Fig. 1. (a) Schematic diagram for generating nonreciprocal transition: two nondegenerate energy levels |a⟩ and |b⟩ are coupled to one another via a coherent interaction Hcoh, and they are also coupled to the same engineered reservoir. (b) Schematic diagram for implementation of a nonreciprocal transition in a cyclic three-level atom (characterized by |a⟩, |b⟩, and |c⟩). A laser field (ΩabeiΦ) is applied to drive the direct transition between the two levels |a⟩ and |b⟩, and they are also coupled indirectly by the auxiliary level |c⟩ through two laser fields (Ωca and Ωcb), where the decay of level |c⟩ is much faster than that of the other two levels, i.e., γc≫max{γa,γb}, so the auxiliary level |c⟩ serves as a engineered reservoir.

Fig. 2. The transition probabilities Tab(t) and Tba(t) are plotted as functions of the time Ωabt for: (a) Φ=π/2, (b) Φ=0, and (c) Φ=−π/2. (d) The isolation I(t) is plotted as a function of time Ωabt for Φ=π/2,0,−π/2. The other parameters are γa=γb=Ωab/10, γc=100Ωab, Ωca=Ωbc=10Ωab, and Δcb=Δca=Δab=0.

Fig. 3. (a) The transition probabilities Tab(t) and Tba(t) and (b) the isolation I(t) are plotted as functions of the synthetic magnetic flux Φ at time Ωabt=1. The other parameters are γa=γb=Ωab/10, γc=100Ωab, Ωca=Ωbc=10Ωab, and Δcb=Δca=Δab=0.

Fig. 4. Schematic of two 1D semi-infinite CRWs connected by a three-level atom characterized by |a⟩, |b⟩, and |g⟩. CRW-a (CRW-b) couples to the three-level atom through the transition |a⟩↔|g⟩ (|b⟩↔|g⟩) with strength ga (gb).

Fig. 5. (a) Scattering flows Iab (black solid curve) and Iba (red dashed curve), (b) Iaa (black solid curve) and Ibb (red dashed curve), are plotted as functions of the wavenumber k/π for ξ/Γ=0.1. (c) Scattering flow Iab is plotted as a function of the wavenumber k/π for different ξ/Γ. (d) The width of the wavenumber Δk for single-photon nonreciprocity is plotted as a function of log10(ξ/Γ) given in Eq. (9). The other parameters are Jba=2Γ, Jab=0, ξ=Γ, Δa=Δb=0, g2=Γξ, ϕ=π/2.