# Decoherence Effects in a Three-Level System under Gaussian Process

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## Abstract

**:**

## 1. Introduction

## 2. Model and Dynamics

#### 2.1. Impact of Local Gaussian Noises

#### 2.2. Coherence Measures

## 3. Main Results

#### 3.1. The Noiseless Classical Field

#### 3.2. A Classical Field with Gaussian Noises

#### 3.2.1. A Classical Field with ${\mathcal{FG}}_{n}$

#### 3.2.2. A Classical Field with ${\mathcal{G}}_{n}$

#### 3.2.3. A Classical Field with ${\mathcal{OU}}_{n}$

#### 3.2.4. A Classical Field with ${\mathcal{PL}}_{n}$

#### 3.3. Relative Dynamics

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Blais, A.; Girvin, S.M.; Oliver, W.D. Quantum information processing and quantum optics with circuit quantum electrodynamics. Nat. Phys.
**2020**, 16, 247–256. [Google Scholar] [CrossRef] - Paesani, S.; Borghi, M.; Signorini, S.; Maïnos, A.; Pavesi, L.; Laing, A. Near-ideal spontaneous photon sources in silicon quantum photonics. Nat. Commun.
**2020**, 11, 2505. [Google Scholar] [CrossRef] [PubMed] - Bennett, C.H.; DiVincenzo, D.P. Quantum information and computation. Nature
**2000**, 404, 247–255. [Google Scholar] [CrossRef] [PubMed] - Sakajo, T.; Yokoyama, T. Discrete representations of orbit structures of flows for topological data analysis. Discret. Math. Algorithms Appl.
**2022**, 2250143. [Google Scholar] [CrossRef] - Hirota, O.; Sohma, M.; Fuse, M.; Kato, K. Quantum stream cipher by the Yuen 2000 protocol: Design and experiment by an intensity-modulation scheme. Phys. Rev. A
**2005**, 72, 022335. [Google Scholar] [CrossRef][Green Version] - Gao, X.; Anschuetz, E.R.; Wang, S.T.; Cirac, J.I.; Lukin, M.D. Enhancing generative models via quantum correlations. Phys. Rev. X
**2022**, 12, 021037. [Google Scholar] [CrossRef] - Di Vincenzo, D.P.; Loss, D. Quantum computers and quantum coherence. J. Magn. Magn. Mater.
**1999**, 200, 202–218. [Google Scholar] [CrossRef][Green Version] - Napoli, C.; Bromley, T.R.; Cianciaruso, M.; Piani, M.; Johnston, N.; Adesso, G. Robustness of coherence: An operational and observable measure of quantum coherence. Phys. Rev. Lett.
**2016**, 116, 150502. [Google Scholar] [CrossRef][Green Version] - Mansour, M.; Dahbi, Z. Entanglement of bipartite partly non-orthogonal-spin coherent states. Laser Phys.
**2020**, 30, 085201. [Google Scholar] [CrossRef] - Mansour, M.; Dahbi, Z.; Essakhi, M.; Salah, A. Quantum correlations through spin coherent states. Int. J. Theor. Phys.
**2021**, 60, 2156–2174. [Google Scholar] [CrossRef] - Abd-Rabbou, M.Y.; Metwally, N.; Ahmed, M.M.A.; Obada, A.S. Decoherence and quantum steering of accelerated qubit–qutrit system. Quantum Inf. Process.
**2022**, 21, 363. [Google Scholar] [CrossRef] - Hu, M.L.; Hu, X.; Wang, J.; Peng, Y.; Zhang, Y.R.; Fan, H. Quantum coherence and geometric quantum discord. Phys. Rep.
**2018**, 762, 1–100. [Google Scholar] [CrossRef][Green Version] - Wang, X.L.; Yue, Q.L.; Yu, C.H.; Gao, F.; Qin, S.J. Relating quantum coherence and correlations with entropy-based measures. Sci. Rep.
**2017**, 7, 12122. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bloch, I. Quantum coherence and entanglement with ultracold atoms in optical lattices. Nature
**2008**, 453, 1016–1022. [Google Scholar] [CrossRef] [PubMed] - Zurek, W.H. Preferred states, predictability, classicality and the environment-induced decoherence. Prog. Theor. Phys.
**1993**, 89, 281–312. [Google Scholar] [CrossRef] - Rahman, A.U.; Javed, M.; Ji, Z.; Ullah, A. Probing multipartite entanglement, coherence and quantum information preservation under classical Ornstein-Uhlenbeck noise. J. Phys. A Math. Theor.
**2021**, 55, 025305. [Google Scholar] [CrossRef] - Rahman, A.U.; Noman, M.; Javed, M.; Ullah, A.; Luo, M.X. Effects of classical fluctuating environments on decoherence and bipartite quantum correlations dynamics. Laser Phys.
**2021**, 31, 115202. [Google Scholar] [CrossRef] - Rahman, A.U.; Ji, Z.X.; Zhang, H.G. Demonstration of entanglement and coherence in GHZ-like state when exposed to classical environments with power-law noise. Eur. Phys. J. Plus
**2022**, 137, 440. [Google Scholar] [CrossRef] - Rahman, A.U.; Zidan, N. Quantum memory assisted entropic uncertainty and entanglement dynamics in classical dephasing channels. arXiv
**2021**, arXiv:2111.11312. [Google Scholar] - Rahman, A.U.; Javed, M.; Kenfack, L.T.; Safi, S.K. Multipartite quantum correlations and coherence dynamics subjected to classical environments and fractional Gaussian noise. arXiv
**2021**, arXiv:2111.02220. [Google Scholar] - Abd-Rabbou, M.Y.; Khan, S.; Shamirzaie, M. Quantum fisher information and quantum coherence of an entangled bipartite state interacting with a common classical environment in accelerating frames. Quantum Inf. Process.
**2022**, 21, 218. [Google Scholar] [CrossRef] - Omri, M.; Abd-Rabbou, M.Y.; Khalil, E.M.; Abdel-Khalek, S. Thermal information and teleportation in two-qutrit Heisenberg XX chain model. Alex. Eng. J.
**2022**, 61, 8335–8342. [Google Scholar] [CrossRef] - Abd-Rabbou, M.Y.; Ali, S.I.; Ahmed, M.M.A. Enhancing the information of nonlinear SU (1, 1) quantum systems interacting with a two-level atom. Opt. Quantum Electron.
**2022**, 548, 548. [Google Scholar] [CrossRef] - Haddadi, S.; Ghominejad, M.; Akhound, A.; Pourkarimi, M.R. Entropic uncertainty relation and quantum coherence under Ising model with Dzyaloshinskii–Moriya interaction. Laser Phys. Lett.
**2021**, 18, 085204. [Google Scholar] [CrossRef] - Rahman, A.U.; Haddadi, S.; Pourkarimi, M.R. Tripartite Quantum Correlations under Power-Law and Random Telegraph Noises: Collective Effects of Markovian and Non-Markovian Classical Fields. Ann. Der Phys.
**2022**, 534, 2100584. [Google Scholar] [CrossRef] - Mallick, K.; Marcq, P. On the stochastic pendulum with Ornstein–Uhlenbeck noise. J. Phys. Math. Gen.
**2004**, 37, 4769. [Google Scholar] [CrossRef][Green Version] - Koutsoyiannis, D. The Hurst phenomenon and fractional Gaussian noise made easy. Hydrol. Sci. J.
**2002**, 47, 573–595. [Google Scholar] [CrossRef] - Benedetti, C.; Paris, M.G. Characterization of classical Gaussian processes using quantum probes. Phys. Lett. A
**2014**, 378, 2495–2500. [Google Scholar] [CrossRef][Green Version] - Toth, G.; Apellaniz, I. Quantum metrology from a quantum information science perspective. J. Phys. A Math. Theor.
**2014**, 47, 424006. [Google Scholar] [CrossRef][Green Version] - Liu, J.; Yuan, H.; Lu, X.M.; Wang, X. Quantum Fisher information matrix and multiparameter estimation. J. Phys. A Math. Theor.
**2019**, 53, 023001. [Google Scholar] [CrossRef] - Javed, M.; Khan, S.; Ullah, S.A. Characterization of classical static noise via qubit as probe. Quantum Inf. Process.
**2018**, 17, 53. [Google Scholar] [CrossRef] - Kenfack, L.T.; Tchoffo, M.; Fai, L.C. Estimation of the disorder degree of the classical static noise using three entangled qubits as quantum probes. Phys. Lett. A
**2019**, 383, 1123–1131. [Google Scholar] [CrossRef] - Lu, X.M.; Wang, X.; Sun, C.P. Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A
**2010**, 82, 042103. [Google Scholar] [CrossRef][Green Version] - Li, N.; Luo, S. Entanglement detection via quantum Fisher information. Phys. Rev. A
**2013**, 88, 014301. [Google Scholar] [CrossRef] - Streltsov, A.; Singh, U.; Dhar, H.S.; Bera, M.N.; Adesso, G. Measuring quantum coherence with entanglement. Phys. Rev. Lett.
**2015**, 115, 020403. [Google Scholar] [CrossRef][Green Version] - Hu, M.; Zhou, W. Enhancing two-qubit quantum coherence in a correlated dephasing channel. Laser Phys. Lett.
**2019**, 16, 045201. [Google Scholar] [CrossRef] - Nirwan, R.S.; Bertschinger, N. Applications of Gaussian process latent variable models in finance. In Proceedings of the SAI Intelligent Systems Conference, London, UK, 5–6 September 2019; Springer: Cham, Switzerland, 2019; pp. 1209–1221. [Google Scholar]
- Lazaro-Gredilla, M.; Titsias, M.K. Variational heteroscedastic Gaussian process regression. In Proceedings of the ICML, Bellevue, WA, USA, 28 June–2 July 2011. [Google Scholar]
- Schwab, D. Efficacy of Gaussian Process Regression for Angles-Only Initial Orbit Determination. Master’s Thesis, Penn State University, State College, PA, USA, 2020. [Google Scholar]
- Sharifzadeh, M.; Sikinioti-Lock, A.; Shah, N. Machine-learning methods for integrated renewable power generation: A comparative study of artificial neural networks, support vector regression, and Gaussian Process Regression. Renew. Sustain. Energy Rev.
**2019**, 108, 513–538. [Google Scholar] [CrossRef] - Ibrahim, S.K.; Ahmed, A.; Zeidan, M.A.E.; Ziedan, I.E. Machine learning methods for spacecraft telemetry mining. IEEE Trans. Aerosp. Electron. Syst.
**2018**, 55, 1816–1827. [Google Scholar] [CrossRef] - Rodrigues, F.; Pereira, F.; Ribeiro, B. Gaussian process classification and active learning with multiple annotators. In Proceedings of the International Conference on Machine Learning, PMLR, Bejing, China, 22–24 June 2014; pp. 433–441. [Google Scholar]
- Taqqu, M.S.; Teverovsky, V.; Willinger, W. Estimators for long-range dependence: An empirical study. Fractals
**1995**, 3, 785–798. [Google Scholar] [CrossRef] - Prasad, S.K.; Aghajarian, D.; McDermott, M.; Shah, D.; Mokbel, M.; Puri, S.; Wang, S. Parallel processing over spatial-temporal datasets from geo, bio, climate and social science communities: A research roadmap. In Proceedings of the 2017 IEEE International Congress on Big Data (BigData Congress), Honolulu, HI, USA, 25–30 June 2017; pp. 232–250. [Google Scholar]
- Pelletier, J.D.; Turcotte, D.L. Long-range persistence in climatological and hydrological time series: Analysis, modelling and application to drought hazard assessment. J. Hydrol.
**1997**, 203, 198–208. [Google Scholar] [CrossRef] - Maxim, V.; Şendur, L.; Fadili, J.; Suckling, J.; Gould, R.; Howard, R.; Bullmore, E. Fractional Gaussian noise, functional MRI and Alzheimer’s disease. Neuroimage
**2005**, 25, 141–158. [Google Scholar] [CrossRef] [PubMed][Green Version] - Paxson, V. Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic. ACM SIGCOMM Comput. Commun. Rev.
**1997**, 27, 5–18. [Google Scholar] [CrossRef] - De Moura, C.E.; Pizzinga, A.; Zubelli, J. A pairs trading strategy based on linear state space models and the Kalman filter. Quant. Financ.
**2016**, 16, 1559–1573. [Google Scholar] [CrossRef] - Blekos, K.; Stefanatos, D.; Paspalakis, E. Performance of superadiabatic stimulated Raman adiabatic passage in the presence of dissipation and Ornstein-Uhlenbeck dephasing. Phys. Rev. A
**2020**, 102, 023715. [Google Scholar] [CrossRef] - Koch, C.P.; Boscain, U.; Calarco, T.; Dirr, G.; Filipp, S.; Glaser, S.J.; Wilhelm, F.K. Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe. EPJ Quantum Tech.
**2022**, 9, 19. [Google Scholar] - Stefanatos, D.; Paspalakis, E. A shortcut tour of quantum control methods for modern quantum technologies. Europhys. Lett.
**2021**, 132, 60001. [Google Scholar] [CrossRef] - Burgess, A.E.; Judy, P.F. Signal detection in power-law noise: Effect of spectrum exponents. J. Opt. Soc. Am. A
**2007**, 24, B52–B60. [Google Scholar] [CrossRef] [PubMed] - Burgess, A.E.; Jacobson, F.L.; Judy, P.F. Human observer detection experiments with mammograms and power-law noise. Med. Phys.
**2001**, 28, 419–437. [Google Scholar] [CrossRef] - Lam, C.H.; Sander, L.M. Surface growth with power-law noise. Phys. Rev. Lett.
**1992**, 69, 3338. [Google Scholar] [CrossRef] - Molina-Garcia, D.; Sandev, T.; Safdari, H.; Pagnini, G.; Chechkin, A.; Metzler, R. Crossover from anomalous to normal diffusion: Truncated power-law noise correlations and applications to dynamics in lipid bilayers. New J. Phys.
**2018**, 20, 103027. [Google Scholar] [CrossRef][Green Version] - Abd-Rabbou, M.Y.; Metwally, N.; Ahmed, M.M.A.; Obada, A.S. Wigner function of noisy accelerated two-qubit system. Quantum Inf. Process.
**2019**, 18, 367. [Google Scholar] [CrossRef][Green Version] - Rossi, M.A.; Benedetti, C.; Paris, M.G. Engineering decoherence for two-qubit systems interacting with a classical environment. Int. J. Quantum Inf.
**2014**, 12, 1560003. [Google Scholar] [CrossRef][Green Version] - Masoomy, H.; Askari, B.; Najafi, M.N.; Movahed, S.M.S. Persistent homology of fractional Gaussian noise. Phys. Rev. E
**2021**, 104, 034116. [Google Scholar] [CrossRef] [PubMed] - Ledesma, S.; Liu, D. Synthesis of fractional Gaussian noise using linear approximation for generating self-similar network traffic. ACM SIGCOMM Comput. Commun. Rev.
**2000**, 30, 4–17. [Google Scholar] [CrossRef] - Luft, M.; Cioc, R.; Pietruszczak, D. Fractional calculus in modelling of measuring transducers. Elektron. Elektrotech.
**2011**, 110, 97–100. [Google Scholar] [CrossRef][Green Version] - Guo, X.; Liu, F.; Tian, X. Gaussian noise level estimation for color image denoising. JOSA A
**2021**, 38, 1150–1159. [Google Scholar] [CrossRef] - Merhav, N.; Guo, D.; Shamai, S. Statistical physics of signal estimation in Gaussian noise: Theory and examples of phase transitions. IEEE Trans. Inf. Theory
**2010**, 56, 1400–1416. [Google Scholar] [CrossRef][Green Version] - Amar, J.G.; Family, F. Scaling of surface fluctuations and dynamics of surface growth models with power-law noise. J. Phys. A Math. Gen.
**1991**, 24, L79. [Google Scholar] [CrossRef] - Kasdin, N.J. Discrete simulation of colored noise and stochastic processes and 1/f/sup/spl alpha//power law noise generation. Proc. IEEE
**1995**, 83, 802–827. [Google Scholar] [CrossRef] - Zhao, M.J.; Ma, T.; Quan, Q.; Fan, H.; Pereira, R. l
_{1}-norm coherence of assistance. Phys. Rev. A**2019**, 100, 012315. [Google Scholar] [CrossRef][Green Version] - Mazzola, L.; Piilo, J.; Maniscalco, S. Frozen discord in non-Markovian dephasing channels. Int. J. Quantum Inf.
**2011**, 9, 981–991. [Google Scholar] [CrossRef] - Benedetti, C.; Paris, M.G.; Buscemi, F.; Bordone, P. Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f
^{α}noise. In Proceedings of the 2013 22nd International Conference on Noise and Fluctuations (ICNF), Montpellier, France, 24–28 June 2013; pp. 1–4. [Google Scholar] - Kenfack, L.T.; Tchoffo, M.; Javed, M.; Fai, L.C. Dynamics and protection of quantum correlations in a qubit–qutrit system subjected locally to a classical random field and colored noise. Quantum Inf. Process.
**2020**, 19, 1–26. [Google Scholar] [CrossRef] - Essakhi, M.; Khedif, Y.; Mansour, M.; Daoud, M. Intrinsic decoherence effects on quantum correlations dynamics. Opt. Quantum Electron.
**2022**, 54, 103. [Google Scholar] [CrossRef] - Benedetti, M.; Garcia-Pintos, D.; Perdomo, O.; Leyton-Ortega, V.; Nam, Y.; Perdomo-Ortiz, A. A generative modeling approach for benchmarking and training shallow quantum circuits. NPJ Quantum Inf.
**2019**, 5, 45. [Google Scholar] [CrossRef][Green Version] - Sweke, R.; Wilde, F.; Meyer, J.J.; Schuld, M.; Fährmann, P.K.; Meynard-Piganeau, B.; Eisert, J. Stochastic gradient descent for hybrid quantum-classical optimization. Quantum
**2020**, 4, 314. [Google Scholar] [CrossRef] - Ji, Z.; Zhang, H.; Wang, H.; Wu, F.; Jia, J.; Wu, W. Quantum protocols for secure multi-party summation. Quantum Inf. Process.
**2019**, 18, 168. [Google Scholar] [CrossRef] - Khedif, Y.; Haddadi, S.; Pourkarimi, M.R.; Daoud, M. Thermal correlations and entropic uncertainty in a two-spin system under DM and KSEA interactions. Mod. Phys. Lett. A
**2021**, 36, 2150209. [Google Scholar] [CrossRef] - Haddadi, S.; Pourkarimi, M.R.; Haseli, S. Relationship between quantum coherence and uncertainty bound in an arbitrary two-qubit X-state. Opt. Quantum Electron.
**2021**, 53, 529. [Google Scholar] [CrossRef] - Khedif, Y.; Daoud, M.; Sayouty, E.H. Thermal quantum correlations in a two-qubit Heisenberg XXZ spin-chain under an inhomogeneous magnetic field. Phys. Scr.
**2019**, 94, 125106. [Google Scholar] [CrossRef] - Yu, T.; Eberly, J.H. Sudden death of entanglement: Classical noise effects. Opt. Commun.
**2006**, 264, 393–397. [Google Scholar] [CrossRef][Green Version] - Kenfack, L.T.; Tchoffo, M.; Fai, L.C.; Fouokeng, G.C. Decoherence and tripartite entanglement dynamics in the presence of Gaussian and non-Gaussian classical noise. Phys. B Condens. Matter
**2017**, 511, 123–133. [Google Scholar] [CrossRef] - Rahman, A.U.; Noman, M.; Javed, M.; Luo, M.X.; Ullah, A. Quantum correlations of tripartite entangled states under Gaussian noise. Quantum Inf. Process.
**2021**, 20, 290. [Google Scholar] [CrossRef] - Rahman, A.U.; Javed, M.; Ullah, A.; Ji, Z. Probing tripartite entanglement and coherence dynamics in pure and mixed independent classical environments. Quantum Inf. Process.
**2021**, 20, 321. [Google Scholar] [CrossRef] - Rahman, A.U.; Noman, M.; Javed, M.; Ullah, A. Dynamics of bipartite quantum correlations and coherence in classical environments described by pure and mixed Gaussian noises. Eur. Phys. J. Plus
**2021**, 136, 846. [Google Scholar] [CrossRef] - Rossi, M.A.; Paris, M.G. Non-Markovian dynamics of single-and two-qubit systems interacting with Gaussian and non-Gaussian fluctuating transverse environments. J. Chem. Phys.
**2016**, 144, 024113. [Google Scholar] [CrossRef] [PubMed][Green Version] - Buscemi, F.; Bordone, P. Time evolution of tripartite quantum discord and entanglement under local and nonlocal random telegraph noise. Phys. Rev. A
**2013**, 87, 042310. [Google Scholar] [CrossRef][Green Version] - Weinstein, Y.S. Tri-partite Entanglement Witnesses and Sudden Death. arXiv
**2008**, arXiv:0812.4612. [Google Scholar] - Hao, Y.; Lian-Fu, W. Correlation dynamics of two-parameter qubit—Qutrit states under decoherence. Chin. Phys. B
**2013**, 22, 050303. [Google Scholar] - Shamirzaie, M.; Khan, S. The Dynamics of Three Different Entropic Measures of Quantum Correlations in Mixed Bipartite State Coupled with Classical Environments. Fluct. Noise Lett.
**2018**, 17, 1850023. [Google Scholar] [CrossRef] - Rahman, A.U.; Khedif, Y.; Javed, M.; Ali, H.; Daoud, M. Characterizing Two-Qubit Non-Classical Correlations and Non-Locality in Mixed Local Dephasing Noisy Channels. Ann. Der Phys.
**2022**, 534, 2200197. [Google Scholar] [CrossRef]

**Figure 1.**The current configuration model depicts the coupling of a three-level system $qt$ exposed to a classical fluctuation field $\mathcal{EL}\left(t\right)$. The system–environment coupling strength $\omega $ is shown by the blue-reddish wavy lines, while the noise’s influence is represented by the yellowish light in the qutrit. The brownish-wavy lines depict system dynamics as defined by the associated environment’s stochastic parameter $\eta \left(t\right)$, with diminishing amplitude showing Gaussian noise-induced dephasing.

**Figure 2.**Time evolution of coherence in a single qutrit system prepared in the time-evolved state ${\rho}_{qt}\left(t\right)$ given in Equation (19) subjected to a noiseless classical channel when (

**a**) $\eta =1$, $0\le \omega \le 1$ and (

**b**) $0\le \eta \le 1$, $\omega =0.5$ against the time evolution parameter t.

**Figure 3.**Time evolution of coherence in a single qutrit system prepared in the time-evolved state ${\rho}_{qt}\left(\tau \right)$ given in Equation (19) when subjected to the classical field generating (

**a**) fractional Gaussian noise when $0\le H\le 1$ and (

**b**) Gaussian noise when $0\le g\le 1$, (

**c**) Ornstein–Uhlenbeck noise when $0\le g\le 1$ and (

**d**) power law noise when $2\le \alpha \le 4$ with $g=1$ against evolution parameter $\tau =3$.

**Figure 4.**Time evolution of (

**a**) purity and (

**b**) von Neumann entropy as functions of H versus $\tau $ in a single qutrit system when subjected to the classical field generating fractional Gaussian noise.

**Figure 5.**Time evolution of (

**a**) purity and (

**b**) von Neumann entropy as functions of H versus $\tau $ in a single qutrit system when subjected to the classical field generating Gaussian noise.

**Figure 6.**Time evolution of (

**a**) purity and (

**b**) von Neumann entropy as functions of H versus $\tau $ in a single qutrit system when subjected to the classical field generating Ornstein–Uhlenbeck noise.

**Figure 7.**Upper Panel: Time evolution of (

**a**) purity and (

**b**) von Neumann entropy as functions of g versus $\tau $ in a single qutrit system when subjected to the classical field generating power law noise when $\alpha =3$. Bottom panel: Time evolution of (

**c**) purity and (

**d**) von Neumann entropy as functions of $\alpha $ versus $\tau $ in a single qutrit system when subjected to the classical field generating power law noise when $g=0.5$.

**Figure 8.**Prolonged preservation of (

**a**) purity and (

**b**) von Neumann entropy as functions of g versus $\tau $ in a single qutrit system under Gaussian (green), Ornstein–Uhlenbeck (blue), and power law noise (red) stemming from the classical field when $g={10}^{-3}$ (non-dashed) and $g={10}^{-2}$ (dashed).

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**MDPI and ACS Style**

Zangi, S.M.; ur Rahman, A.; Ji, Z.-X.; Ali, H.; Zhang, H.-G.
Decoherence Effects in a Three-Level System under Gaussian Process. *Symmetry* **2022**, *14*, 2480.
https://doi.org/10.3390/sym14122480

**AMA Style**

Zangi SM, ur Rahman A, Ji Z-X, Ali H, Zhang H-G.
Decoherence Effects in a Three-Level System under Gaussian Process. *Symmetry*. 2022; 14(12):2480.
https://doi.org/10.3390/sym14122480

**Chicago/Turabian Style**

Zangi, Sultan M., Atta ur Rahman, Zhao-Xo Ji, Hazrat Ali, and Huan-Guo Zhang.
2022. "Decoherence Effects in a Three-Level System under Gaussian Process" *Symmetry* 14, no. 12: 2480.
https://doi.org/10.3390/sym14122480