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 pop- ulation 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 ! which couple to left- and right moving quantized 0 p 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 !0τ1 and !0τ3. 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 Schr¨odingerequation right-moving photonic field, the transformed Hamilto- R r;! l;! approach [61, 71, 72] and numerically solve the model nian reads HI (t) = ~g0 (HI + HI )d!, where using a matrix-product-state algorithm [59, 73{77] as an r;! ! ! y 12 12 −i 2 τ1 12 −i 2 τ alternative to the t-DMRG method in position space [78]. HI = r!(t) σ1 + σ2 e + σ3 e + H.c.; ! ! We report on striking differences between the Markovian l;! y 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 ! 1 2 3 case of non-Markovian excitation exchange, the triply- y y y with r!(t) = r!(0) exp[i(! − !0)t] and l!(t) = excited initial state allows for population trapping, in y l!(0) exp[i(! − !0)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- y y y y ters form a singly-excited dark state together with the R (t) = d!r!(t);L (t) = d!l!(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 y i!0τ y vian treatment. HI (t)=~ = g0 σ1 R (t) + e L (t − τ) + H.c. !0 12 y i 2 τ1 + g0 σ2 R (t − τ1=2)e + H.c. (4) !0 II. MODEL 12 y i 2 τ3 + g0 σ2 L (t − τ3=2)e + H.c. 12 y i!0τ y 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 !0. All three emitters interact (y) (y) dation effects between the excitation exchange, therefore with left- (l! ) and right-moving photons (r! ) in a one- in the Markovian approximation we set R(y)(t − t0) ≈ dimensional waveguide, as depicted in Fig. 1(a). To fo- R(y)(t) and L(y)(t − t0) ≈ L(y)(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 !0 i tant feature for waveguide-based counting experiments. M y 12 i!0τ 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 !0 i bined, one-dimensional waveguide photon-emitter system y 12 12 i 2 τ1 i!0τ 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 y y right- and left-moving fields and no time delay is present H0=~ = !0 σi + d! ! r!r! + l!l! ; (1) i=1 in the interaction. We solve for the system's dynamics in both cases using the time-discrete Schr¨odingerequation where the emitters are treated as two-level systems, with with the time-step size ∆ up to time N∆ in N steps, j1i as the ground state and j2i as the excited state, and cf. Fig. 1(b): ij with σn := jiinnhjj, the flip operator of the n-th emitter. The interaction Hamiltonian describes the emitters inter- j (n)i = UNM/M(n; n − 1)j (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 j (n − 1)i; ~ (n−1)∆ 3 Z X 12 y i!xi=c y −i!xi=c HI = ~g0 σi d! r!e + l!e + 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 j (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 jl1; r1 ··· s; ln; rn; ln+1; rn+1 ··· lN ; rN i; (7) 3 FIG.