Analysis of exceptional surfaces via Doppler- and Zeeman-affected non-Hermitian Hamiltonian in a three-level atomic system

Abdul Majeed; Obaid J. Algahtani; Imdad Ullah; Amir Ali; Dragan Pamucar; Abdul Majeed; Obaid J. Algahtani; Imdad Ullah; Amir Ali; Dragan Pamucar

doi:10.3934/math.2025680

AIMS Mathematics

2025, Volume 10, Issue 7: 15165-15192. doi: 10.3934/math.2025680

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Analysis of exceptional surfaces via Doppler- and Zeeman-affected non-Hermitian Hamiltonian in a three-level atomic system

1.
Department of Physics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia
3.
Department of Mathematics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, Pakistan
4.
Department of Operations Research and Statistics, Faculty of Organizational Sciences, University of Belgrade, Belgrade, Serbia
5.
Department of Industrial Engineering & Management, Yuan Ze University, Taoyuan City 320315, Taiwan
6.
Department of Applied Mathematical Science, College of Science and Technology, Korea University, Sejong 30019, Republic of Korea

Received: 22 November 2024 Revised: 05 February 2025 Accepted: 11 February 2025 Published: 01 July 2025
MSC : 15A18, 45A05, 45C05, 70H05, 78A10, 81-10, 81Q93

The exceptional surface of the Doppler- and Zeeman-affected non-Hermitian Hamiltonian of a three-level atomic medium, driven by one probe and two control fields, was controlled and modified. Multiple exceptional surfaces of the second and third orders were reported based on the real part of eigenstates and the real and imaginary parts of the eigenvalues of the non-Hermitian Hamiltonian. The exceptional surfaces and degeneracy regions were studied with variations in detunings, Rabi frequencies, decay rates, phases, and Doppler and Zeeman widths. The effect of the Doppler shift was obtained by the average of $ \lambda_i(kv) $ and $ S_i(kv) $ over the Maxwellian distribution, while the Zeeman shift was obtained by the average of $ \lambda_i(\mu B) $ and $ S_i(\mu B) $ over the Gaussian distribution. The exceptional surfaces shift with both Zeeman and Doppler shift. The degeneracy increases with an increase in the Zeeman effect. With an increase in Doppler width, the degeneracy in the real part of the eigenvalues also increases but decreases in the imaginary part of eigenvalues. The maximum range of degeneracy of Zeeman-affected exceptional surfaces was investigated with decay rates and Rabi frequencies of the coupled driving fields, while for Doppler-affected exceptional surfaces, it was investigated with detuning and Rabi frequencies. The modified results of exceptional surfaces are useful for hybrid quantum systems, artificial intelligence, entanglement physics, quantum state sensing, and optics.
- exceptional surfaces,
- non-Hermitian Hamiltonian,
- Doppler and Zeeman effect,
- eigenstates,
- eigenvalues,
- three-level atomic system
Citation: Abdul Majeed, Obaid J. Algahtani, Imdad Ullah, Amir Ali, Dragan Pamucar. Analysis of exceptional surfaces via Doppler- and Zeeman-affected non-Hermitian Hamiltonian in a three-level atomic system[J]. AIMS Mathematics, 2025, 10(7): 15165-15192. doi: 10.3934/math.2025680

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Abstract

The exceptional surface of the Doppler- and Zeeman-affected non-Hermitian Hamiltonian of a three-level atomic medium, driven by one probe and two control fields, was controlled and modified. Multiple exceptional surfaces of the second and third orders were reported based on the real part of eigenstates and the real and imaginary parts of the eigenvalues of the non-Hermitian Hamiltonian. The exceptional surfaces and degeneracy regions were studied with variations in detunings, Rabi frequencies, decay rates, phases, and Doppler and Zeeman widths. The effect of the Doppler shift was obtained by the average of $ \lambda_i(kv) $ and $ S_i(kv) $ over the Maxwellian distribution, while the Zeeman shift was obtained by the average of $ \lambda_i(\mu B) $ and $ S_i(\mu B) $ over the Gaussian distribution. The exceptional surfaces shift with both Zeeman and Doppler shift. The degeneracy increases with an increase in the Zeeman effect. With an increase in Doppler width, the degeneracy in the real part of the eigenvalues also increases but decreases in the imaginary part of eigenvalues. The maximum range of degeneracy of Zeeman-affected exceptional surfaces was investigated with decay rates and Rabi frequencies of the coupled driving fields, while for Doppler-affected exceptional surfaces, it was investigated with detuning and Rabi frequencies. The modified results of exceptional surfaces are useful for hybrid quantum systems, artificial intelligence, entanglement physics, quantum state sensing, and optics.

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