T-bulge single photon quantum router with three level system
Liu Lin




Lin Liu Key Laboratory of Low-Dimensional Quantum Structures
and Quantum Control of Ministry of Education, and Department of
Physics, Hunan Normal University, Changsha 410081, China 

              North China Institute of Aerospace
Engineering, Langfang 065000, China 

             liulin@hunnu.edu.cn       
        

T-bulge single photon quantum router with three level system
    Lin Liu
    Received: date / Accepted: date
============================================================




This paper considers the transmission characteristics of incidenting from an infinte coupled-resonator waveguide (CRW) or a semi-infinite CRW, respectively. The Nth cavity of a semi-infinite CRW intersecting with an infinite CRW and a cascade three level system(CTLS) form a T-bulge structure. Due to symmetry breaking boundary, the maximum of transfer rate would reach unity when light incidenting from the semi-infinite CRW, while reach 0.5 for infinite CRW case. The position of the intersection have effect on the location of the extreme point of the transmission coefficients changing with couple strength. 4D figures are used to illustrate the transmission characteristics, which is more visable.






§ INTRODUCTION


    Quantum networks composed of many nodes and channels, have many application in quantum communication. Quantum router, which acting as key component of quantum networks, can be achieved by the optical interactions of single photons and atoms, allowing the distribution of entanglement across the network <cit.>. Measurement-device-independent quantum key can distribute over a 404 km optical fiber <cit.>. The coupling strength can be increased at frequencies near resonances in free space. The employment of a cavity can enhance the coupling strength further <cit.>.

    Zhou et al have made good jobs on quantum router, utilising a controllable two-level system as a quantum switch for the coherent transport of a single photon <cit.>, a cyclic three-level system embedded in the junction of two infinite CRWs forming an X-type quantum router <cit.>, changing the cyclic type into a new type with an inversion center <cit.>. Then make alter both in the atom and the CRWs' structure, a two-level system (TLS) in a T-shaped waveguide made of an infinite CRW and a semi-infinite CRW <cit.>.

    Now, changing into a CTLS couple with two CRWs, the Nth cavity of a semi-infinite CRW intersecting with an infinite CRW, let we see what happens to the transmission characteristics of incidenting from each CRW.

    This paper is organized as follows: In Sec. 2, the model used in this paper is introduced. In Sec. 3 and Sec. 4, the single-photon scattering process is studied for waves incident from different CRWs. Finally, we conclude with a brief summary of the results.


§ THE MODEL



As shown in Fig. 1, coupled resonators on the blue (red) line construct the infinite (semi-infinite) CRW, which is called CRW-a (-b) hereafter. And the Nth cavity of a semi-infinite CRW intersects with an infinite CRW. The CTLS is situated in the node, which is the cross point of the two CRWs.



Hamilton of CTLS is below

    H_T=ω_e| e⟩⟨ e| + ω _f| f⟩⟨ f|,

the CTLS's transition frequency of | g⟩↔| f⟩
 (| f⟩↔| e⟩)
 is ω_f(ω_e).
Single photon hopes among cavities nearby, which lead to photon propagating along the single-mode cavities of the waveguides. Using tight-binding model, the two CRWs are described by the Hamiltonian

    H_C = ∑^+∞_u=-∞[ω_aa^†_ua_u - ξ_a(a^†_ua_u+1 + a^†_u+1a_u)]
    +∑^+∞_v=1[ω_bb^†_vb_v - ξ_b(b^†_vb_v+1 + b^†_v+1b_v)]+ω_cc^†c.

Hamiltonian of interaction between atom and two CRWs is

    H_I=| f⟩⟨ g|
    (g_aa_0+g_bb_N)+| g⟩⟨ f| (g_aa_0^++g_bb_N^+)
    +| e⟩⟨ f| g_cc+| f⟩⟨ e| g_cc^+.

    The system H consist of the two CRWs H_C, the CTLS H_T, and the interaction between atom and two CRWs H_I
    H=H_C+H_T+H_I.
    In single excitation subspace, eigenstate of the full hamiltonian is


    | E⟩=(∑^+∞_u=-∞U_u^aa_u+∑^+∞_v=1U_v^bb_v)
    | g,0_a,0_b,1_c⟩+U_f| f,0_a,0_b,1_c⟩+U_e| e,0_a,0_b,0_c⟩.

    Fock states of CRW-a(-b) and c mode are0_a,1_a,0_b,1_b,0_c,1_c.
    The Schrödinger Equation H| E⟩=E| E⟩ gives rise to a series of coupled stationary equations for all amplitudes, which is the discrete scattering equation for the single-photon propagation in the T-shaped waveguide


    ∑^+∞_u=-∞(E-ω_a-ω_c)U_u^a=∑^+∞
    _u=-∞[-ξ_a(U_u-1^a+U_u+1^a)+δ_u,0(V_aU_0^a+GU
    _N^b)],


    ∑^+∞_v=1(E-ω_b-ω_c)U_v^b=∑^+∞_v=1
    [-ξ_b(U_v-1^b+U_v+1^b)+δ_v,N(V_bU_N^b+GU_0^a)],

where G≡ g_ag_bδ_e/[δ_e(E-ω_f-ω_c)-g_c^2],V_j=a,b≡ g_j^2δ_e/[δ_e(E-ω_f-ω_c)-g_c^2],the detuning between the incident photon frequency and atom transition frequency being δ_e≡ E-ω_e.




 §.§ Single photons incident from the infinite CRW-a



    When the single photons incident from the CRW-a, then CRW-a contains incident light, reflected light and transmitted light, and maybe existing transfer light in CRW-b both upwords and downwords. The CTLS absorbs the photon with wavenumber k incident along the u axis onto the T-bulge shaped waveguide, transitting the CTLS from its ground state to its excited state. Since the excited state is coupled to the continua of states, the excited CTLS will emit a photon spontaneously into the propagating state of either CRW-a or CRW-b. For symmetry breaking, the wave functions between the first site and the junction of the two CRWs have standing wave form depending on N.
   
    U_u^a={[ e^ik_au+re^-ik_au,u < 0;;          te^ik_au,u > 0. ].


    U_v^b={[  t^be^ik_bv,      v > N ;; Asin( k_bN), v=1,2,3..N . ].


The dispersion relations in the two CRWs are E=ω_c+ω_d-2ξ_dcos k_d,d=a,b.Using continue condition, the transmission, reflection and transfer amplitude can be obtained as below, with group velocity v_d=2ξ_dsin k_d,d=a,b.
    Nondimensionalize the the propagating amplitude(in case of g_a,g_b,v_a,δ_e0). In this form, the effect of N is clearer.

    t^b =2isin k_bN/g_av_b/g_bv_a-2g_b/
    g_ae^ik_bNisin k_bN+i[ g_c^2/δ _e-(
    E-ω _s-ω _c) ] v_b/g_ag_b,


    t =-i[ g_c^2/δ _e-( E-ω
    _s-ω _c) ] v_a/g_ag_b-2g_b/g_a
    e^ik_bNisin k_bN/g_av_b/g_bv_a-2g_b/g_a
    e^ik_bNisin k_bN+i[ g_c^2/δ _e-(
    E-ω _s-ω _c) ] v_b/g_ag_b,


    r =-g_av_b/g_bv_a/g_av_b/g_bv_a-2
    g_b/g_ae^ik_bNisin k_bN+i[ g_c^2/δ
    _e-( E-ω _s-ω _c) ] v_b/g_ag_b
    ,


   The transmission coefficients are modulus square of the transmission amplitude, T_b^a=|t^b|^2, T^a=|t|^2, R^a=|r|^2. The total possibility is conserved(T_b^a+ T^a+R^a=1).







Fig2 shows the propagation coefficients change with couple strength g_a,g_b,g_c. The centre of the 4D picture has a cuboid margin so that it's convenient to see the inner side. When g_a is small, T^a is tend to be one with less effect of g_b and g_c, while T_b^a and R^a is tending to be zero. When g_b is small, g_a is between ξ_0 and 6ξ_0, T^a first decreases and then increases when g_c increases, while R^a is vice versa.



 §.§ Single photons incident from the semi-infinite CRW-b



    The single photons is launched into the semi-infinite CRW-b. When the traveling photon arrives at the node of the T bulge node, it is either absorbed by the CTLS or stored in the N-1 FP cavities (reflected by the boundary when N=1). The probability amplitudes in the CRW-a and CRW-b are given by


    U_u^a={[ t_l^ae^-ik_au,         u < 0;;  t_r^ae^ik_au,         u > 0. ].


    U_v^b={[ e^-ikv+r^be^ik_bv,            v > N ;;       Asin( k_bN),       v=1,2,3..N . ].


Using continue condition, obtaining t_l^a=t_l^a≡ t^a, the reflection and transfer amplitude can be obtained as below

    r^b =-g_av_b/g_bv_a-2ig_b/g_a
    e^-ik_bNsin k_bN+i[ g_c^2/δ _e-(
    E-ω _s-ω _c) ] v_b/g_ag_b/
    g_av_b/g_bv_a-2ig_b/g_ae^ik_bNsin k_bN+i[
    g_c^2/δ _e-( E-ω _s-ω _c) ]
    v_b/g_ag_b,


    t^a =2iv_b/v_asin k_bN/g_av_b/g_bv_a
    -2ig_b/g_ae^ik_bNsin k_bN+i[ g_c^2/
    δ _e-( E-ω _s-ω _c) ] v_b/
    g_ag_b


    The photons incidenting from CRW-b go left and right into CRW-a with same possibilities. It's can be verified that possibility is conserved R^b+T_a^b=1,R^b=|r^b|^2,T_a^b=2|t^a|^2.





    From Fig.3, R^b tend to be small when g_a g_b and g_c to be large. But when g_a and g_b are small, R^b is large no matter g_c is large or small. T_a^b is vice versa for the reason that T_a^b=1-R^b.



§ DISCUSSION AND CONCLUSION



The effect of parameter g_a g_b g_c N on the transmission coefficients of incidenting from infinte CRW-a as well as semi-infinite CRW-b are considered in this paper. When light incident from CRW-a-b, the maximum of T_b^a (T_a^b) would be 0.5 (1). This can be predicted for the symmetry breaking boundary. The cavity number N in the junction of the two CRWs have effect on the location of the extreme point of the curve between the transmission coefficients and couple strength.


This work is supported by the National Fundamental Research Program
of China (the 973 Program) under Grant No. 2012CB922103 and the
National Natural Science Foundation of China under Grants No.
11374095, No. 11422540, No. 11434011, No. 11575058.


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