Fig. 1. Interaction of the laser fields and the multi-level atomic systems. (a) Laser coupling scheme, (b) co-propagation, and (c) counter-propagation setups of the probe field and the strong coupling fields.

Fig. 2. Transmission of the probe field in multi-level atomic systems as a function of the probe detuning Δ_p, where the blue solid line stands for the counter-propagation and the red dash-dotted line for co-propagation. (a) Three-level system with Δ₁ = 0, Ω₁ = 40γ; (b) four-level system with Δ₁ = −20γ, Δ₂ = 20γ, Ω₁ = Ω₂ = 40γ; (c) five-level system with Δ₁ = − 20γ, Δ₂ = 0, Δ₃ = 20γ, Ω₁ = Ω₂ = Ω₃ = 40γ; and (d) six-level system with Δ₁ = − 20γ, Δ₂ = −10γ, Δ₃ = 10γ, Δ₄ = 20γ, Ω₁ = Ω₂ = Ω₃ = Ω₄ = 40γ.

Fig. 3. Tunable nonreciprocal frequency bands in the five-level atomic system with Δ₂ = 0, Ω₁ = Ω₂ = Ω₃ = 40γ, Δ₁ = −30γ, Δ₃ = 30γ and (a) Δ₂ = 20γ; (b) Δ₂ = 0γ; (c) Δ₂ = −20γ. The other parameters are the same as in Fig. 2.

Fig. 4. Variation of transmission of probe fields with (a), (b) Δ_p and Ω₀ or (c), (d) Ω₂, where (a), (c) are the results for the counter-propagating probe field and (b), (d) for the co-propagating probe field. In the calculation, Δ₁ = −30γ, Δ₂ = 0, Δ₃ = 30γ, and the other parameters are the same as in Fig. 2.

Fig. 5. Variations of (a) transmissivity T of the counter-propagating probe field and (b) corresponding transmission contrast η with Rabi frequency of the coupling field Ω₂ at the center frequencies of the three nonreciprocal bands, corresponding to the cases of Figs. 4(c) and 4(d).

Fig. 6. Possible atomic systems and laser coupling schemes in experiments, where probe and control fields are with (a) adjacent frequencies by using Zeeman splitting levels and (b) different frequencies.

Fig. 7. Transmission of the probe field in counter-propagating (blue solid line) and co-propagating (red dashed line) directions by using the scheme in Fig. 6(b), where Δ₁ = −40γ, Δ₂ = −10γ, Δ₃ = 10γ, Δ₄ = 40γ, and other parameters are the same as in Fig. 2(d).