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Abstract

This paper theoretically investigates the manipulation of the Photonic Spin Hall Effect (photonic SHE) using a four-level closed coherent control coupling scheme in a cavity. The atomic system is configured to function as a combined Tripod and Lambda (CTL), Lambda $\Lambda$, and $N$ level model by properly adjusting the control field strengths and their relative phases. The system demonstrates multiple transparency windows in the CTL configuration, allowing the tunable photonic SHE to be used over a wider range of probe field detuning. At probe field resonance under the condition of electromagnetic induced transparency (EIT), the $\Lambda$-type system exhibits photonic SHE similar to the CTL system, showing a maximum upper limit of photonic SHE equal to half of the incident beam waist. This upper limit arises due to zero absorption and dispersion. Control field strengths and atomic density do not influence photonic SHE at resonance for both atomic configurations. Our findings reveal that atomic density and strength of control fields significantly influence photonic SHE in the $N$-type model at resonance, offering additional control parameters for tuning photonic SHE. Finally, the results are equally valid and applicable to conventional $\Lambda$-type and N-type atomic systems, making the findings broadly relevant in cavity atomic systems. The results of angular photonic SHE are also discussed.