, 013138. [Google Scholar] [CrossRef] - Abdalla, M.S.; Obada, A.S.-F.; Khalil, E.M.; Mohamed, A.-B.A. Wehrl entropy information and purity of a SC charge qubit interacting with a lossy cavity field. Solid State Commun.
**2014**, 184, 56. [Google Scholar] [CrossRef][Green Version] - Milburn, G.J. Intrinsic decoherence in quantum mechanics. Phys. Rev. A
**1991**, 44, 5401. [Google Scholar] [CrossRef][Green Version] - Mohamed, A.-B.A. Pairwise quantum correlations of a three-qubit XY chain with phase decoherence. Quantum Inf. Process.
**2013**, 12, 1141. [Google Scholar] [CrossRef] - Mohamed, A.-B.A. Bipartite non-classical correlations for a lossy two connected qubit-cavity systems: Trace distance discord and Bell’s non-locality. Quantum Inf. Process.
**2018**, 17, 96. [Google Scholar] [CrossRef] - Courty, J.-M.; Spälter, S.; König, F.; Sizmann, A.; Leuchs, G. Noise-free quantum-nondemolition measurement using optical solitons. Phys. Rev. A
**1998**, 58, 1501. [Google Scholar] [CrossRef] - Barut, A.O.; Girardello, L. New coherent states associated with non-compact groups. Commun. Math. Phys.
**1971**, 21, 41. [Google Scholar] [CrossRef] - Vieira, V.R.; Sacramento, P.D. Generalized Phase-Space Representatives of Spin-J Operators in Terms of Bloch Coherent States. Ann. Phys.
**1995**, 242, 188. [Google Scholar] [CrossRef] - Zyczkowski, K. Localization of Eigenstates & Mean Wehrl Entropy. Physica E
**2001**, 9, 583. [Google Scholar] - Wehrl, A. General properties of entropy. Rev. Mod. Phys.
**1978**, 50, 221. [Google Scholar] [CrossRef] - Mohamed, A.-B.A.; Eleuch, H. Quasi-probability information in a coupled two-qubit system interacting non-linearly with a coherent cavity under intrinsic decoherence. Sci. Rep.
**2020**, 10, 13240. [Google Scholar] [CrossRef] - Van Enk, S.J.; Kimble, H.J. Quantum Information Processing in Cavity-QED. Quant. Inform. Comput.
**2002**, 2, 1. [Google Scholar] - Obada, A.-S.F.; Mohamed, A.-B.A. Erasing information and purity of a quantum dot via its spontaneous decay. Solid State Commun.
**2011**, 151, 1824. [Google Scholar] [CrossRef] - Sanchez-Ruis, J. Improved bounds in the entropic uncertainty and certainty relations for complementary observables. Phys. Lett. A
**1995**, 201, 125. [Google Scholar] [CrossRef]

**Figure 1.**The dependence of the HD on the atomic distribution angles $\theta $ and $\varphi $ at the time $\lambda t=2.112\pi $ is shown for the initial coherent intensity $\alpha =2$ and $k=\frac{1}{2}$, and also for different detuning and decoherence values: $(\delta ,\gamma )=(0,0)$ in (

**a**), $(\delta ,\gamma )=(30\lambda ,0)$ in (

**b**) and $(\delta ,\gamma )=(0,0.3\lambda )$ in (

**c**).

**Figure 2.**Time evolution of Husimi distribution is shown for the initial coherent intensity $\alpha =2$, with detuning and decoherence values: $(\delta ,\gamma )=(0,0)$ (solid curve), $(\delta ,\gamma )=(20\lambda ,0)$ (dashed curve) and $(\delta ,\gamma )=(0,0.1\lambda )$ (dashed-dotted curve).

**Figure 3.**Dynamics of the Wehrl entropy are shown for the initial coherent intensity $\alpha =4$ and $k=\frac{1}{2}$ with different decoherence values $\gamma =0$ (solid plots), $\gamma =0.01\lambda $ (dashed plots) and $\gamma =0.001\lambda $ (dashed-doted plots). The case $\delta =0$ in (

**a**) and $\delta =20\lambda $ in (

**b**).

**Figure 4.**Dynamics of the entropy squeezing are shown for $k=\frac{1}{2}$ and $\alpha =4$ with different decoherence values $\gamma =0$ (solid plots), $\gamma =0.01\lambda $ (dashed plots) and $\gamma =0.001\lambda $ (dashed-dotted plots). The case $\delta =0$ in (

**a**) and $\delta =20\lambda $ in (

**b**).

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**MDPI and ACS Style**

Khalil, E.M.; Mohamed, A.-B.A.; Obada, A.-S.F.; Eleuch, H. Quasi-Probability Husimi-Distribution Information and Squeezing in a Qubit System Interacting with a Two-Mode Parametric Amplifier Cavity. *Mathematics* **2020**, *8*, 1830.
https://doi.org/10.3390/math8101830

**AMA Style**

Khalil EM, Mohamed A-BA, Obada A-SF, Eleuch H. Quasi-Probability Husimi-Distribution Information and Squeezing in a Qubit System Interacting with a Two-Mode Parametric Amplifier Cavity. *Mathematics*. 2020; 8(10):1830.
https://doi.org/10.3390/math8101830

**Chicago/Turabian Style**

Khalil, Eied. M., Abdel-Baset. A. Mohamed, Abdel-Shafy F. Obada, and Hichem Eleuch. 2020. "Quasi-Probability Husimi-Distribution Information and Squeezing in a Qubit System Interacting with a Two-Mode Parametric Amplifier Cavity" *Mathematics* 8, no. 10: 1830.
https://doi.org/10.3390/math8101830