Single-photon nonreciprocal excitation transfer with non-Markovian retarded effects
aa r X i v : . [ qu a n t - ph ] F e b Single-photon nonreciprocal excitation transfer with non-Markovian retarded effects
Lei Du, Mao-Rui Cai, Jin-Hui Wu, Zhihai Wang, and Yong Li
1, 2, 3, ∗ Beijing Computational Science Research Center, Beijing 100193, China Center for Quantum Sciences and School of Physics,Northeast Normal University, Changchun 130024, China Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University, Changsha 410081, China (Dated: February 9, 2021)We study at the single-photon level the nonreciprocal excitation transfer between emitters coupledwith a common waveguide. Non-Markovian retarded effects are taken into account due to thelarge separation distance between different emitter-waveguide coupling ports. It is shown that theexcitation transfer between the emitters of a small-atom dimer can be obviously nonreciprocal byintroducing between them a coherent coupling channel with nontrivial coupling phase. We prove thatfor dimer models the nonreciprocity cannot coexist with the decoherence-free giant-atom structurealthough the latter markedly lengthens the lifetime of the emitters. In view of this, we furtherpropose a giant-atom trimer which supports both nonreciprocal transfer (directional circulation) ofthe excitation and greatly lengthened lifetime. Such a trimer model also exhibits incommensurateemitter-waveguide entanglement for different initial states in which case the excitation transfer ishowever reciprocal. We believe that the proposals in this paper are of potential applications inlarge-scale quantum networks and quantum information processing.
I. INTRODUCTION
Waveguide quantum electrodynamics (QED) studiesinteractions between atoms and various one-dimensionalopen waveguides. It provides an excellent platform forachieving strong light-matter interactions due to thestrong transverse confinement on the electromagneticfields [1, 2]. Different from cavity QED systems, whereatoms are commonly coupled with a single or multiplediscrete modes in bounded spaces, atoms can interactwith a continuum of modes in waveguides, similar tothose of a thermal reservoir [2]. In view of this, many dis-advantages presented in cavities can be evaded in waveg-uide QED systems, such as limited bandwidth of emit-ted photons and stochastic release of cavities [3]. Sincethe first experimental realization in 2007 [4], waveguideQED has brought out a great deal of advances, e.g., chiralphoton-atom interactions [5–7], single-photon routers [8–10], and topologically induced unconventional quantumoptics [11] to name a few. In particular, waveguide-mediated interactions between far apart atoms (res-onators), which can be tailored to be either coherent ordissipative, exhibit important applications in achievinglarge-scale quantum networks [12–17].In waveguide QED, atoms are commonly regardedas point-like dipoles because their sizes are in generalmuch smaller than the wavelengths of the waveguidemodes they interact with. Recent experiments showthat such an approximation is no longer valid when(artificial) atoms interact with a surface acoustic wavewhose wavelength can be even much smaller than mi-crowave photons [18–20]. Moreover, it is also possible ∗ [email protected] to couple a single atom with bent waveguides at two ormore points separated by distances much larger than onewavelength. Such configurations are referred to as gi-ant atoms [21], which can exhibit striking effects suchas frequency-dependent decays and Lamb shifts [21, 22],chiral emission [23, 24], and oscillating bound states [25].Recently, decoherence-free interactions between braidedgiant atoms are theoretically proposed [26] and experi-mentally verified [27], where the giant atoms are immuneto emitting photons to the waveguide yet they still in-teract effectively with each other. Moreover, giant-atomstructures have also been extended to two or higher di-mensions with optical lattices of cold atoms [28]. Theseseminal works provide new inspirations for many applica-tions in quantum simulating and quantum computation.It has been shown that non-Markovian retarded effectsarising from the large separation distances between, forinstance, a single atom and the waveguide end [29–31],different coupling ports of a giant atom [32], and far apartatoms [33–35] can markedly modify the dynamics. Whenthe traveling time of photons or phonons in the waveg-uide between different atom-waveguide coupling chan-nels is large enough compared with the inverse of theatomic relaxation rate, the dynamics can exhibit promi-nent non-Markovianity and thus can not be predicted bycommon Markovian treatments. This suggests that non-Markovian retarded effect should be taken into accountwhen considering nonlocal couplings that are inevitablein many large-scale systems (e.g., quantum networks).In particular, a counterintuitive phenomenon referred toas “superradiant paradox” arises if the separation L be-tween atoms satisfies l c / < d < l c , where l c is the co-herent length of photons emitted from the emitters tothe waveguide [33, 34]. As the number of atom increases,the non-Markovianity is shown to be non-negligible evenfor small separations [36]. Such non-Markovian dynam- FIG. 1. Schematic illustration of (a) small-atom dimer andeffective energy levels and (b) giant-atom trimer with braidedcoupling ports. ics in single-photon waveguide QED can be solved semi-analytically via a real-space approach [37] and fully ana-lytically via a diagrammatic method [38].In this paper, we focus on nonreciprocal excitationtransfer between emitters in waveguide QED systemswith considerable non-Markovian retarded effects. Westart by considering a simple dimer model where twosmall emitters couple with each other via both direct andindirect (waveguide mediated) interactions. By introduc-ing a nontrivial coupling phase (synthetic magnetic flux)for the direct coupling terms, nonreciprocal excitationtransfer can be achieved and the nonreciprocity is also de-pendent on the phase accumulation of traveling photonsin this case. To lengthen the duration of the nonrecipro-cal phenomenon, we also propose a giant-atom trimer inwhich the excitation exhibits directional circulation andgreatly suppressed dissipation. Dimer models, however,are proved to be incapable of supporting nonreciprocaltransfer as long as the emitters become “decoherence-free”. Moreover, we demonstrate that the entanglementbetween emitters and the waveguide modes can be quiteincommensurate in the case of trivial coupling phasewhen the single excitation is initially prepared in differentemitters, although the excitation transfer is reciprocal inthis case.
II. MODEL AND EQUATIONS
We first consider a simple model composed of two iden-tical small emitters and a U-type waveguide (the term“small” here means that each emitter is coupled withthe waveguide at only one point, in contrast to giant-atom models). As shown in Fig. 1, emitters a and b areside-coupled with the waveguide at x = x and x = x , respectively. In addition, we consider a direct interac-tion between a and b by assuming that they are spatiallyclose together. Hereafter, we refer to this model as thesmall-atom dimer for simplicity. The Hamiltonian of thesmall-atom dimer can be written as ( ~ = 1) H = H e + H w + H int , (1)where H e = ω ( σ + a σ a + σ + b σ b ) and H w = P k ω k p † k p k arethe free Hamiltonians of the emitters and the waveguide,respectively. Here σ + j and σ j ( j = a, b ) are respectivelythe raising and lowering operators of emitter j with ω the transition frequency between the ground state | g i andexcited state | e i . p † k and p k are respectively the creationand annihilation operators of the traveling photons in thewaveguide with wave vector k and frequency ω k . The dis-persion relation of the waveguide can be approximatelygiven by ω k = v g | k | if ω is far away from the cut-offfrequency [39]. Under the rotating-wave approximation,the interaction Hamiltonian is written as H int = X k [ g k p k ( σ + a + e ikd σ + b ) + h.c. ]+( Jσ + a σ b + h.c. ) , (2)where g k and J are the emitter-waveguide and emitter-emitter coupling coefficients, respectively, d = | x − x | is the separation distance between the two emitters.With the Hamiltonian above, the state at time t in thesingle-excitation manifold can be given by | ψ ( t ) i = X k u k ( t ) e − iω k t p † k | g, g, i + X j = a,b c j ( t ) σ + j e − iω t | g, g, i , (3)where c j ( t ) and u k ( t ) are the probability amplitudes ofexciting emitter j to the excited state and creating aphoton with wave vector k in the waveguide, respec-tively. | g, g, i denotes the vacuum state of the systemwith both emitters in the ground state and no photonin the waveguide. By solving the Schr¨odinger equationand eliminating the waveguide modes, one can obtainthe time-delayed equations of probability amplitudes (seemore details in Appendix A) dc a ( t ) dt = − γ e iφ c b ( t − dv g )Θ( t − dv g ) − κ + γ c a ( t ) − iJc b ( t ) ,dc b ( t ) dt = − γ e iφ c a ( t − dv g )Θ( t − dv g ) − κ + γ c b ( t ) − iJ ∗ c a ( t ) , (4)where γ = 2 | g k | /v g is the spontaneous emission rate ofthe emitters to the waveguide with v g the group velocityand k = ω /v g [33]. φ = k d is the phase accumulationof photons traveling from one emitter to another through (d) (e) (f) (a) (b) (c) FIG. 2. Dynamic evolutions of populations P b, (blue solid)and P a, (red dashed) for (a) η = 0 .
56, (b) η = 0 . η =0 . θ = π/
4, (e) θ = π/
2, (f) θ = π . Here we assume θ = 0 in (a)-(c) and η = 0 .
56 in (d)-(f). Other parametersare ω /γ = 112 . κ/γ = 8 . × − , and | J | /γ = 0 . the waveguide and Θ( t ) is the Heaviside step function. κ denotes the loss of the emitters due to other decay chan-nels, which can be much smaller than γ experimentally.Equation (4) shows that the waveguide introduces botha decay channel for each emitter and a retarded indirectcoupling between them. In Markovian limit d/v g → t → ∞ , Eq. (4) can be ap-proximately regarded as simultaneous differential equa-tions, while as d increases gradually, the non-Markovianretarded effect becomes more and more dominant suchthat the dynamic evolution can markedly deviate fromthe Markovian expectation.Note that the coupling phase θ of the direct interactionshould be considered (i.e., J = | J | e iθ ), which can not beremoved by any gauge transformation in the presence ofthe waveguide mediated coupling. As will be shown inthe following, it plays a crucial role for achieving nonre-ciprocal excitation transfer. Experimentally, such a cou-pling phase can be achieved via an ac driving in eachemitter [40–43]. For whispering gallery mode resonators,one can also use an anti-resonant linker to introduce anoptical path imbalance ∆ x in opposite directions betweenthe two resonators, such that the effective coupling phasereads θ = 2 π ∆ x/λ with λ the resonant wavelength of theresonators [44–47]. III. NONRECIPROCAL EXCITATIONTRANSFER
Now we consider two initial states | ψ (0) i = | e, g, i and | ψ (0) i = | g, e, i (either a or b is initially preparedin the excited state) to compare the excitation transfersfrom a to b and from b to a , respectively. This can be doneby focusing on the dynamic evolutions of the probabilities P b, and P a, , where P j,n ( j = a, b ; n = 1 ,
2) denotes thepopulations | c j | of emitter j with initial state | ψ n (0) i .Moreover, we define η = dγ/v g as the separation dis-tance between the emitters normalized by the coherencelength [32–34]. Therefore the relation between phase φ and the time delay reads φ = ω d/v g = ω η/γ . Forexample, if emitters a and b are two identical supercon-ducting qubits (artificial atoms) with ω / π = 3 . γ/ π = 29 . φ = { , . , } π for η = { . , . , . } , respectively.We first consider the case of trivial coupling phase θ = 0 and plot in Figs. 2(a)-2(c) the dynamical evolu-tions of P b, and P a, with η = 0 . η = 0 . η = 0 . φ cannot lead to nonreciprocitysince it does not break the time-reversal symmetry ofthe Hamiltonian. Note that the populations decay muchslower when φ is an integer multiple of π . This is reminis-cent of the Fabry-P´erot bound states in the continuum(BICs) in Markovian limit, which shows that one of theeigenstates becomes lossless if φ = mπ ( m is an arbitraryinteger) [48, 49]. For general initial states consideredhere, one can find from Eq. (4) that the waveguide in-duced indirect coupling is purely dissipative (i.e., iγe iφ / φ = mπ , which servesas an effective gain and thus suppresses the decay of theemitters [50]. Moreover, the populations decay in an os-cillating form for φ = mπ because the excitation bouncesbetween the emitters back and forth.On the other hand, Figs. 2(d) and 2(e) exhibit obviousnonreciprocal excitation transfer within a certain timerange due to the nontrivial coupling phase ( θ = mπ ),which breaks the time-reversal symmetry of the Hamil-tonian. In particular, the optimal nonreciprocal trans-fer can be achieved for θ = π/ | P b, ( t ) − P a, ( t ) | during the evolution maximizes for θ = π/ θ has noimpact on dynamics for t < d/v g because it can alwaysbe gauged away in the absence of the retarded coupling(denoted by the red dashed line). This is also why P b, and P a, coincide with each other in the beginning. Onceeach emitter meets the retarded feedback coming fromthe other one at t = d/v g , an additional transfer pathbetween two emitters is formed such that the two pathscan interfere with each other (see the black solid and reddashed lines in the energy-level diagram) and the interfer-ence effects of opposite directions are generally differentfor θ = mπ . Figure 2(f) shows that the transfer becomesreciprocal again for θ = π , which attributes to the recov-ered time-reversal symmetry.Although the time-reversal symmetry of the system isbroken by tuning the coupling phase θ , the nonreciprocityis also dependent on φ in the case of θ = mπ . As shownin Figs. 3(a)-(c), one can observe nearly inverse nonre-ciprocal transfer for φ = 20 π and φ = 21 π ( η = 0 . η = 0 . φ = 20 . π ( η = 0 . (a) (b) (c) (d) FIG. 3. (a)-(c): Dynamic evolutions of populations P b, (bluesolid) and P a, (red dashed) for (a) η = 0 .
56, (b) η = 0 . η = 0 . P b, and P a, with different values of η corresponding to φ = 2 mπ .Here we assume θ = π/ trivial. This is because the coupling phases are effectivelyshifted from ± θ to ± θ − φ by removing φ in the indirectcoupling terms, implying that the moduli of the overallcouplings of opposite directions are effectively swappedin the case of φ = (2 m + 1) π . Note that the nonrecip-rocal behaviors are not exactly inverse for φ = 20 π and21 π due to different time delays before which the emittersdecay exponentially [33].We point out that η determines the non-Markovianityof the system such that it also affects the onset timeand the optimal effect of the nonreciprocal transfer. Asshown in Fig. 3(d), the onset time of the nonrecipro-cal transfer is exactly t = d/v g (i.e., γt = η ). More-over, all values of η chosen here correspond to φ = 2 mπ ( φ = { π, π, π } for η = { . , . , . } ), whichyield the optimal nonreciprocal transfer for θ = π/ η in-creases. In other words, the retarded effect puts off theonset and suppresses the degree of the nonreciprocal ex-citation transfer.Note that in the absence of external inputs, the popu-lations of both emitters should fall to zero rapidly and thenonreciprocal phenomenon can only be observed withina short-lived duration, as shown in Figs. 2 and 3. Ithas been shown that giant atoms (self-interference res-onators) can be completely decoupled from the waveg-uide and thus no longer emit photons to it [26, 27, 52].However, this generally makes the emitters isolated suchthat they can hardly interact with each other if they arespatially separated. Thanks to the braided structure pro-posed in Refs. [26, 27], decoherence-free couplings can beachieved between far apart giant atoms, i.e., the spon-taneous emissions of the atoms to the waveguide can be completely suppressed while the indirect coupling be-tween them is nonvanishing. Nevertheless, we would liketo point out that a dimer model with such a braidedstructure is unable to demonstrate nonreciprocal trans-fer, although the lifetime of the emitter can be markedlyextended in this case (see more details in Appendix B). IV. DIRECTIONAL EXCITATIONCIRCULATION IN A GIANT-ATOM TRIMER
As discussed in Sec. II and Appendix B, nonreciprocalexcitation transfer is not allowed in dimer models with“decoherence-free” giant atoms (the quotation mark heremeans that the giant atoms are not exactly decoherence-free due to the retarded self-interference effects), al-though the lifetime of the emitters can be markedlylengthened. In view of this, we extend the braided struc-ture by introducing the third emitter c (with the raisingand lowering operators denoted by σ + c and σ c , respec-tively). As shown in Fig. 1(b), emitters a and c arecoupled with the waveguide via the same two ports lo-cated at x = x and x = x , respectively, while emitter b couples with the waveguide at x = x and x = x , re-spectively. The four coupling ports are arranged in thebraided manner to suppress the spontaneous emission tothe waveguide and obtain nonvanishing indirect coupling.We assume that the coupling ports are evenly spaced(i.e., x − x = x − x = x − x = d ) and all emitter-waveguide couplings are identical. Moreover, a and c cou-ple directly with each other in terms of | J | e iθ σ + a σ c + h.c. .For simplicity, we refer to this structure as the giant-atomtrimer hereafter. In this case, the effective equations ofthe probability amplitudes are written as dc a ( t ) dt = − γ D ,b + D ,b ) − γ ( D ,a + D ,c ) − ( κ + γ ) a ( t ) − i ( | J | e iθ − iγ ) c c ( t ) ,dc b ( t ) dt = − γ D ,a + D ,c ) + D ,a + D ,c ] − ( κ + γ ) b ( t ) − γD ,b ,dc c ( t ) dt = − γ D ,b + D ,b ) − γ ( D ,a + D ,c ) − ( κ + γ ) c ( t ) − i ( | J | e − iθ − iγ ) c a ( t ) , (5)where D n,l = c l ( t − nd/v g ) e inφ Θ( t − nd/v g ) ( n = 1 , , l = a, b, c ) with subscript n corresponding to time delay nd/v g . Clearly, the overall coupling between a and c becomes asymmetric for θ = mπ .We plot in Figs. 4(a) and 4(b) the dynamic evo-lutions of the populations P b, and P a, , respectively,where P j, ( j = a, b ) denotes the population | c j | of emitter j with the initial state | ψ (0) i = | e, g, g, i ( | ψ (0) i = | g, e, g, i ). We find that in the giant-atomtrimer, nonreciprocal excitation transfer emerges againfor θ = mπ due to the asymmetric overall coupling be-tween a and c . Different from the behaviors in Figs. 2 (d) (c) (a) (b) FIG. 4. (a) and (b): Dynamic evolutions of populations P b, (blue solid) and P a, (red dashed) for (a) θ = 0 and (b) θ = π/
2. (c) and (d): Dynamic evolutions of populations of allthree emitters with (c) θ = π/ | ψ (0) i , (d) θ = π/ | ψ (0) i . Other parameters are thesame as those in Fig. 3(a) except for | J | /γ = 1 ( | J | and θ denote in this case the amplitude and phase of the directcoupling coefficient between a and c , respectively). and 3, P b, and P a, oscillate here with a fixed phasedifference. To understand this difference, we also plot inFigs. 4(b) and 4(c) the evolutions of the populations of allthree emitters in the case of θ = π/
2, with initial states | ψ (0) i and | ψ (0) i , respectively. One can find that forboth initial states, the excitation exhibits a directionalcirculation along the same direction of a → c → b → a after t = d/v g , which is a signature of broken time-reversal symmetry that cannot be observed for θ = mπ .That is to say, the single excitation initially prepared inemitter b is preferentially transferred to a while that ini-tially prepared in emitter a shows a preferential transferto c . Note that circulations along the opposite direc-tion a → b → c → a can be achieved for θ = − π/ b for | ψ (0) i and a for | ψ (0) i ) can be excited efficiently withindifferent durations for the two initial states. Note thatsince emitter b is not requested to interact directly with a or c in this case, the trimer model can be used for non-reciprocal excitation transfer between far apart emitters,which is of potential applications in large-scale quantumnetworks.Finally, we focus on the dynamic evolutions of thegiant-atom trimer in long-time limit. We first plot inFigs. 5(a)-5(d) the evolutions of the populations of allthree emitters with different values of θ and initial states.Two major differences between the cases of θ = π/ θ = 0 can be found: (i) for θ = π/ θ = 0 the pop- FIG. 5. (a)-(d): Dynamic evolutions of populations of allthree emitters with (a) θ = π/ | ψ (0) i ,(b) θ = π/ | ψ (0) i , (c) θ = 0 and initialstate | ψ (0) i , (d) θ = 0 and initial state | ψ (0) i . (e) and (f):Dynamic evolutions of linear entropies and total populationswith (e) θ = π/ θ = 0. Other parameters are thesame as those in Fig. 4. ulation of b exhibits obviously different evolution fromthose of a and c ; (ii) for θ = π/ θ = 0,i.e., the long-time evolutions are quite different for ini-tial states | ψ (0) i and | ψ (0) i [see Figs. 5(c) and 5(d)].Physically, this is because for θ = π/
2, the giant-atomtrimer exhibits directional excitation circulation as men-tioned above, yet the oscillation amplitudes of all threepopulations minish gradually due to both the other de-cay channels and the non-Markovian retarded effect (asdiscussed in Appendix B, the decoherence of the emit-ters to the waveguide can be exactly suppressed in caseof κ → d/v g → θ = 0, the excitation can be transferredsimultaneously from one emitter to both the other two.The transfer probability from b to a and that from b to c should be equal because the two paths are identical.However, the transfer probability from a ( c ) to b andthat from a ( c ) to c ( a ) are unequal because the twopaths are quite different [the overall coupling between a ( c ) and b is purely coherent with time delays d/v g and3 d/v g while that of a and c contains both coherent anddissipative parts with time delays 0 or 2 d/v g ]. For ini-tial state | ψ (0) i , the excitation is apt to be transferredfrom a to c rather than from a to b while it comes backequiprobably from b to a and from b to c . As a result, theexcitation tends to bounce between a and c in long-timelimit. For initial state | ψ (0) i , the populations of a and c are always identical because the excitation initialized in b is transferred between b and a or between b and c withidentical probabilities since the beginning.We point out that the entanglement between the emit-ters and the waveguide mode can be markedly incommen-surate for initial states | ψ (0) i and | ψ (0) i in the case of θ = 0. This can be verified by calculating the linearentropy S = 1 − Tr( ρ ), which estimates here the corre-lation between the emitters and the electromagnetic fieldin the waveguide [33, 57]. ρ denotes the reduced densitymatrix of the emitters, which can be obtained by taking atrace over the waveguide states, i.e., ρ = Tr w [ | ψ ( t ) ih ψ ( t ) | ]with | ψ ( t ) i given in Eq. (3). We plot in Figs. 5(e) and5(f) the linear entropies S (corresponding to initial state | ψ (0) i ) and S ′ (corresponding to initial state | ψ (0) i )for θ = π/ θ = 0, respectively. It can be found thatfor θ = 0 the evolutions of linear entropy can be quitedifferent by initially exciting different emitters ( a or b )to the excited state while for θ = π/ P tot = P a, + P b, + P c, and P ′ tot = P a, + P b, + P c, for θ = π/ θ = 0, respec-tively. As expected, the total populations exhibit similarbehaviors with those of the linear entropies [33]. Sucha result is reminiscent of the nonreciprocal entanglementdemonstrated in Ref. [58], which reveals that the con-ditions of nonreciprocal entanglement and transport arenot necessary the same. However, we refer to the phe-nomenon here as incommensurate entanglement to dis-tinguish it from the phenomena stemming from brokentime-reversal symmetry. V. CONCLUSIONS
In summary, we have studied the nonreciprocal ex-citation transfer in the presence of nonnegligible non-Markovian retarded effects by considering two waveg-uide QED models, i.e., a small-atom dimer and a giant-atom trimer. Both models exhibit nonreciprocal single-excitation transfer if a coherent coupling channel withnontrivial coupling phase is introduced, while the waveg-uide induced phase accumulation does not break thetime-reversal symmetry and thereby cannot result in non-reciprocity solely. The retarded effects, which are in-evitable in the presence of nonlocal couplings, are shownto put off the onset and suppress the degree of the non-reciprocal transfer. In particular, we have demonstratedthat the giant-atom trimer supports both nonreciprocalexcitation transfer (in a directionally circulatory man-ner) and greatly suppressed decoherence of the emitters,which cannot be achieved in dimer models with simi-lar decoherence-free structures. Moreover, incommensu-rate emitter-waveguide entanglement has been revealedwhen different emitters of the giant-atom trimer are ini-tially excited, whose condition is independent of the time-reversal symmetry. The results in this paper may inspireapplications based on large-scale quantum networks dueto the rapid progress in relevant experimental platforms.
ACKNOWLEDGMENTS
L. Du thanks D.-W. Xiao and Q.-S. Zhang for help-ful discussions. This work was supported by the Sci-ence Challenge Project (Grant No. TZ2018003), theNational Key R&D Program of China (Grant No.2016YFA0301200), and the National Natural ScienceFoundation of China (Grants No. 11774024, No.11875011, No. 12047566, No. 12074030, No. 12074061,and No. U1930402).
Appendix A: Derivation of Eq. (4)
In this appendix, we show in detail the derivation ofEq. (4) in the main text. With the Hamiltonians andthe single-excitation wave function given in Eqs. (1)-(3)and solving the schr¨odinger equation, one can obtain theequations of the probability amplitudes dc a ( t ) dt = − i X k g k c ( k, t ) e − i ( ω k − ω ) t − iJc b ( t ) ,dc b ( t ) dt = − i X k g k c ( k, t ) e ikd e − i ( ω k − ω ) t − iJ ∗ c a ( t ) ,du k ( t ) dt = − ig ∗ k e i ( ω k − ω ) t [ c a ( t ) + e − ikd c b ( t )] . (A1)By substituting the formal solution of u k ( t ) u k ( t ) = − i Z tt dt ′ g ∗ k e i ( ω k − ω ) t ′ [ c a ( t ′ )+ e − ikd c b ( t ′ )] (A2)into the dynamic equations of c a ( t ) and c b ( t ), we obtain dc a ( t ) dt = − iJc b ( t ) − π Z tt dt ′ e iω ( t − t ′ ) Z + ∞−∞ dk × | g k | [ c a ( t ′ ) + e − ikd c b ( t ′ )] e − iω k ( t − t ′ ) ,dc b ( t ) dt = − iJ ∗ c a ( t ) − π Z tt dt ′ e iω ( t − t ′ ) Z + ∞−∞ dk × | g k | [ e ikd c a ( t ′ ) + c b ( t ′ )] e − iω k ( t − t ′ ) . (A3)Considering that both ω k and g k are even functions of k , i.e., ω ( k ) = ω ( − k ) and g ( k ) = g ( − k ), one can changethe variable that is being integrated, i.e., dc a ( t ) dt = − iJc b ( t ) − Z tt dt ′ e iω ( t − t ′ ) Z + ∞ dω | g ω | πv g × [2 c a ( t ′ ) + 2 cos ( kd ) c b ( t ′ )] e − iω ( t − t ′ ) ,dc b ( t ) dt = − iJ ∗ c a ( t ) − Z tt dt ′ e iω ( t − t ′ ) Z + ∞ dω | g ω | πv g × [2 cos ( kd ) c a ( t ′ ) + 2 c b ( t ′ )] e − iω ( t − t ′ ) , (A4) FIG. B1. Schematic illustration of braided giant-atom dimer. where g ω is the emitter-waveguide coupling coefficient asa function of ω k . Here and hereafter, we denote ω ( k )by ω for simplicity. Assuming that | g ( ω ) | /v g = γ/ ω is far away from the cut-off frequency of the waveguidesuch that its dispersion can be approximately linearizedas ω = ω + ν if [39, 53], we have dc a ( t ) dt = − γ π Z + ∞−∞ dν Z tt dt ′ e − iν ( t − t ′ ) { c a ( t ′ )+ cos [( k + νv g ) d ] c b ( t ′ ) } − iJc b ( t ) ,dc b ( t ) dt = − γ π Z + ∞−∞ dν Z tt dt ′ e − iν ( t − t ′ ) { c b ( t ′ )+ cos [( k + νv g ) d ] c a ( t ′ ) } − iJ ∗ c a ( t ) , (A5)where k = ω /v g . According to the definition of thedelta function R dωe iωt = 2 πδ ( t ), Eq. (A5) can be sim-plified as Eq. (4) in the main text by including the decay κ for each emitter. Note that we have dropped the contri-bution of δ ( t ′ − t − d/v g ) at this step because it is centeredoutside the the range of integral, i.e., t ′ − t − d/v g is neg-ative for t ′ ∈ [0 , t ]. This is always true as long as d is notexactly zero, even if d is very small. Appendix B: Extended lifetime with braidedgiant-atom structure
In this appendix, we aim to prove that two giant atomswith decoherence-free indirect coupling cannot exhibitnonreciprocal transfer even in the presence of a directcoupling between them. As shown in Fig. B1, bothemitters a and b couple with the waveguide twice inthe braided manner, i.e., emitter a ( b ) couples with thewaveguide at x = x and x = x ( x = x and x = x ).The emitter-waveguide couplings are assumed to be iden-tical (i.e., g ) and the coupling ports are evenly spaced bydistance d . Such a braided structure can be implemented,for example, with a S-type waveguide [26, 27]. To distin-guish from the small-atom dimer in Fig. 1(a), we refer tothis model as the giant-atom dimer. Once again, we con-sider the direct coupling | J | e ± iθ between a and b . In thiscase, the effective equations of the probability amplitudes (a) (b) FIG. B2. Dynamic evolution of populations P b, (blue solid)and P a, (red dashed) with (a) η = 0 .
154 ( φ = 5 . π ) and(b) η = 0 .
014 ( φ = 0 . π ). Other parameters are the same asthose in Fig. 3. c a and c b are written as dc a ( t ) dt = − γc a ( t ) − γ D ,b + D ,b ) − γD ,a − i | J | e iθ c b ( t ) ,dc b ( t ) dt = − γc b ( t ) − γ D ,a + D ,a ) − γD ,b − i | J | e − iθ c a ( t ) , (B1)where φ = k d = ω η and D n,l = c l ( t − nd/v g ) e inφ Θ( t − nd/v g ) ( n = 1 , , l = a, b ). Here we have neglectedother decay channels for simplicity which can be experi-mentally much weaker than the spontaneous emission tothe waveguide.We plot in Fig. B2(a) the dynamic evolutions of P b, and P a, by numerically calculating Eq. (B1). The sepa-ration distance d between adjacent coupling ports is welltailored such that 2 φ = π . According to Ref. [26, 27],the emitters become decoherence-free in this case yet canstill interact with each other in the absence of the directcoupling. However, as show in Fig. B2(a), the excita-tion transfer becomes reciprocal in this case, althoughthe coupling phase θ = π/ θ ,which always show reciprocal transfer). Moreover, we canfind that in the presence of considerable time delay, theexcitation transferred between two emitters still shows adamped oscillation although the damping is much weakerthan that in Fig. 2(b). This is because the emitters arenot exactly decoherence-free due to the retarded self in-terference effects. As η (i.e., d ) decreases gradually, thenon-Markovian retarded effect becomes more and morenegligible such that the emitters tends to be completelydecoherence-free, as shown in Fig. B2(b).The results above can be understood as follows. InMarkovian limit ( d/v g → g eff = − i γ (3 e iφ + e iφ ), which is purely real in the caseof φ = ( m + 1 / π . On the other hand, the direct cou-pling coefficients | J | e ± iθ possess identical real and op-posite (vanishing) imaginary parts as long as θ = mπ ( θ = mπ ). In view of this, the overall coupling between a and b is always reciprocal due to the identical strength(modulus) for both directions. For the non-Markoviancase here, this can be seen from the Laplace transforma-tion of Eq. (B1), i.e., s ˜ c a ( s ) − c a (0) = − γ (1 + e ϕ ) ˜ c a ( s ) − i | J | e iθ ˜ c b ( s ) − γ e ϕ + e ϕ ) ˜ c b ( s ) ,s ˜ c b ( s ) − c b (0) = − γ (1 + e ϕ ) ˜ c b ( s ) − i | J | e − iθ ˜ c a ( s ) − γ e ϕ + e ϕ ) ˜ c a ( s ) , (B2)where ˜ c j ( s ) ( j = a, b ) is the Laplace transformationof c j ( t ) and ϕ = iφ − sd/v g . Eq. (B2) shows a pairof complex conjugate overall coupling coefficients in the s -domain, implying that the excitation transfer should be reciprocal in the absence of other mechanisms thatmay induce nonreciprocity (such as optical Sagnac ef-fects [59]). 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