Pronounced Non-Markovian Features in Multiply-Excited, Multiple-Emitter Waveguide-QED: Retardation-Induced Anomalous Population Trapping

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Pronounced non-Markovian features in multiply-excited, multiple-emitter waveguide-QED: Retardation-induced anomalous population trapping

Alexander Carmele,1 Nikolett Nemet,1, 2 Victor Canela,1, 2 and Scott Parkins1, 2 1Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 2Dodd-Walls Centre for Photonic and Quantum Technologies, New Zealand (Dated: February 5, 2020) The Markovian approximation is widely applied in the field of quantum optics due to the weak frequency dependence of the vacuum field amplitude, and in consequence non-Markovian effects are typically regarded to play a minor role in the optical electron-photon interaction. Here, we give an example where non-Markovianity changes the qualitative behavior of a quantum optical system, rendering the Markovian approximation quantitatively and qualitatively insufficient. Namely, we study a multiple-emitter, multiple-excitation waveguide quantum-electrodynamic (waveguide-QED) system and include propagation time delay. In particular, we demonstrate anomalous population trapping as a result of the retardation in the excitation exchange between the waveguide and three initially excited emitters. Allowing for local phases in the emitter-waveguide coupling, this population trapping cannot be recovered using a Markovian treatment, proving the essential role of non-Markovian dynamics in the process. Furthermore, this time-delayed excitation exchange allows for a novel steady state, in which one emitter decays entirely to its ground state while the other two remain partially excited.

I. INTRODUCTION 68]. In this work, we employ the matrix-product state representation to study exactly this regime, the multiple- excitation and multiple-emitter limit. We focus, in One-dimensional (1D) waveguide-QED systems are at- particular, on the three-emitter and three-photon case, tractive platforms for engineering light-matter interac- treating the emitters as two-level systems, which couple tions and studying collective behavior in the ongoing to the left- and right-moving photons and thereby inter- efforts to construct scalable quantum networks [1–12]. act with each other, subject to time delays associated Such systems are realized in photonic-like systems includ- with the propagation time of photons between emitters ing photonic crystal waveguides [13–19], optical fibers [58, 69, 70]. We choose throughout the paper the triply- [20–24], or metal and graphene plasmonic waveguides excited state as the initial state and compare the re- [25–28]. Due to their one-dimensional structure, long- laxation dynamics in the Markovian and non-Markovian distance interactions become significant [3, 5, 29]. As a result of these interactions mediated by left- and right- moving quantized electromagnetic fields, strongly entan- (a) gled dynamics and collective, cooperative effects related to Dicke sub- and superradiance emerge [1, 6, 12, 17, 22– 24, 30–37]. In the framework of standard quantum optics, these systems are widely explored in the Markovian, single- (b) emitter or single-excitation limit [9, 31, 38–41]. Such lim- its can be described by a variety of theoretical methods including real-space approaches [5, 38, 42, 43], a Green’s function approach [44–47], Lindblad master equations

arXiv:1910.13414v3 [quant-ph] 4 Feb 2020 [48, 49], input-output theory [50–54], and the Lippmann- Schwinger equation [55–57]. Already in these regimes, exciting features have been predicted. For example, strong photon-photon interactions can in principle be engineered, allowing for quantum computation protocols using flying qubits (propagating photons) and multilevel atoms [5, 11, 56, 58, 59]. Furthermore, bound states FIG. 1. Scheme of the simulated waveguide QED system. (a) in the continuum are addressed via a joint two-photon The system consists of three identical emitter with transition pulse, showing that excitation trapping via multiple- frequency ω which couple to left- and right moving quantized 0 √ photon scattering can occur without band-edge effects light fields via the decay constant γ. (b) Due to the delay or cavities [7, 43, 55, 60]. −1 (in the scheme two time steps τ1 = τ3 = 2∆ = 2γ /10 Beyond the single-excitation and/or single-emitter and τ = 4∆) a closed loop is formed between the first and limit, the Markovian approximation becomes question- third emitter interacting with their respective past bins. The able and the aforementioned methods problematic [61– interaction strongly depends on the phases ω0τ1 and ω0τ3. 2 cases. To compare both scenarios on the same footing, applying a time-independent phase shift to the left- and we employ the quantum stochastic Schrödingerequation right-moving photonic field, the transformed Hamilto- R r,ω l,ω approach [61, 71, 72] and numerically solve the model nian reads HI (t) = ~g0 (HI + HI )dω, where using a matrix-product-state algorithm [59, 73–77] as an r,ω ω ω † 12 12 −i 2 τ1 12 −i 2 τ alternative to the t-DMRG method in position space [78]. HI = rω(t) σ1 + σ2 e + σ3 e + H.c., ω ω We report on striking differences between the Markovian l,ω † 12 −i τ 12 −i τ3 12 H = l (t) σ e 2 + σ e 2 + σ + H.c., (2) and non-Markovian description. First, we find that in the I ω 1 2 3 case of non-Markovian excitation exchange, the triply- † † † with rω(t) = rω(0) exp[i(ω − ω0)t] and lω(t) = excited initial state allows for population trapping, in † lω(0) exp[i(ω − ω0)t], and τ = (τ1 + τ3)/2, cf. App. A. strong contrast to the Markovian description. Second, In the following, the left- and right-moving excitations time-delayed excitation exchange allows for anomalous are treated collectively: population trapping, in which one emitter relaxes com- Z Z pletely into its ground state while the two other emit- † † † † ters form a singly-excited dark state together with the R (t) = dωrω(t),L (t) = dωlω(t). (3) waveguide field in between. No local phase combination in the Markovian case allows for such anomalous popula- Given these definitions, the non-Markovian interaction tion trapping, rendering the non-Markovian description Hamiltonian reads: qualitatively and quantitatively different from a Marko- NM 12 † iω0τ † vian treatment. HI (t)/~ = g0 σ1 R (t) + e L (t − τ) + H.c. ω0 12 † i 2 τ1 + g0 σ2 R (t − τ1/2)e + H.c. (4)

ω0 II. MODEL 12 † i 2 τ3 + g0 σ2 L (t − τ3/2)e + H.c.

12 † iω0τ † To demonstrate the importance of retardation-induced + g0 σ3 R (t − τ)e + L (t) + H.c. . effects and the underlying non-Markovian dynamics, we In the following, we compare the Markovian with the choose a system consisting of three identical emitters non-Markovian case. The Markovian case neglects retar- with transition frequency ω0. All three emitters interact (†) (†) dation effects between the excitation exchange, therefore with left- (lω ) and right-moving photons (rω ) in a one- in the Markovian approximation we set R(†)(t − t0) ≈ dimensional waveguide, as depicted in Fig. 1(a). To fo- R(†)(t) and L(†)(t − t0) ≈ L(†)(t). In this approxima- cus on the retardation-induced effects, we neglect out-of- tion, only the local phases but not the retardation in plane losses which inevitably enforce a fully thermalized, the amplitude are taken into account. Consequently, the trivial steady-state in the ground state, and render the Markovian interaction Hamiltonian reads: non-Markovian effects a transient, nevertheless impor- h ω0 i tant feature for waveguide-based counting experiments. M † 12 iω0τ 12 i 2 τ3 12 HI (t)/~ = g0 L (t) σ1 e + σ2 e + σ3 + H.c. The Hamiltonian governing the free evolution of the com- h ω0 i bined, one-dimensional waveguide photon-emitter system † 12 12 i 2 τ1 iω0τ 12 + g0 R (t) σ1 + σ2 e + e σ3 + H.c. , (5) reads: 3 Z where the emitters interact with time-local collective X 22 † † right- and left-moving fields and no time delay is present H0/~ = ω0 σi + dω ω rωrω + lωlω , (1) i=1 in the interaction. We solve for the system’s dynamics in both cases using the time-discrete Schrödingerequation where the emitters are treated as two-level systems, with with the time-step size ∆ up to time N∆ in N steps, |1i as the ground state and |2i as the excited state, and cf. Fig. 1(b): ij with σn := |iinnhj|, the flip operator of the n-th emitter. The interaction Hamiltonian describes the emitters inter- |ψ(n)i = UNM/M(n, n − 1)|ψ(n − 1)i (6) acting with right and left moving photons at the emitters’ " # i Z n∆ positions: NM/M 0 0 = exp − HI (t )dt |ψ(n − 1)i, ~ (n−1)∆ 3 Z X 12 † iωxi/c † −iωxi/c HI = ~g0 σi dω rωe + lωe + H.c., where ∆ is small enough to minimize the error in the i=1 Suzuki-Trotter expansion [59, 73–77], and the evolution is taken either in the Markovian (M) or in the non- where we have assumed a frequency-independent cou- Markovian limit (NM). Here, the wavefunction is in pling of the emitters to the quantized light field. The po- MPS form: sition of the second emitter is chosen as x2 = 0, leading to X x1 = −d1/2 = −cτ1/2 for the first and x3 = d2/2 = cτ3/2 |ψ(n)i = L[l1]R[r1] ··· S[s]L[ln]R[rn]1[ln+1]1[rn+1] ... for the third emitter, with c the speed of light in the s,l1···lN waveguide. After transforming into the interaction pic- r1···rN ture with respect to the free evolution Hamiltonian, and |l1, r1 ··· s, ln, rn, ln+1, rn+1 ··· lN , rN i, (7) 3

FIG. 2. Algorithm to compute the next time step via MPS FIG. 3. The impact of different phase choices ϕ := in case of a Markovian dynamics, i.e. we set R(†)(n − m) ≈ (ϕ1, ϕ2, ϕ3) = ([0, 2π], 0, [0, 2π]) in the atom-waveguide cou- R(†)(n) and L(†)(n − m) ≈ L(†)(n) and m∆ = τ. plings in the Markovian limit with photon operators: R(t − t0) = R(t) and L(t − t0) = L(t), on the integrated reservoir population in the steady state. If the system is initialized in the triply-excited state (green line), for all choices of phases, where we assume vacuum input states for m > n with all excitation is radiated into the reservoir. If the system is [lm] [lm] [rm] [rm] identity tensor L = 1 and R = 1 in the time initialized in a superposition of doubly-(orange line) or singly- bins for the right- and left-moving field corresponding to excited states (red line), the only case where all excitation is a vacuum input state from left and right, and the indices radiated into the reservoir is when all phases are a multiple s count the degrees of freedom for the emitter system, of 2π. and ri, li the number of excitations in the time bin for the left- and right-moving field. In the following, we assume dim[s] = 23 = 8, and choose a time-step size to guarantee namics using the quantum stochastic Schrödingerequa- that the dimension of the reservoir excitation does not tion ignoring time-delay effects to give a Markovian evo- exceed dim[ri] = dim[li] = 3. However, all results have lution. also been calculated with dim[ri] = dim[li] = 4 to prove In Fig. 2 the simulation protocol is depicted. In the convergence. time-discrete basis, the time-local Hamiltonian in the To efficiently simulate the multiple-emitter, multiple- matrix-product-operator MPO(n) form acts only on the excitation case, we employ the matrix-product-state present reservoir bins n and the system state s due to the technique described in [59, 73–77], and choose a collec- Markovian approximation. After applying the evolution tive basis for the flip operators of the emitters to al- operator via contracting the physical indices (in the cor- low for entangled initial states: |ijki = |(i − 1)22 + responding color code, red for the system bin, green for 1 0 12 (j − 1)2 + (k − 1)2 i, which leads to, e.g., σ1 ≡ right moving field, blue for left moving field) in step (a), |0ih4| + |2ih6| + |1ih5| + |3ih7|, or |222i = |7i the triply- an entangled system-reservoir matrix is created. To write excited state. We discuss in the corresponding sections the MPS in the canonical form, a Schmidt value decom- the simulation protocol in detail. position is performed, and the entangled system-reservoir state is expressed as a matrix product (c), after which the next MPO(n+1) can be applied. In this manner, the III. MARKOVIAN LIMIT: NO TIME DELAY steady state can be calculated step by step in an efficient and less memory-consuming way. Due to the Marko- We start our investigation in the Markovian limit, vian approximation, i.e. we set R(†)(n − m) ≈ R(†)(n) (†) (†) and calculate the system’s dynamics with UM (Eq. and L (n − m) ≈ L (n) and m∆ = τ, no rear- (6)) and the initial state |ψ(0)i = |7i until the ranging of the MPS is necessary. The simulations are steady state is reached. The Markovian case allows√ done, in the Markovian and non-Markovian case, until for a master-equation treatment with g0 = 2πγ the steady state is reached. In the MPS representation, [61–68, 79]. Tracing out the left- and right-moving (Eq. (7)), the steady state is reached when the applica- photons leads to a collective jump operator, J := tion of the MPO leads to the identical tensor combination √ 12 12 12 γ σ1 exp[iϕ1] + σ2 + σ3 exp[iϕ3] . The phases ϕi for every time step after the Schmidt value decomposi- can be chosen individually via local unitary transforma- tion and swapping procedure, i.e. the MPO and subse- tions, or they arise from the spatial position without quent re-arranging only acts as an index-shifting operator taking the finite distance into account in the evolution ln, rn → ln+1, rn+1. [6, 70]. In the following, we nevertheless solve the dy- In Fig. 3, the phase dependence of the integrated reser- 4

FIG. 4. Algorithm to compute the next time step via MPS in case of retardation of the left- (blue) and right-moving quantized field (green). The emitter 2 (middle) interacts with a time delay of 2∆ with the right-moving field, taking emitter 1 as point of reference for the right-moving field, and emitter 2 also interacts with the emitted left-moving field of emitter 3, taking emitter 3 as reference point for the left-moving field, resulting in a delay of 4∆. The dashed squares couple to the system s at time n∆. Since the canonical form of the MPS needs to be maintained, swapping procedures need to be applied where the orthogonality center is swapped from n − 6 next to the system.

s voir excitation I in the steady state is plotted for differ- a dark state with Hl−m|Di = 0. Therefore, with cor- PN−f † † responding phase differences, such dark states can be ent initial states: I = n=0 R (n)R(n) + L (n)L(n) with N as the number of time steps to reach the steady driven via individual decays and lead to dark state pop- state and f = (τ1 + τ3)/∆. The phases are permu- ulation, or population trapping, e.g. [3, 6, 23, 24, 44, tated by changing ϕ1, ϕ3 between 0 to 2π. If the system 73, 80, 81]. We conclude that, within the Markovian is initialized in the triply-excited state |ψ(0)i = |7i, the treatment, we find that either all emitters relax into phases have no impact at all on the steady-state values their ground state or none. For systems initialized in and all excitation will eventually be radiated into the the triply-excited state, excitation trapping cannot be reservoirs on the left of emitter one and on the right of achieved. And for the superradiant, symmetric singly- emitter three, leading for all phase permutations to the excited and doubly-excited initial states, the emitters un- integrated reservoir occupation of 3, cf. Fig. 3 (green dergo complete decay only in the case of vanishing phase line). difference. We show now that including retardation and In contrast to the triply-excited case, the steady states back-action effects changes this picture completely. of the emitters initially in a superposition of singly- (|1i + |2i + |4i, red line) and doubly-excited states (|3i + |5i + |6i, orange line) are strongly influenced by IV. NON-MARKOVIAN DYNAMICS: the choice of phases. For those initial states, only if the SYMMETRIC TIME DELAY. phase difference vanishes, ϕ1 = 2π = ϕ3, is all radia- tion emitted into the reservoir. For all other phase com- In the non-Markovian case, the MPO is not only act- binations, population trapping occurs, and all emitters ing on the present reservoir bins n but also on the past have a finite probability to be found in the excited state. left- and right-moving reservoir bins n − m for m > 0. In Population trapping is created due to the fact that the in- Fig. 4, the simulation protocol is schematically explained dividual decay of the emitters allows to populate a dark for the case when emitter 2 (middle) interacts with a time state of the Hamiltonian. In the two-emitter case, the delay of τ3/2 = 2∆ with the emitted right-moving field standard light-matter Hamiltonian can be written in the of emitter 1 and with a time delay of τ1/2 = 4∆ with the collective basis as: left-moving field from emitter 3. Correspondingly, emitter 1 interacts with the present reservoir bin n of the s X 12 21 Hl−m = ~g0 (σi + σi ) (8) right-moving (green) and n − 6 of the left-moving field i=1,2 (blue) to provide for a time delay τ = 6∆. In Fig. 4, the ≡ ~g0 [|22ih21| + |21ih11| + |22ih12| + |12ih11| + h.c.] dotted squares are those with which the MPO at time √ step n interacts, and the orthogonality center is initially and therefore with |Di = (|12i − |21i)/ 2 leads to such in n − 6 of the left-moving field (blue). For efficiency, 5

0.1 0,1 Emitter 1 0.01 γτ=0.5 Emitter 2 0,01 Emitter 3 0.1 1 10 1 Emitter 1

Population 0,001 Emitter 2 0.1 Emitter 3 0,0001 0.01 γτ=1.25 1 10

Population τ t/ 3 0.1 1 10 1

0.1 FIG. 6. The dynamics of the emitter populations if all emitters are initially in their excited state: |ψ(0)i = |7i for a phase 0.01 γτ=6.25 choice of ω0(τ1 + τ3) = 3π, and τ1 = 2τ3. Emitter 3 (green line) decays completely into its ground state while emitter 1 0.1 1 10 t/τ (black line) and 2 (orange line) remain in a partially excited state. This steady state is impossible to reach in the Marko- FIG. 5. The dynamics of the emitter populations for different vian treatment if no additional interactions are included. feedback lengths with phase ω0τ/2 = 2π for a system initially in the triply-excited state. Already short feedback times (τ = 2ns, i.e. γτ = 0.5) lead to population trapping in contrast to the Markovian case. For longer feedback (τ = 25ns, i.e. γτ = 6.25), slowly decaying oscillations become visible. cay exponentially as expected before the first excitation with a neighboring emitter takes place t ∈ [0, τ/2]. From the dotted squares need to be arranged in the MPS next this moment on (indicated with a dashed line), the de- to each other to avoid a memory-consuming contraction cay is slowed down considerably due to the re-excitation of the MPS without gain of information. To ascertain and re-emission dynamics. For longer times, the emit- the normalization, the swapping procedures start from ter starts to partially interact with its own ”past” and left to right, cf. Fig. 4(a-b). The swapping guarantees after several round trips the population in the emitter that the essential entanglement in between the reservoir is stabilized and a dark state is formed for this partic- bins is preserved. Every swap creates a new orthogonal- ular chosen phase. We emphasize that this anomalous ity center, but in (c) the n − 6 left-moving bin (blue) is population trapping depends on the presence of left- and swapped through the whole MPS next to the system s right-moving photons and correspondingly to a feedback and creates a well-defined orthogonality center before the effect induced by those. Emitter two (green line) has, for MPO is applied. After applying the MPO, the resevoir longer delays (middle and lower panel), a slightly higher bins are arranged in the previous, canonical form but the population than emitters one and three due to excita- orthogonality center is left at n − 5 (blue). In this pro- tions from both the left and right emitters (black and tocol, the matrix product state remains in the canonical orange line). For very long feedback, i.e., γτ = 6.25, a form and a fast numerical simulation is possible. regime where the feedback phase ceases to have a strong We initialize the emitters in the triply-excited state influence, we observe an interesting oscillatory behavior in the emitter populations due to the feedback and finite and assume first symmetric delays, i.e. τ1 = τ3. In the case of quantum coherent feedback, the delay between the excitation, which settles eventually to a small but finite excitation exchanges introduces a corresponding phase steady-state value. [59, 73–75, 82–87]. In the following, we assume a transition frequency of the emitters to yield: ω τ/2 = 2π, 0 This example shows that allowing even for a short i.e. exp[±iω τ] = 1 = exp[±iω τ/2]. We show now 0 0 retardation and back-action time, the dynamics of the that the evolution under the influence of a finite delay emitter populations changes qualitatively and quantita- in between the emission events together with subsequent tively. Population trapping from an initial triply-excited back-actions from the previous emissions of each emit- emitter state cannot be recovered just with local phases ter lead to population trapping in strong contrast to the in the Hamiltonian. This impossibility is lifted due to Markovian case. a time-delayed coherent feedback mechanism. We em- In Fig. 5, the emitter populations in the presence of phasize that for very long delay the need for a particular 22 coherent quantum feedback are shown, e.g., σ1 = choice of ω0τ/2 = 2π is partially lifted, and it takes di- P4 2 i=1 |h2i−1|ψ(t)i| . Even for short feedback in compari- vergingly long for the emitter population to decay. How- son to the decay time, i.e., γτ = 0.5, population trapping ever, for long delay times γτ 1, the population is also is observed (upper panel). Emitter one (black dotted) trapped in the reservoir between the emitters and the and three (orange solid line) exhibit the same dynamics absolute stored population in the emitter system is ex- due to the symmetry of the system. Both start to de- ponentially small. 6

V. NON-MARKOVIAN DYNAMICS: state, which interact via left- and right-moving photons. ASYMMETRIC TIME DELAY. We compared the Markovian and the non-Markovian case, i.e., without and with time delay in propagation Until now, we have discussed symmetric time delay between them. In the Markovian case, only a local phase τ1 = τ2, or |x1| = |x3|. Asymmetric time delay pro- is taken into account but no delayed amplitude in the vides a further example how the full non-Markovian and re-emission events. We recovered the well-known results, quantized description of many-excitation dynamics in that the triply-excited state decays, independent of waveguide-QED deviates qualitatively from the Marko- phase choice, while the doubly- and singly-superradiant vian treatment. As shown above, in the Markovian treat- superposition state shows population trapping for any ment either all emitters remain partially excited or none non-vanishing phase differences. In strong contrast, a of them do. For the triply-excited state, no population non-Markovian excitation exchange results in population trapping occurs, and for other initial states the system trapping even if the system is initialized in the triply- is not able to reach a steady state with only one emitter excited state. Furthermore, quantum feedback allows in the ground state and the other emitters partially ex- for states in which two emitters form a superposition cited. We show now that the non-Markovian description state together with a part of the reservoir, whereas the with asymmetric time delay allows for another example third emitter relaxes entirely into the ground state; of anomalous population trapping, where one emitter de- a state that is not possible to realize in a Markovian cays completely into its ground state whereas the other setup if only local phases in the jump operators, and no emitters have a finite probability to be found in the ex- additional interactions, are assumed. These examples cited state. prove the significance of time delay in many-emitter, In Fig. 6, we choose a phase ω0τ = 3π and delay times many-excitation systems and the possibility of entirely γτ1 = 1 between the left (1) and middle emitter (2), new physics beyond the Markovian regime in the and γτ3 = 0.5 between the middle and the right emit- steady-state and 1D (or β → 1) limit considered here, ter (3). Excitingly, this setup allows for the right emitter whereby the emitters radiate purely into the (detectable) (green line) to decay entirely to its ground state while the waveguide modes, which is a regime already in reach of left (black line) and middle emitters (orange line) form a various waveguide-QED platforms [88, 89]. dark state together with the waveguide field in between and exhibit population trapping. This effect results from AC gratefully acknowledges support from the Deutsche the asymmetric delay between left- and right emission Forschungsgemeinschaft (DFG) through the project events. For t < τ3, all emitters radiate unperturbed into B1 of the SFB 910, and from the European Unions the reservoir. For τ3 < t < τ1, the left emitter (black Horizon 2020 research and innovation program under the line) continues to radiate unperturbed whereas the mid- SONAR grant agreement no. [734690]. The calculations dle and right emitters start to interact with the emit- were performed using the ITensor Library [90]. ted photons. Due to symmetry, both the right and middle emitters exhibit the same decay behavior for t < τ1. This picture changes for larger times, as now the middle Appendix A: Hamiltonian of the multiple-emitter emitter’s field starts to constructively interfere with the waveguide-QED system right-moving photons from emitter one. Emitter three interacts with its own past emission and decays faster, The free evolution Hamiltonian of the combined one- while emitters one and two start to form a superposition dimensional waveguide photons and emitters system reads: state. After several roundtrip times, γt & 15, emitter three has decayed, and no emission takes place. Z X 22 † † Interestingly, a necessary condition for this feature to H0/~ = ωiσi + dω ω rωrω + lωlω , (A1) happen is asymmetric feedback. A symmetric feedback i=1,2,3 τ1 = τ3 exhibits, as in the Markovian case, only fi- where the emitters are treated as two-level systems with nite population in all emitters, or none. This effect de- |1i as the ground state and |2i as the excited state, and pends only on the destructive and constructive interfer- ij 12 σn = |iinnhj| for the n-th emitter, with σn = |1innh2| ence between left- and right-moving photons. For differ- the de-excitation operator of the n-th emitter. The in- ent ϕ = ω0τ, a different positioning needs to be chosen. teraction Hamiltonian consists of the emitter interacting Quantity γτ determines the extent of population trap- with right- and left-moving photons at the emitter’s po- ping between emitter one and two, but not the qualita- sition: tive effect. Z X 12 † iωxi/c † −iωxi/c HI /~ = σi dω gi(ω) rωe + lωe i=1,2,3 VI. CONCLUSION Z X 21 ∗ −iωxi/c iωxi/c + σi dω gi (ω) rωe + lωe . We have investigated a waveguide-QED system con- i=1,2,3 sisting of three emitters initialized in the triply-excited (A2) 7

p The positions of the atoms are assumed to be centered γi/(2π). New operators are introduced: around x = 0, so in the case of three atoms, the mid- Z √ dle atom is located at x0 = 0, the ﬁrst (from left) † † i(ω−ω2)t R (t) = dωrωe / 2π, (A6) atom at x1 = −d1/2 = −cτ1/2, and the third atom at x3 = d3/2 = cτ3/2. Choosing a rotating frame cor- Z √ L†(t) = dωl† ei(ω−ω2)t/ 2π. (A7) responding to the middle atoms frequency ω2 and the ω reservoir modes of the left and right-moving photons, the Given these deﬁnitions, the interaction Hamiltonian total Hamiltonian reads: reads:

22 22 22 22 HI (t)/~ = δ1σ1 + δ3σ3 HI / = δ1σ + δ3σ (A3) √ ~ 1 3 12 † iω2τ † Z + γ1 σ1 R (t) + e L (t − τ) + H.c. X 12 ∗ † iωτi/2 −i(ω2−ω)t + σi dω gi (ω)rωe e + H.c. (A8) √ i=1,2,3 12 † iω2τ1/2 + γ2 σ2 R (t − τ1/2)e + H.c. (A9) X Z + σ12 dω g∗(ω)l† e−iωτi/2e−i(ω2−ω)t + H.c. , √ i i ω + γ σ12L†(t − τ /2)eiω2τ3/2 + H.c. i=1,2,3 2 2 3 √ 12 † iω2τ † + γ3 σ3 R (t − τ)e + L (t) + H.c. , with δ = ω − ω and δ = ω − ω . For convenience, 1 1 2 3 3 2 where τ = (τ1 + τ3)/2. In the following, we use the we transform this Hamiltonian. The left atom interacts notation for the collective states corresponding to |ijki = without delay with the right-moving field. Also, we want |(i − 1)22 + (j − 1)21 + (k − 1)20i, which leads to, e.g., the right atom to interact with the left-moving field with- 12 σ1 = |0ih4| + |2ih6| + |1ih5| + |3ih7| . out delay: To achieve this, we apply unitary transforma- To solve the corresponding Schrödingerequation, we tions: switch to a time-discrete evolution picture, and integrate from 0 to ∆ as the first time step from |ψ(0)i. Z iτ1 U = exp − dωωr† r → U r† U † = r† e−iωτ1/2 |ψ(n)i = U(n, n − 1)|ψ(n − 1)i 3 2 ω ω r ω r ω " Z n∆ # Z i 0 0 iτ3 = exp − H (t )dt |ψ(n − 1)i (A10) U = exp − dωωl† l → U l† U † = l† e−iωτ3/2. I 1 2 ω ω l ω l ω ~ (n−1)∆ We can now introduce in the discrete time-bin basis, the Now, the interaction Hamiltonian reads: collective bath operators as: Z n∆ Z dt ∆R†(n) = dωr† ei(ω−ω2)t √ , (A11) ω 2π∆ HI (t)/~ = (A4) (n−1)∆ 22 22 Z n∆ Z dt = δ1σ1 + δ3σ3 † † i(ω−ω2)t ∆L (n) = dωlωe √ . (A12) Z (n−1)∆ 2π∆ 12 ∗ † † −iω(τ1+τ3)/2 −i(ω2−ω)t + dω σ1 g1 (ω) rω + lωe e Given the time bin dynamics, we can now study dif- (A5) ferent cases. Due to the richness of the model, we fo- 12 ∗ † −iωτ1/2 † −iωτ3/2 −i(ω2−ω)t cus in the following only on the identical emitter case: + σ2 g2 (ω) rωe + lωe e γi = γ and ω1 = ω2 = ω3= ω0. In this case, we 12 ∗ † −iω(τ3+τ1)/2 † −i(ω2−ω)t have as free parameters only: γ, τ1, τ3, and the feed- + σ3 g3 (ω) rωe + lω e + H.c. . back phase ω0∆/2 which determines all phases in the matrix-product-operator (MPO). For example, if we set φ = ω0(τ1 + τ3)/2 = 2π, in its feedback interaction the In the following, the left and right-moving excitations are middle atom is subjected to the phase: φ1 = φ/(1+τ1/τ3) treated collectively, and we assume the coupling elements and φ3 = φ/(1+τ3/τ1). For the symmetric case it follows to be constants with respect to frequency, i.e., gi(ω) = obviously: φ1,3 = φ/2.

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