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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 ᵢ (kv) and Sᵢ (kv) over the Maxwellian distribution, while the Zeeman shift was obtained by the average of ᵢ (B) and Sᵢ (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.
Majeed et al. (Wed,) studied this question.