• Photonics Research
  • Vol. 9, Issue 5, 879 (2021)
Xunwei Xu1、2、*, Yanjun Zhao3, Hui Wang4, Aixi Chen5、8, and Yu-Xi Liu6、7
Author Affiliations
  • 1Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China
  • 2Department of Applied Physics, East China Jiaotong University, Nanchang 330013, China
  • 3Key Laboratory of Opto-electronic Technology, Ministry of Education, Beijing University of Technology, Beijing 100124, China
  • 4Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan
  • 5Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
  • 6Institute of Microelectronics, Tsinghua University, Beijing 100084, China
  • 7Frontier Science Center for Quantum Information, Beijing 100084, China
  • 8e-mail: aixichen@zstu.edu.cn
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    DOI: 10.1364/PRJ.412904 Cite this Article Set citation alerts
    Xunwei Xu, Yanjun Zhao, Hui Wang, Aixi Chen, Yu-Xi Liu. Nonreciprocal transition between two nondegenerate energy levels[J]. Photonics Research, 2021, 9(5): 879 Copy Citation Text show less

    Abstract

    Stimulated emission and absorption are two fundamental processes of light–matter interaction, and the coefficients of the two processes should be equal. However, we will describe a generic method to realize the significant difference between the stimulated emission and absorption coefficients of two nondegenerate energy levels, which we refer to as a nonreciprocal transition. As a simple implementation, a cyclic three-level atom system, comprising two nondegenerate energy levels and one auxiliary energy level, is employed to show a nonreciprocal transition via a combination of synthetic magnetism and reservoir engineering. Moreover, a single-photon nonreciprocal transporter is proposed using two one-dimensional semi-infinite coupled-resonator waveguides connected by an atom with nonreciprocal transition effect. Our work opens up a route to design atom-mediated nonreciprocal devices in a wide range of physical systems.

    1. INTRODUCTION

    According to Einstein’s phenomenological radiation theory [1], the absorption coefficient should be equal to the stimulated emission coefficient between two nondegenerate energy levels. When the spontaneous emission can be neglected, a two-level system undergoes optical Rabi oscillations under the action of a coherent driving electromagnetic field [2]. However, can we make the absorption coefficient different from the stimulated emission coefficient for the transition between two energy levels with different eigenvalues, i.e., nonreciprocal transition between two nondegenerate energy levels? The answer is yes. In this paper, we describe a generic method to realize a nonreciprocal transition between two nondegenerate energy levels, and we show that the absorption and stimulated emission coefficients can be controlled via a combination of synthetic magnetism and reservoir engineering.

    Theoretical research has shown that [3] a combination of synthetic magnetism and reservoir engineering can be used to implement nonreciprocal photon transmission and amplification in coupled photonic systems, and this has been confirmed by a recent experiment [4]. Based on a similar mechanism, many different schemes for nonreciprocal photon transport are proposed theoretically [58] and implemented experimentally [912]. Synthetic magnetism is an effective approach to achieve nonreciprocal transport of uncharged particles, such as photons [1316] or phonons [17,18], for potential applications in simulating quantum many-body phenomena [1925] and creating devices robust against disorder and backscattering [2630]. Reservoir engineering [31] has been a significant subject for generating useful quantum behavior by specially designing the couplings between a system of interest and a structured dissipative environment, such as cooling mechanical harmonic oscillators [32]; synthesizing quantum harmonic oscillator states [33]; and generating state-dependent photon blockades [34], stable entanglement between two nanomechanical resonators [35,36], and squeezed states of nanomechanical resonators [3739].

    In this paper, we introduce the concept of nonreciprocity to investigate the transitions between different energy levels and generalize the general strategy for nonreciprocal photon transmission [3] to atomic systems to achieve a nonreciprocal transition between two nondegenerate energy levels. As a simple implementation, a cyclic three-level atom system, comprising two nondegenerate energy levels and one auxiliary energy level, is employed to show a nonreciprocal transition via a combination of synthetic magnetism and reservoir engineering.

    In application, the atomic systems with nonreciprocal transitions allow one to generate nonreciprocal devices. In this paper, a single-photon nonreciprocal transporter is proposed in a system of two one-dimensional (1D) semi-infinite coupled-resonator waveguides (CRWs) connected by an atom based on the nonreciprocal transition effect. The nonreciprocal transition effect provides a new routine to design atom-mediated nonreciprocal devices in a variety of physical systems.

    2. GENERAL METHOD FOR NONRECIPROCAL TRANSITION

    (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.

    Figure 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., γcmax{γa,γb}, so the auxiliary level |c serves as a engineered reservoir.

    The second method is through coupling to a common engineered reservoir. A dissipative interaction Hdis between the two levels can be obtained by adiabatically eliminating the engineered reservoir. The effective Hamiltonian for the dissipative interaction Hdis can be written in a non-Hermitian form as Hdis=iγ(|ab|+|ba|) with positive real strength γ. This dissipative version of interaction can be implemented by an auxiliary energy level, which is damping much faster than the two levels. The details of the realization will be shown in the next section.

    Based on the two distinct methods, the total Hamiltonian for the interaction between the two levels is Hcoh+dis=(Ωiγ)|ab|+(Ω*iγ)|ba|.When Ω=iγ and Ω*Ω, there is only transition |a|b but |b|a. Instead, when Ω*=iγ and Ω*Ω, there is only transition |b|a but |a|b.

    3. NONRECIPROCAL TRANSITION WITH CYCLIC THREE-LEVEL TRANSITION

    To make the method more concrete, we show how to implement nonreciprocal transition in a cyclic three-level atom as depicted in Fig. 1(b). We consider a cyclic three-level atom (|a, |b, and |c) driven by three classical coherent fields (at rates Ωij, frequencies νij, phases ϕij, with i,j=a,b,c) that is described by a Hamiltonian (see Appendix A): H=(Δabiγa)|aa|iγb|bb|+(Δcbiγc)|cc|+(ΩabeiΦ|ab|+Ωcb|cb|+Ωca|ca|+H.c.),where Δij=ωijνij (i,j=a,b,c), ωij is the frequency difference between levels |i and |j; γi (i=a,b,c) are the decay rates. We assume that νab=νcbνca, so the detuning Δab=ΔcbΔca. The synthetic magnetic flux Φϕabϕcb+ϕca is the total phase of the three driving fields around the cyclic three-level atom and independent of the local redefinition of states |i. The time-reversal symmetry of the system is broken when we choose the phase Φnπ (n is an integer) even without spontaneous emissions (γa=γb=γc=0), and this is one of the key ingredients for nonreciprocal transition. In addition, we assume that the decays satisfy the conditions min{ωca,ωcb}γcmax{Ωca,Ωcb,γa,γb}, so that level |c serves as an engineered reservoir.

    In order to show the nonreciprocal transition between levels |a and |b intuitively, we can derive an effective Hamiltonian by eliminating level |c (the engineered reservoir) adiabatically (see Appendix B) under the assumption that γcmax{γa,γb}. Then an effective Hamiltonian only including levels |a and |b is given by Heff=(ΔaiΓa)|aa|+(ΔbiΓb)|bb|+Jab|ab|+Jba|ba|,with the detunings ΔaΔabΩca2Δcb/(γc2+Δcb2) and ΔbΩcb2Δcb/(γc2+Δcb2), effective decay rates Γaγa+Ωca2γc/(γc2+Δcb2) and Γbγb+Ωcb2γc/(γc2+Δcb2), and effective coupling coefficients JabΩabeiΦiΩcaΩcb(γciΔcb)γc2+Δcb2,JbaΩabeiΦiΩcaΩcb(γciΔcb)γc2+Δcb2.The effective coupling coefficients Jab and Jba include two terms: the first term comes from the coherent driving field and is dependent on the synthetic magnetic flux Φ, and the second term is induced by the auxiliary level |c. Under the resonant condition Δab=Δca=Δcb=0, the second term becomes purely imaginary, i.e., iΩcaΩcb/γc, and the effective Hamiltonian is the same as Eq. (2). Under the resonant conditions, the perfect nonreciprocal transition, i.e., Jab=0 and Jba0 (or Jba=0 and Jab0), is obtained when Φ=π/2 (or Φ=π/2) with Ωab=ΩcaΩcb/γc. More generally, we have |Jab|<|Jba| for 0<Φ<π and |Jab|>|Jba| for π<Φ<0.

    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.

    Figure 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.

    (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.

    Figure 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.

    4. SINGLE-PHOTON NONRECIPROCAL TRANSPORT

    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).

    Figure 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).

    The efficiency for nonreciprocity transport can be described by the scattering flow [4043] Ill for a single photon from CRW-l to CRW-l (l=a,b). The detailed calculations of the scattering flow Ill can be found in Appendix C. Nonreciprocal single-photon transport appears when IbaIab, which implies that the scattering flow from CRW-a to CRW-b is not equal to that along the opposite direction.

    (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.

    Figure 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.

    Now let us discuss the width of the wavenumber for single-photon nonreciprocity; see Appendix E. We define the width of the wavenumber Δk for single-photon nonreciprocity as the full width at half-maximum (FWHM) by setting Iba=1/2 for k=khalf[0,π/2): Δkπ2khalf.Under the conditions |Jba|=2Γ and g2=Γξ, there is a maximum FWHM for single-photon nonreciprocity at ξ=Γ/2, and the maximum FWHM Δkmax0.81π is obtained with khalf=arcsin(221222) in excellent agreement with Figs. 5(c) and 5(d).

    5. CONCLUSIONS AND DISCUSSION

    In summary, we have shown theoretically that nonreciprocal transition can be observed between two nondegenerate energy levels. A general method has been presented to realize nonreciprocal transition between two nondegenerate energy levels based on a combination of synthetic magnetism and reservoir engineering. As a simple example, we explicitly show an implementation involving an auxiliary energy level, i.e., a cyclic three-level atom system. The generic method for realizing a nonreciprocal transition can be applied to design nonreciprocal phonon devices. A single-photon nonreciprocal transporter has been proposed by the nonreciprocal transition effect. The atom-mediated nonreciprocal devices based on the nonreciprocal transition are suitable for applications in building hybrid quantum networks.

    To realize a nonreciprocal transition with a cyclic three-level atom, one ingredient is breaking the symmetry of the potential of the atom. The cyclic three-level transition has been proposed and observed in chiral molecules [4451]. In addition, the potential of the atom can also be broken by applying an external magnetic field. We can consider a qubit circuit composed of a superconducting loop with three Josephson junctions [52,53] that encloses an applied magnetic flux Φe=fΦ0 (Φ0h/2e is the superconducting flux quantum, where h is Planck’s constant and fΦe/Φ0 is the reduced magnetic flux; e is the charge quantity of one electron). When the reduced magnetic flux f is a half-integer, the potential of the artificial atom is symmetric, and the interaction Hamiltonian has odd parity. However, when f is not a half-integer, the symmetry of the potential is broken, and the interaction Hamiltonian does not have well-defined parity. In this case, transitions can occur between any two levels.

    Alternatively, cyclic transitions in a three-level atom can be realized by a single nitrogen-vacancy (NV) center embedded in a mechanical resonator [54]. Three eigenstates (|0 and |±1) of the spin operator along the NV’s symmetry axis z (i.e., Sz|m=m|m) are selected as a three-level atom [55,56]. The two degenerate levels |±1 can be split by applying an external magnetic field along z. We can use microwave magnetic fields to drive the transitions between |0 and |±1; the magnetic dipole-forbidden transition |+1|1 can be driven by a time-varying strain field through the mechanical resonator [57,58].

    Besides the implementations in a cyclic three-level atom, the nonreciprocal transition can also be implemented in the other physical systems, such as a four-level atom system [59], two qubits in a one-dimensional waveguide [60], and even qubit arrays [61]. The nonreciprocal transition can be extended to explore lasing without inversion [6265], quantum nonreciprocal physics [6668], and topological phases [69] in a single multilevel atom or qubit array.

    APPENDIX A: HAMILTONIAN FOR CYCLIC THREE-LEVEL ATOM

    We consider a cyclic three-level atom (|a?, |b?, and |c?) driven by three classical coherent fields (at rates Ωij, phases ?ij, frequencies νij, with i,j=a,b,c and νcb=νab+νca) that is described by a Hamiltonian given by H?=(ωab?iγa)|a??a|?iγb|b??b|+(ωcb?iγc)|c??c|+(Ωabei?abe?iνabt|a??b|+Ωcbei?cbe?iνcbt|c??b|+Ωcaei?cae?iνcat|c??a|+H.c.),where ωij is the frequency difference between levels |i? and |j?, and the three levels can decay to the other levels with the decay rates γi(i=a,b,c).

    In the rotating frame respect to the operator W=e?i(νab|a??a|+νcb|c??c|)t, we have H=W?H?W+idW?dtW=(Δab?iγa)|a??a|?iγb|b??b|+(Δcb?iγc)|c??c|+Ωabei?ab|a??b|+Ωcbei?cb|c??b|+Ωcaei?ca|c??a|+H.c.,with the detuning Δijωij?νij(i,j=a,b,c). By local redefinition of the eigenstates, i.e.,?ei?cb?b|?b| and e?i?ca|a?|a?, the Hamiltonian can be rewritten as Eq.?(2) in the main text with the synthetic magnetic flux Φ?ab??cb+?ca.

    APPENDIX B: ADIABATIC ELIMINATION

    We will derive the effective Hamiltonian Eq.?(3) by eliminating the level |c? (the engineered reservoir) adiabatically. The state vector for these three levels at time t can be written as |ψ?=A(t)|a?+B(t)|b?+C(t)|c?.The coefficients |A(t)|2, |B(t)|2, and |C(t)|2 denote occupying probabilities in states |a?, |b?, and |c?, respectively. Then the dynamical behaviors for the coefficients can be obtained by the Schr?dinger equation, i.e.,?i|ψ?=H|ψ?, given by A˙(t)=(?iΔab?γa)A(t)?iΩabeiΦB(t)?iΩcaC(t),B˙(t)=?γbB(t)?iΩabe?iΦA(t)?iΩcbC(t),C˙(t)=(?iΔcb?γc)C(t)?iΩcaA(t)?iΩcbB(t).Under the assumption that the decay of the state |c? is much faster than decay of the states |a? and |b? with the conditions min{ωca,ωcb}?γc?max{Ωca,Ωcb,γa,γb}, we can adiabatically eliminate level |c? with C˙(t)=0 as C(t)=?iΩcaγc+iΔcbA(t)?iΩcbγc+iΔcbB(t).By substituting Eq.?(B5) into Eqs.?(B2) and (B3), then the dynamical equations of A(t) and B(t) become A˙(t)=?[i(Δab?Ωca2Δcbγc2+Δcb2)+(γa+Ωca2γcγc2+Δcb2)]A(t)?[iΩabeiΦ+ΩcaΩcb(γc?iΔcb)γc2+Δcb2]B(t),B˙(t)=?[?iΩcb2Δcbγc2+Δcb2+(γb+Ωcb2γcγc2+Δcb2)]B(t)?[iΩabe?iΦ+ΩcaΩcb(γc?iΔcb)γc2+Δcb2]A(t).Physically, the dynamic equations in Eqs.?(B6) and (B7) correspond to the Schr?dinger evolution of the effective Hamiltonian Eq.?(3) in the main text.

    APPENDIX C: SCATTERING FLOW

    To study the nonreciprocal single-photon transport, we discuss the scattering of a single photon in the system with the total Hamiltonian in the rotating reference frame with respect to Hrot as Htot=l=a,bHl+H?eff+Hint,where the Hamiltonians Hl, H?eff, and Hint are given in Eqs.?(6)–(8) in the main text. As the total number of photons in the system is a conserved quantity (without dissipation), we consider the stationary eigenstate of a single photon in the system as |E?=j=0+[ua(j)aj?+ub(j)bj?]|g,0?+A|a,0?+B|b,0?,where |0? indicates the vacuum state of the 1D semi-infinite CRWs, ul(j) denotes the probability amplitude in the state with a single photon in the jth cavity of the CRW-l, and A (B) denotes the probability amplitude in the atom state |a? (|b?). Substituting the stationary eigenstate |E? in Eq.?(C2) and the total Hamiltonian Htot into the eigenequation Htot|E?=E|E?, we can obtain the coupled equations for the probability amplitudes as Δaua(0)?ξaua(1)+gaA=Eua(0),Δbub(0)?ξbub(1)+gbB=Eub(0),?iΓaA+gaua(0)+JabB=EA,?iΓbB+gbub(0)+JbaA=EB,Δlul(j)?ξl[ul(j+1)+ul(j?1)]=Eul(j),with j>0 and l=a,b.

    If a single photon with energy E is incident from the infinity side of CRW-l, the ?-type three-level atom will result in photon scattering between different CRWs or photon absorption by the dissipative of the atom. The general expressions of the probability amplitudes in the CRWs (j0) are given by ul(j)=e?iklj+slleiklj,ul(j)=slleiklj,where sll denotes the single-photon scattering amplitude from CRW-l to CRW-l (l,l=a,b). Substituting Eq.?(C8) or Eq.?(C9) into Eq.?(C7), the eigenvalue of the semi-infinite CRW-l in the rotating reference frame is given by [40] E=Δl?2ξl?cos?kl,0<kl<π,where kl is the wavenumber of the single photon in the CRW-l. Without loss of generality, we assume that ξl>0 and 0<kl<π for semi-infinite CRW-l.

    Now let us derive the scattering amplitudes for single-photon scattering by the atom with nonreciprocal transition. By solving Eqs.?(C5) and (C6), the coefficients A and B can be expressed by A=(E+iΓb)gaua(0)+Jabgbub(0)(E+iΓa)(E+iΓb)?JbaJab,B=(E+iΓa)gbub(0)+Jbagaua(0)(E+iΓa)(E+iΓb)?JbaJab.Substituting A and B into Eqs.?(C3) and (C4), we have (Δa?E+Δˉa)ua(0)+Jabub(0)=ξaua(1),(Δb?E+Δˉb)ub(0)+Jbaua(0)=ξbub(1),with the effective coupling strengths Jll and frequency shifts Δˉl induced by the ?-type three-level atom defined by Jab=Jabgagb(E+iΓa)(E+iΓb)?JbaJab,Jba=Jbagagb(E+iΓa)(E+iΓb)?JbaJab,Δˉa=(E+iΓb)ga2(E+iΓa)(E+iΓb)?JbaJab,Δˉb=(E+iΓa)gb2(E+iΓa)(E+iΓb)?JbaJab.When a single photon is input from CRW-a, we have ua(j)=e?ikaj+saaeikaj and ub(j)=sbaeikbj, and the scattering amplitudes saa and sba satisfy the following equations: (ξae?ika+Δˉa)saa+Jabsba=?ξaeika?Δˉa,Jbasaa+(ξbe?ikb+Δˉb)sba=?Jba.Similarly, when a single photon is input from CRW-b, we have ub(j)=e?ikbj+sbbeikbj and ua(j)=sabeikaj, and the scattering amplitudes sab and sbb satisfy the following equations: (ξae?ika+Δˉa)sab+Jabsbb=?Jab,Jbasab+(ξbe?ikb+Δˉb)sbb=?ξbeikb?Δˉb.Equations?(C19)–(C22) can be expressed concisely in matrix form as LS=R,with the scattering matrix S=(saasabsbasbb)and coefficient matrices L=(ξae?ika+ΔˉaJabJbaξbe?ikb+Δˉb),R=?(ξaeika+ΔˉaJabJbaξbeikb+Δˉb).The solutions of Eq.?(C23) are given by saa=JabJba?(ξaeika+Δˉa)(ξbe?ikb+Δˉb)(ξae?ika+Δˉa)(ξbe?ikb+Δˉb)?JabJba,sba=2iξaJba?sin?ka(ξae?ika+Δˉa)(ξbe?ikb+Δˉb)?JabJba,sab=2iξbJab?sin?kb(ξae?ika+Δˉa)(ξbe?ikb+Δˉb)?JabJba,sbb=JabJba?(ξae?ika+Δˉa)(ξbeikb+Δˉb)(ξae?ika+Δˉa)(ξbe?ikb+Δˉb)?JabJba.To quantify the efficiency for nonreciprocity transport, we define the scattering flow [4143] of a single photon from CRW-l to CRW-l as Ill|sll|2ξl?sin?klξl?sin?kl,where ξl?sin?kl (ξl?sin?kl) is the group velocity in the CRW-l (CRW-l).

    APPENDIX D: PERFECT SINGLE-PHOTON NONRECIPROCITY

    In this section, we will derive the conditions for perfect nonreciprocal single-photon transport, i.e.,?Iab=0 and Iba=1, analytically. For simplicity, we assume that the two semi-infinite CRWs have the same parameters, i.e.,?ξξa=ξb, kka=kb, gga=gb, and they are coupled to the atom resonantly with Δa=Δb=0 and ΓΓa=Γb. Iab=0 can be obtained by setting Jab=0 or Jab=0. In this case, we have Iba=|sba|2=|2Jbag2ξ?sin?k[ξe?ik(?2ξ?cos?k+iΓ)+g2]2|2.So the condition for Iba=1 is |sin?k|=(|Jba|?Γ)g2±Θ4(g2?ξ2)ξ,with Θ=(|Jba|?Γ)2g4?4(g2?ξ2)[4(ξ2?g2)ξ2+Γ2ξ2+g4].As a simple example, the maximum scattering flow Iba=1 can be obtained at the maximum group velocity |sin?k|=1, with |Jba|=(g2+Γξ)22g2ξ.Furthermore, if g2=Γξ, then we have |Jba|=2Γ.

    APPENDIX E: MAXIMUM FULL WIDTH AT HALF-MAXIMUM

    We will derive the maximum full width at half-maximum (FWHM) for perfect nonreciprocal single-photon transport. The half-maximum of the scattering flow Iba is given by Iba=|sba|2=|2Jbag2ξ?sin?khalf[ξe?ik(?2ξ?cos?khalf+iΓ)+g2]2|2=12.Under the conditions that |Jba|=2Γ and g2=Γξ, we have 2(Γ?ξ)ξ|sin?khalf|2+(1?22)Γ2|sin?khalf|+2(ξ?Γ)ξ+Γ2=0.Defining ηξ/Γ and ζ4(1?η)η, Eq.?(E2) can be rewritten as |sin?khalf|=?(1?22)±(1?22)2?ζ(2?ζ)ζ.The condition for maximum width Δkmax is ddη|sin?khalf|=d|sin?khalf|dζdζdη=0,which is satisfied with η=12?ξ=Γ2.That is to say, the maximum width Δkmax is obtained at ξ=Γ/2 with |sin?khalf|=22?1?22?2,and the maximum FWHM Δkmax is Δkmaxπ?2?arcsin?(22?1?22?2)0.81π.

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    Xunwei Xu, Yanjun Zhao, Hui Wang, Aixi Chen, Yu-Xi Liu. Nonreciprocal transition between two nondegenerate energy levels[J]. Photonics Research, 2021, 9(5): 879
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