Multiphoton pulses interacting with multiple emitters in a one-dimensional waveguide
aa r X i v : . [ qu a n t - ph ] J u l Multiphoton pulses interacting with multiple emitters in a one-dimensional waveguide
Zeyang Liao ∗ , Yunning Lu , and M. Suhail Zubairy School of Physics, Sun Yat-sen University, Guangzhou 510275, the People’s Republic of China Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy,Texas A & M University, College Station, TX 77843-4242, USA
We derive a generalized master equation for multiphoton pulses interacting with multiple emittersin a waveguide-quantum electrodynamics system where the emitter frequency can be modulated.Based on this theory, we can calculate the real-time dynamics of the collective interacting emittersdriven by an incident photon pulse which can be vacuum, coherent states, Fock states or theirsuperpositions. We also derive generalized input-output relations to calculate the reflectivity andtransmissivity of the incident field and the output photon pulse shapes can also be calculated. Ourtheory here can find important applications in the researches of waveguide-based quantum systems.
PACS numbers: 42.50.Nn, 42.50.Ct, 32.70.Jz
I. INTRODUCTION
In recent years, a major goal in quantum optics andquantum information is to build a large scale of quan-tum network from single quantum levels [1]. However, itis a big challenge to manufacture a single quantum sys-tem which contains a large number of qubits. In contrast,it is relatively feasible to build a small quantum systemwith high accurate controls and then connect these smallsystems into a large scale quantum network by certainquantum channels. Waveguide quantum electrodynam-ics (QED), which studies the interaction between emit-ters and waveguide photons, is a very good system forrealizing large scale quantum network and has been at-tracted extensive studies in the past two decades [2–4] .The emitter-photon interaction can be significantly en-hanced in the reduced dimension and the emission of anemitter to the waveguide modes can be nearly unit [5]which can find important applications in the high effi-cient single photon sources [6], single photon detection[7], and atom cavity [8–11]. The emitter-emitter inter-action mediated by the one-dimension (1D) waveguidephotons can be long-ranged which provides a unique sys-tem for studying many-body physics [12, 13] and long-ranged quantum information transfer [14, 15]. Due tothe confinement of transverse field, the photon modesin a quasi-1D waveguide can have intrinsic direction de-pendent longitudinal angular momentum [16] which isextremely suitable for studying chiral quantum optics[17–23]. The waveguide-QED theory can be applied toa number of systems under extensive studies currentlysuch as the photonic line defects coupling to quantumdots [24], cold atoms trapped along the alligator waveg-uide [25], superconducting qubits interacting with themicrowave transmission lines [26, 27], and the plasmonicnanowire coupling to quantum emitters [28].The theory of single photon transport in waveguide- ∗ [email protected] QED is the basics for studying this system and has beenextensively studied in the past two decades. Shen andFan used the real-space Hamiltonian together with theBethe-ansatz to study the stationary properties of sin-gle photon scattering by a single quantum emitter in a1D waveguide [29] and this method is then extended tomulti-emitter systems [30] and multi-level systems wheremany interesting effects can occur such as photon fre-quency conversion [31–33], the electromagnetic inducedtransparency (EIT) effects [34], single photon transistor[35] and single photon switch [36]. In addition to thestationary spectrum, the real-time dynamics of the emit-ter system is also very interesting because the emittersare important units for quantum information processingand storage. Chen et al. applied the wavefunction ap-proach to study the dynamics of a single photon pulseinteracting with a single emitter [37]. We generalizedthe wavefunction approach to the multiple identical [38]and non-identical [39] emitters case from which we canstudy many interesting collective many-body physics andquantum information applications such as quantum statepreparation [40] and waveguide-based quantum sensing[41]. Recently, Dinc et al. developed an analyticalmethod based on Bethe-ansatz approach to study thetime dynamics of a single photon transport problem [42].Compared with single photon transport, the multi-photon scattering can have more interesting physicsbut its calculation is also much more complicated.The Bethe-ansatz approach can be extended to calcu-late the few-photon scattering problem where photon-photon bound states can occur [43–46]. However, whenthis method is generalized to more than two photons,the calculation becomes extremely cumbersome [47–50]. Alternative methods such as Lehmann-Symanzik-Zimmermann reduction method [51, 52] , the Green func-tion decomposition of multiple particle scattering matrix[53], the input-output formalism [54–57], the Feynmandiagrams [58], and the SLH formalism [59, 60] have alsobeen proposed. In these stationary calculations, the pho-ton is usually assumed to be a plane wave and the real-time dynamics of the emitter is usually ignored. Based onHeisenberg-Langevin approach, Domokos et al. studiedthe coherent photon pulse scattering by a single emitterin a waveguide [61]. Kony and Gea-Banacloche gener-alized the wavefunction approach to study the one- andtwo-photon scattering by two emitters in 1D waveguide[62]. In 2015, Caneva et al. used the effective Hamil-tonian approach to derived a master equation to calcu-late the emitter dynamics driven by a coherent photonpulse [63], and this method can be generalized to var-ious systems [64, 65]. For the continuous-mode Fockstates input, Gheri et al. derived a mater equation tostudy the dynamics of a single emitter driven by a singleand two photon wavepacket [66] and Baragiola general-ized this method to the general N-photon case based onthe It¯o Langevin approach [67]. In their studies, theymainly focused on the single quantum system where themany-body interaction between the emitters is not con-sidered. In 2018, we derived a master equation to studythe dynamics of multiple emitters driven by continuoussqueezed vacuum field in 1D waveguide [68] and foundthat steady-state population inversion of multiple Ξ-typeemitters can occur in this system [69].In this article, we consider a multiphoton pulse inter-acting with multiple emitters coupled with 1D waveguide.Since the input photon spectrum is not flat, we can notuse the usual way to derive a master equation using whitenoise limit. Instead, we generalize the method shown inRef. [66] to a more general multi-photon-multi-emittercase where the emitter frequency can be modulated andthe effects of non-waveguide modes can also be included.In addition, we also derived a general input-output the-ory to calculate the reflection and transmission proper-ties of the system. The theory here can be applied tocalculate transport of a large class of field like the vac-uum, coherent states, Fock states and their superposi-tions which can have broad applications in the studyingof waveguide-QED system.This article is arranged as follows. In Sec. II, we de-rive a generalized master equation for the emitter dy-namics and the generalized input-output theory to studythe scattering field properties. In Sec. III, we apply thistheory to the cases of coherent state input, the singleand general N photon input. Finally, we summarize ourresults.
II. MULTIPHOTON SCATTERING THEORY
In this section, we first derive a generalized mas-ter equation for general multiphoton transport in 1Dwaveguide-QED system. Then we derive the generalizedinput-output relations of this system and use it to cal-culate the reflection and transmission properties of thefield.
FIG. 1: Multiphoton pulses interacting with multiple emittersin a 1D waveguide.
A. Generalized master equation for arbitraryphoton input
The model we studied is shown in Fig. 1 where aphoton pulse containing multiple photons is injected intoa 1D waveguide coupling to N a emitters with arbitraryspatial distributions. Here, we consider a general casewhere the emitters can have time modulating frequen-cies and they can couple to both the waveguide and non-waveguide photon modes. It is convenient to work in therotating frame with the original emitter frequency ω a .The total Hamiltonian of the system and reservoir fieldsin the rotating frame is given by H ( t ) = ~ N a X j =1 ε j ( t ) σ zj + ~ X k ∆ ω k a † k a k + ~ X ~q λ ∆ ω ~q λ a † ~q λ a ~q λ + ~ N a X j =1 X k ( g jk e ikz j σ + j a k + H.c. )+ ~ N a X j =1 X ~q λ ( g j~q λ e i~q · ~r j σ + j a ~q λ + H.c. ) . (1)The physical meaning of each term in the Hamiltonianis as follows. The first term is the emitter Hamilto-nian with time-dependent modulating frequency ε j ( t )where j = 1 , , · · · , N a . σ zj and σ + j ( σ − j ) are the zthcomponent and the raising (lowering) Pauli operatorsof the j th emitter. The second term is the Hamilto-nian of the waveguide photons with the detuning fre-quency ∆ ω k = ω k − ω a and a k ( a † k ) is the annihilation(creation) operator of the waveguide photon mode withfrequency ω k . The third term is the non-guided reser-voir field Hamiltonian where a ~q λ ( a † ~q λ ) is the annihila-tion (creation) operator of the non-guided photon modewith frequency ω ~q λ and ∆ ω ~q λ = ω ~q λ − ω a . The fourthterm is the emitter-waveguide photon interaction termwith g jk = ~µ j · ~E k ( ~r j ) / ~ being the coupling strength.The last term is the interaction between the emittersand the non-guided reservoir field with coupling strength g j~q λ = ~µ j · ~E ~q λ ( ~r j ) / ~ .According to the Heisenberg equation, the dynamicsof an arbitrary emitter operator O s is given by˙ O S ( t ) = i N a X j =1 ε j ( t )[ σ zj ( t ) , O S ( t )]+ i N a X j =1 X k (cid:16) g jk e ikz j [ σ + j ( t ) , O S ( t )] a k + g j ∗ k e − ikz j a † k [ σ − j ( t ) , O S ( t )] (cid:17) + i N a X j =1 X ~q λ (cid:16) g j~q λ e i~q · ~r j [ σ + j ( t ) , O S ( t )] a ~q λ + g j ∗ ~q λ e − i~q · ~r j a † ~q λ [ σ − j ( t ) , O S ( t )] (cid:17) , (2)and the dynamics of the field operators are given by˙ a k ( t ) = − i ∆ ω k a k − i N a X j =1 g j ∗ k e − ikz j σ − j ( t ) , (3)˙ a † k ( t ) = i ∆ ω k + i N a X j =1 g jk e ikz j σ + j ( t ) , (4)˙ a ~q λ ( t ) = − i ∆ ω ~q λ a ~q λ − i N a X j =1 g j ∗ ~q λ e − i~q · ~r j σ − j ( t ) , (5)˙ a † ~q λ ( t ) = i ∆ ω ~q λ a † ~q λ + i N a X j =1 g j~q λ e i~q · ~r j σ + j ( t ) . (6)By formally integrating Eqs. (3-6), we can obtain a k ( t ) = a k (0) e − i ∆ ω k t − i N a X j =1 g j ∗ k e − ikz j Z t σ − j ( t ′ ) e i ∆ ω k ( t ′ − t ) dt ′ , (7) a † k ( t ) = a † k (0) e i ∆ ω k t + i N a X j =1 g jk e ikz j Z t σ + j ( t ′ ) e − i ∆ ω k ( t ′ − t ) dt ′ , (8) a ~q λ ( t ) = a ~q λ (0) e − i ∆ ω ~qλ t − i N a X j =1 g j ∗ ~q λ e − i~q · ~r j Z t σ − j ( t ′ ) e i ∆ ω ~qλ ( t ′ − t ) dt ′ , (9) a † ~q λ ( t ) = a † ~q λ (0) e i ∆ ω ~qλ t + i N a X j =1 g j~q λ e i~q · ~r j Z t σ + j ( t ′ ) e − i ∆ ω ~qλ ( t ′ − t ) dt ′ , (10)from which we can see that the field at time t is theinterference between the incident field and the emittedfield by the emitters. Inserting Eqs. (7-10) into Eq. (2)and using the Weisskopf-Wigner approximation we can obtain (see Appendix A)˙ O S ( t ) = i N a X j =1 ε j [ σ zj ( t ) , O S ( t )]+ i N a X j =1 r Γ j σ + j ( t ) , O S ( t )][ a j ( t ) + b j ( t )]+ i N a X j =1 r Γ j a † j ( t ) + b † j ( t )][ σ − j ( t ) , O S ( t )]+ X jl Λ jl [ σ + j ( t ) , O S ( t )] σ − l ( t ) − X jl Λ ∗ jl σ + l ( t )[ σ − j ( t ) , O S ( t )] , (11)where a j ( t ) = q v g π R ∞ e ikz j a k (0) e − iδω k t dk describes theabsorption of the incident waveguide photons and b j ( t ) = q v g π R R R e i~q λ · ~r j a ~q λ (0) e − iδω ~qλ t d ~q λ is the absorption ofthe incident nonguided photons. The collective interac-tion between the emitters can be calculated as [39]Λ jl = p Γ j Γ l e ik a | z jl | + 3 √ γ j γ l h sin φ − ik a r jl + (1 − φ )( 1( k a r jl ) + i ( k a r jl ) ) i e ik a | r jl | , (12)where the first term is the effective interaction mediatedby the waveguide photons and the second term is theusual dipole-dipole interaction induced by the non-guidedreservoir fields. | r jl | = | ~r j − ~r l | is the distance betweenthe jth and lth emitters and | z jl | = | ~z j − ~z l | the distance inthe zth direction. Γ j = 4 π | g jk a | /v g is the decay rate dueto the waveguide vacuum field and γ j is the spontaneousdecay rate due to the nonguided photon modes. φ isthe angle between the direction of the transition dipolemoment and the waveguide direction.From Eq. (11), we can derive a corresponding mas-ter equation for the emitters. Since T r S + R [ O S ( t ) ρ ] = T r S [ O S ρ S ( t )] where ρ S ( t ) = T r R [ ρ ( t )] is the emitter sys-tem density operator, by time derivation on both sideswe have T r S [ O S ˙ ρ S ( t )] = T r S + R [ ˙ O S ( t ) ρ ] and from Eq.(11) we can obtain (see Appendix A)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j σ + j , ρ ′ j ( t )] − i N a X j =1 r Γ j σ − j , ρ ′ † j ( t )] − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − X jl Re(Λ jl )[ σ + j σ − l ρ S ( t ) + ρ S ( t ) σ + j σ − l − σ − l ρ S ( t ) σ + j ] . (13)Here, we consider the case that the incident phontons arefrom the waveguide photons and the nonguided reser-voir field is initially in the vacuum. Since a ~q λ | i =0, we have b j ( t ) ρ (0) = 0. Therefore ρ ′ j ( t ) = T r R [ U ( t ) a j ( t ) ρ (0) U † ( t )] is due to the contribution of theincident waveguide photons. This is the main equationof this section. The first term in Eq. (13) is the fre-quency modulation term. The second and third termsdescribe the excitation and deexcitation due to the inci-dent photon field. The forth term describes the dipole-dipole interactions between the emitters induced by theguided and nonguided vacuum field. The last term is thecollective dissipation due to the guided and nonguidedvacuum fluctuation. However, we should note that Eq.(13) itself is in general not closed because we have thenew operators like ρ ′ j ( t ) and ρ ′ † j ( t ). In some special cases,Eq. (13) is closed. For example, if there is not externaldriving field, the second and third terms disappear andthe equation is closed from which the emitter excitationtransport can be studied. Another example is that if theincident field is a coherent field or superposition of coher-ent fields, the ρ ′ j ( t ) and ρ ′ † j ( t ) terms can then be reducedto a complex number multiplying ρ S ( t ) and Eq. (13) be-comes closed again from which the full dynamics of theemitters driven by a coherent field can be calculated. Ingeneral cases such as the Fock state input, we have torepeat the above procedures to derive equations for ρ ′ j ( t )until all the equations are closed. B. The generalized input-output theory
In the previous subsections, we derive the master equa-tions for the emitter system which allows to calculatethe real dynamics of the emitters for an arbitrary photonwavepacket input. In this subsection, we derive the gen-eralized input-output relations of this system by express-ing the output field operators as the function of inputoperators and the system operators. Together with themaster equations derived in the previous subsection, wecan then study the reflection and transmission propertiesof this system.If we integrate Eq. (3) from t to t f where t f > t , wecan obtain a k ( t ) = a k ( t f ) e i ∆ ω k ( t f − t ) + i N a X j =1 g j ∗ k e − ikz j Z t f t σ − j ( t ′ ) e i ∆ ω k ( t ′ − t ) dt ′ . (14)Comparing Eq. (7) with Eq. (14) it is readily to obtainthat a k ( t f ) e i ∆ ω k ( t f − t ) = a k (0) e − i ∆ ω k t − i N a X j =1 g jk e − ikz j Z t f σ + j ( t ′ ) e i ∆ ω k ( t ′ − t ) dt ′ . (15) We can define the following input-output operators [70] a Rin ( t ) = r v g π Z ∞ a k (0) e − i ∆ ω k t dk, (16) a Lin ( t ) = r v g π Z −∞ a k (0) e − i ∆ ω k t dk, (17) a Rout ( t ) = r v g π Z ∞ a k ( t f ) e iδkz N e − i ∆ ω k ( t − t f ) dk, (18) a Lout ( t ) = r v g π Z −∞ a k ( t f ) e − iδkz e − i ∆ ω k ( t − t f ) dk, (19)where z is the position of the left most emitter and z N is the position of the right most emitter. Since the rightoutput field propagates freely after scattering by th rightmost emitter and the left output field propagates freelyafter scattering by the first emitter, phase factors e iδkz N and e − iδkz are added in the definitions of the right andleft output operators, respectively [63]. From Eq. (15)we can obtain the generalized input-output relations (seeAppendix B) a Rout ( t ) = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j e − ik z j σ − j ( t ) , (20) a Lout ( t ) = a Lin ( t + z /v g ) − i N a X j =1 r Γ j e ik z j σ − j ( t ) , (21)where z Nj = z N − z j . From these two generalizedinput-output relations we can calculate the propertiesof the scattering field of this system. We can definethe instant field intensity propagating to the right andto the left at time t by r ( t ) = h a R + out ( t ) a Rout ( t ) i and l ( t ) = h a L + out ( t ) a Lout ( t ) i , respectively, which are given by r ( t ) = h a R + in ( t − z N /v g ) a Rin ( t − z N /v g ) i− N a X j =1 r Γ j e ik z j h σ + j ( t ) a Rin ( t − z N /v g )]+ X jl √ Γ i Γ l e ik ( z j − z l ) h σ + j ( t ) σ − l ( t ) i , (22) l ( t ) = h a L + in ( t + z /v g ) a Lin ( t + z /v g ) i− N a X j =1 r Γ j e − ik z j h σ + j ( t ) a Lin ( t + z /v g )]+ X jl √ Γ i Γ l e ik ( z l − z j ) h σ + j ( t ) σ − l ( t ) i . (23)On the right hand side of Eqs. (22) and (23), the firstterms are the incident field intensities, the second termsare the absorption and stimulated emission of the system,and the last terms are the spontaneous emission of thesystem. From r ( t ) and l ( t ), we can obtain the pulse shapepropagating to the right and to the left after the scatter-ing process. The field intensity reflected to the left andthe right in the whole scattering process are then givenby I R = R ∞ r ( t ) dt and I L = R ∞ l ( t ) dt . Supposing thatthe photon pulse is initially propagating to the right, thereflectivity of the pulse is then given by R = I L I R + I L , (24)and the transmissivity T = 1 − R .The scattering power spectrum can be usually obtainedfrom the two-time correlation function of the output fieldoperator S ( ω ) = Z ∞ Z ∞ h a + out ( t ) a out ( t ) i e iω ( t − t ) dt dt , (25)where the average is over the initial state of the whole sys-tem. According to the generalized input-output relationshown in Eqs. (20) and (21), a out ( t ) can be expressedas the summation of the input field operator a in ( t ) andthe emitter operators σ − j ( t ). The results when a in ( t )operator acts on the initial state can be readily workedout. Usually, the two-time average of the emitter opera-tors h σ + j ( t ) σ − l ( t + τ ) i can be calculated from the masterequation according to the quantum regression theorem[71]. However, to apply the quantum regression theoremto calculate the two-time correlation function, it usuallyrequires that the reservoir field does not change signifi-cantly. This condition may not be very well satisfied inthe waveguide-QED system because the waveguide pho-ton can be significantly absorbed by the emitters espe-cially near the resonance frequency. Therefore, direct useof the quantum regression theorem to numerically calcu-late the spectrum here may cause some errors and needto be treated carefully. However, at the plane wave limit,an alternative strategy can be used to calculate the scat-tering spectrum. If the incident photon pulse has a verynarrow bandwidth, we can calculate its reflectivity andtransmissivity from the above discussions. After calculat-ing the reflectivity and transmissivity for each frequency,we can then obtain the whole scattering power spectrumof the system at the plane wave limit. III. APPLICATION TO DIFFERENT PHOTONWAVEPACKETS
In this section, we take the coherent states and theFock states as example to show how to apply the theorywe developed in the previous section to study the emitterdynamics and the field scattering property.
A. Coherent state wavepacket
We first consider the case when the incident fieldis a coherent state. Suppose that the incident fieldis a continuous-mode coherent state with wavefunction | Ψ cs i = Π k | α k i where | α k i = e −| α k | / ∞ X n k =0 ( α k ) n k √ n k ! | n k i . (26)The average photon for the k th mode ¯ n k = | α k | .Since a k | Ψ cs i = α k | Ψ cs i , the operator ρ ′ j ( t ) = T r R [ U ( t ) a j ( t ) ρ (0) U † ( t )] = α j ( t ) ρ S ( t ) where α j ( t ) = r v g π Z ∞−∞ e ikz j e − iδω k t α k dk (27)is the real-time incident coherent photon pulse. There-fore, the operator ρ ′ j ( t ) is reduced to a number multi-plying the system density operator ρ S ( t ). The masterequation shown in Eq. (13) then becomes˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j α j ( t ) σ + j + α ∗ j ( t ) σ − j , ρ S ( t )]+ i X jl Im(Λ jl )[ ρ S ( t ) , σ + j σ − l ] + L [ ρ S ( t )] , (28)where L [ ρ S ( t )] = − P jl Re(Λ jl )[ σ + j σ − l ρ S ( t ) + ρ S ( t ) σ + j σ − l − σ − l ρ S ( t ) σ + j ] describes the collectivedissipation process. Equation (28) is the master equa-tion of the waveguide-QED system when the incidentphoton pulse is in a coherent state.The master equation shown in Eq. (28) is itself a closedequation from which we can calculate the real-time dy-namics of the emitters for arbitrary coherent pulse in-put. Here, without loss of generality we assume that thephoton pulse has Gaussian shape throughout this paper.Supposing that the incident coherent field has a Gaussianpulse shape with average photon number ¯ n , its spectrumcan be written as α k = √ ¯ nπ / √ ∆ e − ( k − k ) / e − ikz , (29)where z is the initial central peak position of the pulseand k is the wavevector corresponding to the centralfrequency of the photon pulse. When k > k < n = P k ¯ n k = R ∞−∞ | α k | dk . For the rightpropagating incident pulse (i.e., k > α Rj ( t ) = p ¯ n ∆ v g π / e − ∆2( zj − vgt )22 e ik a z j e i ∆ k ( z j − v g t ) . (30)For the left propagating incident pulse (i.e., k < α Lj ( t ) = p ¯ n ∆ v g π / e − ∆2( zj vgt )22 e − ik a z j e − i ∆ k ( z j + v g t ) , (31) E x c it a ti on p r ob a b ilit y t atom 1 n=1 atom 2 n=1 atom 1 n=20 atom 2 n=20 P u l s e s h a p e ( a r b . un it s ) t R(n=1) T(n=1) R(n=20) T(n=20) two atom: / =1 two atom: / =5 R n n FIG. 2: Coherent state input interacting with two emitters. (a) emitter excitation as a function of time for two different averagephoton number ( n = 1 and n = 20). ∆ / Γ = 1. (b) Reflected and transmitted pulse shapes after the scattering for the sameparameters as (a). (c) The average reflected photon number as a function of average incident photon number for two differentpulse width (∆ / Γ = 1 and ∆ / Γ = 5). For comparison, the results for one emitter are also shown as the blue dashed (∆ / Γ = 1)and red dashed-dotted (∆ / Γ = 5) lines. The emitter distance d = 0 . λ a for all three figures. where z j = z j − z and ∆ k = | k | − k a is the detuningbetween the center frequency of the pulse and the emittertransition frequency.The numerical results for the coherent state input areshown in Fig. 2 where the coherent state is scatteredby two emitters. We assume that the distance betweenthese two emitters is 0 . λ a where λ a = 2 π/k a . Theexcitations of the two emitters as a function time for twodifferent incident average photon number ( n = 1 and n =30) are shown in Fig. 2(a). When the average incidentphoton number is small, e.g. ¯ n = 1, both emitters arefirst excited and then deexcited as the coherent pulsepassing through. However, when the average incidentphoton number is large, e.g. ¯ n = 20, the excitations ofboth emitters can have oscillations which is the signatureof Rabi oscillations.The corresponding reflected and transmitted photonpulse shapes are shown in Fig. 2(b). When the av-erage photon number is small, the reflected pulse (redsolid line) has a single peak and the transmitted pulse(blue dashed line) has two peaks due to the interferencebetween the incident photon and the reemitted photon.When the average photon number is large, most photonsare transmitted (olive dashed line) and only a very smallpart of the photons are reflected (orange solid line). Thisis because the large pulse can saturate the emitter ex-citation and only a very small part of photons can beabsorbed. Here, the reflection photon pulse can have twopeaks instead of one peak due to the Rabi oscillationswhich does not occur when the photon number is small.For a coherent pulse with finite time duration, the av-erage photon number reflected by the emitters may besaturated. Here, we also study the average reflected pho-ton number ¯ n R as a function of average incident photonnumber ¯ n in for two fixed pulse spectrum widths (∆ = Γand ∆ = Γ / . λ a .When the pulse width is about Γ, the average reflectedphoton number increases quickly first as ¯ n in increasesbut then it increases extremely slowly when ¯ n in is largedue to the saturation effect (blue line with open circles in Fig. 2(c)). It is also noted that when the incident pho-ton number is large, the average reflected photon numbercan be larger than two despite that there are only twoemitters. This is because the incident pulse is not shortenough to saturate the emitters immediately. When theincident pulse duration is shorter, i.e. the incident pulsehas broader spectrum (e.g., ∆ = 5Γ), ¯ n R first increasesand then oscillates as ¯ n in increases (red line with solidcircles in Fig. 2(c)) due to the stimulated emission. Theaverage reflected photon number is obviously less than2 because the shorter pulse can saturated the emittersquickly. For comparison, we also plot the results whenthere is only a single emitter in the system (blue dashedline and the red dashed-dotted line). We can see thattheir behaviors are similar but the average reflected pho-ton number for two emitters is larger than that of thesingle emitter. When the pulse width is much smallerthan the decay rate of the emitters, the average photonbeing reflected by a single emitter is less than one whichcan be used to produce single photon sources [72]. B. Single photon wavepacket
Compared with the coherent state input, the calcula-tion of Fock state input is more involved mostly becauseof its quantum nature. The theory developed in Sec. IIcan be also applied for the arbitrary Fock state input.In this subsection, we consider the simplest case whereonly single photon pulse is incident. For the single pho-ton pulse case, we have developed a dynamical transporttheory for calculating the real-time evolution of the sys-tem based on the wavefunction approach [38, 39]. Here,we show that the master equation developed here can bealso applied to the single photon pulse case.Suppose that the incident photon is a single photonwavepacket described by the wavefunction | Ψ F i = Z ∞−∞ α ( k ) a † k | i dk, (32)where R ∞−∞ | α ( k ) | dk = 1. Since a k | Ψ F i = α k | i , the E x c it a ti on p r ob a b ilit y t atom 1 atom 2 t P u l s e s h a p e ( a r b . un it s ) R T -4 -2 0 2 40.00.20.40.60.81.0(c) R e f l ec ti v it y a nd t r a n s m i ss i v it y R T
FIG. 3: A single photon wavepacket interacting with two emitters. (a) emitter excitation as a function of time. ∆ / Γ = 1.(b) Reflected and transmitted pulse shapes after the scattering for the same parameters as (a). (c) The reflectivity andtransmissivity as a function of photon frequency. In all three figures, d = 0 . ρ ′ j ( t ) term in Eq. (13) is then given by ρ ′ j ( t ) = T r R [ U ( t ) a j ( t ) ρ (0) U + ( t )] = α j ( t ) ρ S ( t ) , (33)where α j ( t ) is given by Eq. (27) and we define a newoperator ρ S ( t ) = T r R [ U ( t ) ρ S (0) ⊗ | ih Ψ F | U + ( t )]. If wedefine ρ S ( t ) = T r R [ U ( t ) ρ S (0) ⊗| Ψ F ih Ψ F | U † ( t )], we havefrom Eq. (13) that˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j α j ( t ) σ + j ( t ) , ρ S ( t )] − i N a X j =1 r Γ j α ∗ j ( t ) σ − j ( t ) , ρ S † ( t )] − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (34)where L [ ρ S ( t )] = P jl Re(Λ jl )[ σ + j σ − l ρ S ( t ) + ρ S ( t ) σ + j σ − l − σ − l ρ S ( t ) σ + j ] is the collective dissi-pation term. ρ S ( t ) is not a valid density matrix becauseit is traceless but it satisfies ρ S † = ρ S . Since a newoperator ρ S ( t ) appears, Eq. (34) is itself not a closedequation and we need to derive an extra equation for ρ S ( t ).The dynamical equation for ρ ( t ) can be derived usingsimilar procedures as deriving ρ S ( t ) shown in Sec. II andit is given by (see Appendix C)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j α ∗ j ( t )[ σ − j , ρ S ( t )] − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (35)where ρ S ( t ) = T r R [ U ( t ) ρ S ⊗ | ih | U † ( t )] is another den-sity matrix describing the evolution of the system when initially there is no photon. Using similar procedure, itis not difficult to obtain that˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (36)where we see that no new density operator appears, sothe equations are now closed.Hence, the master equation for the single photon stateinput consists of three cascaded equations as given byEqs. (34-36) where a single equation is needed in thecoherent state input. The dynamics of the emitters forarbitrary single photon pulse input can then be calcu-lated from these three equations. The time evolution ofthe average value of an arbitrary emitter operator O ( t )can be then calculated as h O ( t ) i = T r S [ Oρ S ( t )].One numerical example is shown in Fig. 3 wherewe consider a single photon wavepacket interacting withtwo emitters. Here, we assume that the single photonwavepacket has a Gaussian spectrum as shown in Eq.(29) and the distance between emitters is λ a /
8. Theemitter excitation as a function of time is shown in Fig.3(a). Due to the collective interaction, the first emittercan have much higher excitation probability than that ofthe second one and the excitation of the second emitterhas a Rabi-like oscillations which does not occur when theincident photon pulse is in a coherent state with ¯ n in = 1.This is due to the interference between two excitationchannels, i.e., the excitation of the incident photon andthe excitation by the first excited emitter. In the coher-ent state input, this interference is concealed.The reflected and transmitted photon pulse shapes areshown in Fig. 3(b) from which we can see that the trans-mitted pulse has multiple peaks due to the quantum in-terference between the incident photon and the reemit-ted photons by the two emitters. The visibility of theoscillation is much larger than that in the coherent stateinput. The reflected and transmitted spectra when theincident single photon is a plane wave are shown in Fig.3(c) where we can see the asymmetric Fano-like struc-ture. This is caused by the interference between two E x c it a ti on p r ob a b ilit y t atom 1 atom 2 R T P u l s e s h a p e ( a r b . un it s ) t FIG. 4: Two photon wavepacket interacting with two emit-ters. (a) emitter excitation as a function of time. ∆ / Γ = 1.(b) Reflected and transmitted pulse shapes after the scatter-ing for the same parameters as (a). In both figures, λ a / emission channels, i.e., the emission from the two collec-tive excited states √ ( | eg i ± | ge i ) which have differentenergy shifts and decay rates. The results shown in Fig.3 are consistent with the results we calculated based on the wavefunction approach [38]. C. N-photon wavepacket
In addition to the single photon Fock state, we can alsoderive generalized master equations for the multi-photonFock state input. Compared with the single-photon in-put, the calculation of multi-photon Fock state input ismore complicated. We first consider a relative simplesubset which is the direct generalization of the single pho-ton wavepacket, i.e., | N α i = 1 √ N ! h Z ∞−∞ dkα ( k ) a † k i N | i , (37)where we have the normalization condition R ∞−∞ | α k | dk = 1. A general N-photon wavepacketcan be always decomposed to the superposition of thewavefunction shown in Eq. (37). For the wavepacketshown in Eq. (37), we can have a k | N α i = a k √ N ! h Z ∞−∞ dk ′ α ( k ′ ) a + k ′ i N | i = √ N α ( k ) | N − α i . (38)In general, we have the relation a k | m α i = √ mα ( k ) | m − α i and therefore a k ρ s (0) ⊗| m α ih n α | = √ mα ( k ) ρ s (0) ⊗| m − α ih n α | . (39)Using the similar procedure to derive Eq. (13), wecan derive a ladder set of dynamical equations for theN-photon wavepacket input which is given by˙ ρ Smn ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ Smn ( t )] − i N a X j =1 r Γ j {√ mα j ( t )[ σ + j , ρ Sm − ,n ( t )] + √ nα ∗ j ( t )[ σ − j ( t ) , ρ Smn − ( t )] }− i X jl Im(Λ jl )[ σ + j σ − l , ρ Smn ( t )] − L [ ρ Smn ( t )] , (40)where ρ Smn ( t ) = T r R [ U ( t ) ρ S (0) ⊗ | m ih n | U + ( t )] and 0 ≤ m, n ≤ N . Considering that ρ Smn = ρ S † nm , ( N + 1)( N +2) / N is the total incident photon number. Forexample, three master equations are needed for single-photon input , while six master equations for two-photon input. For the general N-photon input state, we can al-ways decompose it into the superposition of the form ofEq. (37) and then we can follow the same procedure toderive a set of closed master equations.Taking the two-photon input as an example, the mas-ter equations are given by˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j {√ α j ( t )[ σ + j , ρ S ( t )] + √ α ∗ j ( t )[ σ − j ( t ) , ρ S ( t )] }− i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (41)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j { α j ( t )[ σ + j , ρ S ( t )] + √ α ∗ j ( t )[ σ − j ( t ) , ρ S ( t )] }− i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (42)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j { α j ( t )[ σ + j , ρ S ( t )] + α ∗ j ( t )[ σ − j ( t ) , ρ S ( t )] }− i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (43)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j {√ α ∗ j ( t )[ σ − j ( t ) , ρ S ( t )] } − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (44)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j { α ∗ j ( t )[ σ − j ( t ) , ρ S ( t )] } − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (45)˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − L [ ρ S ( t )] , (46)and we have ρ Snm = ρ S † mn . Hence, for two-photonwavepacket, six cascaded master equations are requiredto calculate the dynamics of the system.A numerical example is shown in Fig. 4 where we con-sider two-photon interacting with two emitters. Similarto the single-photon case, we also assume that the two-photon pulse has a Gaussian spectrum and the distancebetween the emitters is λ a /
8. Compared with the singlephoton case, the emitter excitation in the two-photoncase is larger and both excitations increase first and thendecrease which is similar to the coherent state input (Fig.4(a)). Different from the single photon case, the emitter2 does not have Rabi-oscillation like structure. This ismainly because the double excited state | ee i can also bepopulated in the two photon cases and it can cover theinterference effect like in the single photon case. Thecorresponding reflected and transmitted pulse shapes areshown in Fig. 4 (b) from which we can see that theyare similar to those in the single photon case but thetransmitted pulse has only a small oscillation in the twophoton case. D. The effects of pulse width
In the stationary scattering theory, the incident fieldis usually assumed to be a plane wave. In practical ex- periments, the incident light is always a pulse with finiteduration and finite bandwidth. Here, our theory allowsus to study the effects of the pulse widths.Taking the single emitter as an example, we investi-gate the average reflected photon number as a functionof pulse widths for different input photon states. The re-sults are shown in Fig. 5(a). For all four incident pulses,¯ n R decreases when the pulse spectrum width increases(i.e., the pulse duration becomes shorter) due to the sat-uration effects. When the pulse has a white spectrum(i.e, the pulse duration is extremely short), almost nophoton will be reflected for both the coherent state in-puts and the Fock state inputs because most photon fre-quencies are far detuned from the resonance frequency.In contrast, when the pulse spectrum is extremely nar-row (i.e., the pulse is at the plane wave limit) and itsfrequency is in resonance with the emitter transition fre-quency, almost all of the incident photons will be reflectedfor both the Fock state inputs and the coherent state in-puts. When the pulse spectrum width is finite, the Fockstate input can have larger reflectivity than that of thecoherent state input with the same average incident pho-ton number. For the same pulse width, the pulse with¯ n in = 1 has larger reflectivity than that of the pulse with¯ n in = 2 due to saturation effects.The reflectivity and transmissivity by a single emitteras a function of detuning frequency for the Fock state0 Pulse width/ Fock-1 Coherent-1 Fock-2 Coherent-2 R n -4 -2 0 2 40.00.20.40.60.81.0 R & T R(coherent) T(coherent) R(Fock) T(Fock) (b)
FIG. 5: (a) The average reflected photon number by a singleemitter as a function of pulse width for four different inci-dent photon states. Fock-1: a single photon state; Fock-2:two photon state; Coherent-1: coherent state with 1 averagephoton number; Coherent-2: coherent state with 2 averagephoton number. The orange dashed line is the correspondingreflectivity for the two photon state. The olive dashed-dottedline is the corresponding reflectivity of the Coherent-2 state.(b) The reflectivity and transmissivity as a function of inci-dent photon frequency when there is only a single emitter fortwo different kinds of input quantum state. Both incidentpulses have very narrow spectrum width at the plane wavelimit. input and the coherent state input at the plane wavelimit are shown in Fig. 5(b). It is seen that the re-flectivity and transmissivity are exactly the same for theFock state input and the coherent state input at the planewave limit. When the incident frequency is resonant withthe emitter transition frequency, it will be completely re-flected due to quantum interference. When the photonfrequency is large detuned from the emitter frequency, itcan pass through the emitter without being scattering.The widths of the reflectivity and transmissivity dependon the emitter decay rate. Therefore, the reflectivity andtransmissivity for a certain frequency is a property of thewaveguide-QED system, and it does not depend on thephoton statistics of the incident photons. However, for anincident photon with finite spectrum width, the reflectiv-ity and transmissivity can strongly depend on the pulsewidth and the photon statistics of the incident photons. E x c it a ti on p r ob a b ilit y t atom 1 atom 2 atom 3 atom 4 atom 5 (a) P u l s e s h a p e ( a r b . un it s ) t R T w/o modulation t E x c it a ti on p r ob a b ilit y w modulation (c) P u l s e s h a p e ( a r b . un it s ) t R w mod. T w mod. R w/o mod. T w/o mod.
FIG. 6: (a) Emitter excitation as a function of time for fiveemitters. The distance between nearest emitters is 0 . λ a .The pulse spectrum width ∆ = Γ. (b) The reflected andtransmitted pulse shapes for the same parameters as (a). (c)Emitter excitation as a function of time for a single emitterwith frequency modulation (solid red line). For comparison,the excitation without modulation is also shown as the blackdashed line. ε ( t ) = 10Γ sin(10Γ t ). The pulse spectrum width∆ = Γ. (d) The reflected and transmitted pulse shapes forthe same parameters as (c). E. Multiple emitters and frequency modulation
In addition to the one or two emitters, our theory canbe applied to calculate the interacting of photon pulsewith arbitrary number of emitters until the computa-tion power is saturated. Here, we take five emitters withnearest neighbor distance 0 . λ a as an example. The ex-citation probability for the five emitters as a function oftime is shown in Fig. 6(a) where we assume that theincident photon pulse is in a coherent state with aver-age photon number 1. We can see that the first emitterhas the largest excitation probability, but it is quicklydeexcited and can transfer its energy to the other emit-ters. The other emitters have smaller excitation prob-abilities, but they can oscillate and last for a period oftime much longer than the decay time of single emitterand the incident pulse duration. This is because the col-lective subradiant states can be formed due to the emitterinteractions and they can be populated by the incidentphoton pulse. The corresponding reflected and transmit-ted photon pulses are shown in Fig. 6(b). Most energyis reflected and the reflected pulse has a major peak. Incontrast the transmitted pulse has multiple peaks due toquantum interference between the incident field and thereemitted fields by the emitters.Our theory also allow us to calculate the transport dy-namics when the emitter frequencies are externally mod-ulated. As an example, we consider a single emitter in-teracting with a coherent photon pulse. The emitter’sfrequency is modulated such that ε ( t ) = 10Γ sin(10Γ t ).1The emitter excitation as a function of time is shown asthe red solid line in Fig. 6(c). For comparison, the re-sult without frequency modulation is also plotted as theblack dashed line. It is seen that the excitation with mod-ulation is smaller and has some small oscillations. Thecorresponding scattering pulses are shown in Fig. 6(d)where the red solid line is the reflected pulse and the bluedashed line is the transmitted pulse. We can see that thereflected pulse has small modulations, while the trans-mitted pulse has very significant modulations. In com-parison, the scattering pulses without modulations havesmooth shapes (black solid and dashed lines). Hence, byfrequency modulation we can realize complicated photonpulse shaping. IV. SUMMARY
In this article we derive master equations for multi-photon interacting with multiple emitters coupled to 1Dwaveguide. Our theory can be applied to calculate thetransport of arbitrary incident photon wavepackets withvery general states of light such as coherent state, Fockstate and their superpositions. It can also be used to cal-culate the scattering of multiple emitters with randomdistribution and even with external frequency modula-tion. We compare the dynamics of emitters and scat-tering pulse shapes when the incident photon pulses arecoherent state, single photon state or multiple photon states. With finite incident pulse width, different statesof light can induce different system dynamics and differ-ent scattering properties. The average reflected photonnumber by a single emitter decreases when the incidentpulse duration is shorter for both the coherent state inputand the Fock state input, but the Fock state input canhave higher average reflected photon number than thatof the coherent state input with the same average pho-ton number. This result can be useful for single photongeneration. At the plane wave limit, the reflectivity andtransmissivity of the waveguide-QED system for a certainfrequency are the same and do not depend on the statis-tics of the incident photons. Our theory also allows tostudy the scattering properties of a photon pulse by emit-ters with frequency modulations which can be used forphoton pulse shaping. Thus, the theory developed herecan become an important basics for studying the many-body physics and quantum information applications inthe waveguide-QED system.
V. ACKNOWLEDGEMENT
This research is supported by startup grants (No.74130-18841222 and No. 74130-31610033) from Sun Yat-sen University and the Key R&D Program of GuangdongProvince (Grant No. 2018B030329001). The research ofMSZ is supported by a grant from King Abdulaziz cityfor Science and Technology (KACST).2
Appendix A: Derivation of Eq. (13)
On inserting Eqs. (7-10) in the main text into Eq. (2), we can obtain˙ O S ( t ) = i N a X j =1 ε j ( t )[ σ zj ( t ) , O S ( t )]+ i N a X j =1 X k g jk e ikz j e − i ∆ ω k t [ σ + j ( t ) , O S ( t )] (cid:16) a k (0) − i N a X l =1 g l ∗ k e − ikz l Z t σ − l ( t ′ ) e i ∆ ω k t ′ dt ′ (cid:17) + i N a X j =1 X k g j ∗ k e − ikz j e i ∆ ω k t (cid:16) a † k (0) + i N a X l =1 g lk e ikz l Z t σ + l ( t ′ ) e − i ∆ ω k t ′ dt ′ (cid:17) [ σ − j ( t ) , O S ( t )]+ i N a X j =1 X ~q λ g j~q λ e i~q · ~r j e − i ∆ ω ~qλ t [ σ + j ( t ) , O S ( t )] (cid:16) a ~q λ (0) − i N a X l =1 g l ∗ ~q λ e − i~q · ~r l Z t σ − l ( t ′ ) e i ∆ ω ~qλ t ′ dt ′ (cid:17) + i N a X j =1 X ~q λ g j ∗ ~q λ e − i~q · ~r j e i ∆ ω ~qλ t (cid:16) a † ~q λ (0) + i N a X l =1 g l~q λ e i~q · ~r l Z t σ + l ( t ′ ) e − i ∆ ω ~qλ t ′ dt ′ (cid:17) [ σ − j ( t ) , O S ( t )]= i N a X j =1 ε j ( t )[ σ zj ( t ) , O S ( t )]+ i N a X j =1 X k g jk e ikz j e − i ∆ ω k t [ σ + j ( t ) , O S ( t )] a k (0) + i N a X j =1 X k g j ∗ k e − ikz j e i ∆ ω k t a † k (0)[ σ − j ( t ) , O S ( t )]+ i N a X j =1 X ~q λ g j~q λ e i~q · ~r j e − i ∆ ω ~qλ t [ σ + j ( t ) , O S ( t )] a ~q λ (0) + i N a X j =1 X ~q λ g j ∗ ~q λ e − i~q · ~r j e i ∆ ω ~qλ t a † ~q λ (0)[ σ − j ( t ) , O S ( t )]+ N a X jl X k g jk g l ∗ k e ik ( z j − z l ) e − i ∆ ω k t [ σ + j ( t ) , O S ( t )] Z t σ − l ( t ′ ) e i ∆ ω k t ′ dt ′ − N a X jl X k g jk g l ∗ k e − ik ( z j − z l ) e i ∆ ω jk ( t ) t Z t σ + l ( t ′ ) e − i ∆ ω k t ′ dt ′ [ σ − j ( t ) , O S ( t )]+ N a X jl X ~q λ g j~q λ g l ∗ ~q λ e i~q · ( ~r j − ~r l ) e − i ∆ ω ~qλ t [ σ + j ( t ) , O S ( t )] Z t σ − l ( t ′ ) e i ∆ ω ~qλ t ′ dt ′ − N a X jl X ~q λ g j ∗ ~q λ g l~q λ e − i~q · ( ~r j − ~r l ) e i ∆ ω ~qλ t Z t σ + l ( t ′ ) e − i ∆ ω ~qλ t ′ dt ′ [ σ − j ( t ) , O S ( t )] . (A1)According to the Weisskopf-Wigner approximation, we have [39] X k g jk g l ∗ k e ik ( z j − z l ) e − i ∆ ω k t e i ∆ ω k t ′ = p Γ j Γ l e ik a | z jl | e iε j ( t ) t − iε l ( t ′ ) t ′ δ [ t ′ − ( t − | z jl | v g )] , (A2) X k g jk g l ∗ k e − ik ( z j − z l ) e i ∆ ω k t e − i ∆ ω k t ′ = p Γ j Γ l e − ik a | z jl | e − iε j ( t ) t + iε l ( t ′ ) t ′ δ [ t ′ − ( t − | z jl | v g )] , (A3) X ~q λ g j~q λ g l ∗ ~q λ e i~q · ( ~r j − ~r l ) e − i ∆ ω ~qλ t e i ∆ ω ~qλ t ′ = Ω jl e iε j ( t ) t − iε l ( t ′ ) t ′ δ [ t ′ − ( t − | r jl | v g )] , (A4) X ~q λ g j ∗ ~q λ g l~q λ e − i~q · ( ~r j − ~r l ) e i ∆ ω ~qλ t e − i ∆ ω ~qλ t ′ = Ω ∗ jl e − iε j ( t ) t + iε l ( t ′ ) t ′ δ [ t ′ − ( t − | r jl | v g )] , (A5)where Γ i = Lv g | g ik | with g ik = q Γ i v g L and Ω jl = √ γ j γ l [sin φ − ik a r jl + (1 − φ )( k a r jl ) + i ( k a r jl ) )] e ik a r jl with r jl = | ~r j − ~r l | .3To proceed, we assume that the emitters are close such that z ij /v g ≪ / Γ, we can approximate that σ − j ( t − z jl v g ) ≈ σ − j ( t ) in the rotating frame. Indeed, this is the usual case. For example, if v g ∼ m/s and Γ ∼ Hz , we requirethat the distance between the emitters z ij ≪ m which is the usual case. By doing this approximation, Eq. (13) thenbecomes˙ O S ( t ) = i N a X j =1 ε j [ σ zj ( t ) , O s ( t )] + i N a X j =1 r Γ j σ + j ( t ) , O S ( t )][ a j ( t ) + b j ( t )] + i N a X j =1 r Γ j a † j ( t ) + b † j ( t )][ σ − j ( t ) , O S ( t )]+ X jl Λ jl [ σ + j ( t ) , O S ( t )] σ − l ( t ) − X jl Λ ∗ jl σ + l ( t )[ σ − j ( t ) , O S ( t )] , (A6)where a j ( t ) = q v g π R ∞ e ikz j a k (0) e − iδω k t dk is the absorption of the incident waveguide photons and b j ( t ) = q v g π R R R e i~q λ · ~r j a ~q λ (0) e − iδω ~qλ t d ~q λ is the absorption of the incident nonguided photons. The collective interactionbetween the emitters is given by [39]Λ jl = p Γ j Γ l e ik a | z jl | + 3 √ γ j γ l φ − ik a r jl + (1 − φ )( 1( k a r jl ) + i ( k a r jl ) )] e ik a | r jl | . (A7)From Eq. (A6), we can derive a corresponding master equation for the emitters. Since T r S + R [ O S ( t ) ρ ] = T r S [ O S ρ S ( t )] where ρ S ( t ) = T r R [ ρ ( t )], we have T r S [ O S ˙ ρ S ( t )]= T r S + R [ ˙ O S ( t ) ρ ]= i N a X j =1 ε j ( t ) T r S + R { [ σ zj ( t ) , O S ( t )] ρ } + i N a X j =1 r Γ j T r S + R { [ σ + j ( t ) , O S ( t )][ a j ( t ) + b j ( t )] ρ } + i N a X j =1 r Γ j T r S + R { [ a † j ( t ) + b † j ( t )][ σ − j ( t ) , O S ( t )] ρ } + X jl Λ jl T r S + R { [ σ + j ( t ) , O S ( t )] σ − l ( t ) ρ } − X jl Λ ∗ jl T r S + R { σ + l ( t )[ σ − j ( t ) , O S ( t )] ρ } = − i N a X j =1 ε j ( t ) T r S { O S [ σ zj , ρ S ( t )] } + i N a X j =1 r Γ j T r S { O S [ ρ jin ( t ) , σ + j ] } + i N a X j =1 r Γ j T r S { O S [ ρ j † in ( t ) , σ − j ] }− X jl T r S { O S [ σ + j σ − l ρ S ( t ) − σ − l ρ S ( t ) σ + j ] } − X jl Λ ∗ jl T r S { O S [ ρ S ( t ) σ + l σ − j − σ − j ρ S ( t ) σ + l ] } , (A8)where ρ jin ( t ) = T r R { U ( t )[ a j ( t ) + b j ( t )] ρ (0) U † ( t ) } is the contribution from the incident sources. In this paper, weconsider that the incident photon is coming from the waveguide photons and the non-guided modes are initially inthe vacuum. Since a ~q λ (0) | i = 0, we have b j ( t ) ρ (0) = 0 and therefore ρ jin ( t ) = T r R { U ( t ) a j ( t ) ρ (0) U † ( t ) } is due to thecontribution of the incident waveguide photons. Comparing both size of Eq. (A8), we can obtain the master equationfor the system density matrix given by˙ ρ S ( t ) = − i N a X j =1 ε j ( t )[ σ zj , ρ S ( t )] − i N a X j =1 r Γ j σ + j , ρ jin ( t )] − i N a X j =1 r Γ j σ − j , ρ j † in ( t )] − i X jl Im(Λ jl )[ σ + j σ − l , ρ S ( t )] − X jl Re(Λ jl )[ σ + j σ − l ρ S ( t ) + ρ S ( t ) σ + j σ − l − σ − l ρ S ( t ) σ + j ] , (A9)which is the master equation shown in Eq. (13) in the main text.4 Appendix B: Derivation of the input-output relations
From Eqs. (15-19) in the main text, we can obtain a Rout ( t ) = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j v g π Z t f σ − j ( t ′ ) dt ′ r v g π Z ∞ e iδkz N e − ikz j e i ∆ ω k ( t ′ − t ) dk = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j v g π r v g π e − ik a z j Z t f σ − j ( t ′ ) dt ′ Z ∞ e iδk ( z N − z j ) e iδkv g ( t ′ − t ) dk = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j v g π r v g π e − ik a z j Z t f σ − j ( t ′ ) dt ′ Z ∞− k e iδkz Nj e iδkv g ( t ′ − t ) dδk = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j v g π r v g π e − ik a z j Z t f σ − j ( t ′ ) dt ′ Z ∞−∞ e iδkz Nj e iδkv g ( t ′ − t ) dδk = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j v g π r v g π e − ik a z j Z t f σ − j ( t ′ ) 2 πv g δ ( t ′ − t + z Nj /v g ) dt ′ = a Rin ( t − z N /v g ) − i N a X j =1 r Γ j e − ik a z j σ − j ( t − z Nj /v g ) ≈ a Rin ( t − z N /v g ) − i N a X j =1 r Γ j e − ik z j σ − j ( t ) , (B1)where z Nj = z N − z j . Similarly, we have a Lout ( t ) = a Lin ( t + z /v g ) − i N a X j =1 r Γ j v g π Z t f σ − j ( t ′ ) dt ′ r v g π Z −∞ e − iδkz e − ikz j e i ∆ ω k ( t ′ − t ) dk = a Lin ( t + z /v g ) − i N a X j =1 r Γ j v g π r v g π e ik a z j Z t f σ − j ( t ′ ) dt ′ Z ∞ e iδkz j e iδkv g ( t ′ − t ) d ( − k )= a Lin ( t + z /v g ) − i N a X j =1 r Γ j v g π r v g π e ik a z j Z t f σ − j ( t ′ ) dt ′ Z ∞− k e iδkz j e iδkv g ( t ′ − t ) dδk = a Lin ( t + z /v g ) − i N a X j =1 r Γ j v g π r v g π e ik a z j Z t f σ − j ( t ′ ) dt ′ Z ∞−∞ e iδkz j e iδkv g ( t ′ − t ) dδk = a Lin ( t + z /v g ) − i N a X j =1 r Γ j v g π r v g π e ik a z j Z t f σ − j ( t ′ ) 2 πv g δ ( t ′ − t + z j /v g ) dt ′ = a Lin ( t + z /v g ) − i N a X j =1 r Γ j e ik a z j σ − j ( t − z j /v g ) ≈ a Lin ( t + z /v g ) − i N a X j =1 r Γ j e ik a z j σ − j ( t ) . (B2)Eqs. (B1) and (B2) are the input-output relations of the system from which we can calculate the field scatteringproperties of this system.5 Appendix C: Derivation of Eq. (35)
We can derive a dynamical equation for ρ ( t ) using similar method as deriving ρ S ( t ). Since T r S + R [ O S ρ ( t )] = T r S + R [ O S ( t ) ρ (0)] where ρ (0) = ρ S (0) ⊗ | ih Ψ F | , we have T r S + R [ O S ˙ ρ ( t )]= T r S + R [ ˙ O S ( t ) ρ (0)]= i N a X j =1 ε j ( t ) T r S + R { [ σ zj ( t ) , O s ( t )] ρ (0) } + i N a X j =1 r Γ j T r S + R { [ σ + j ( t ) , O S ( t )] a jin ( t − z j /v g ) ρ (0) } + i N a X j =1 r Γ j T r S + R { a j † in ( t − z j /v g )[ σ − j ( t ) , O S ( t )] ρ (0) } + X jl Λ jl T r S + R { [ σ + j ( t ) , O S ( t )] σ − l ( t ) ρ } − X jl Λ ∗ jl T r S + R { σ + l ( t )[ σ − j ( t ) , O S ( t )] ρ } = − i N a X j =1 ε j ( t ) T r S + R { [ σ zj , ρ ( t )] O s } − i N a X j =1 r Γ j α ∗ j ( t − z j /v g ) T r S + R { O S [ σ − j , ρ ( t )] }− X jl Λ jl T r S + R { O S [ σ + j σ − l ρ ( t ) − σ − l ρ ( t ) σ + j ] } − X jl Λ ∗ jl T r S + R { O S [ ρ ( t ) σ + l σ − j − σ − j ρ ( t ) σ + l ] } , (C1)where ρ ( t ) = U ( t ) ρ S ⊗ | ih | U † ( t ). 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