Collisional picture of quantum optics with giant emitters

The effective description of the weak interaction between an emitter and a bosonic field as a sequence of two-body collisions provides a simple intuitive picture compared to traditional quantum optics methods as well as an effective calculation tool of the joint emitter-field dynamics. Here, this collisional approach is extended to many emitters (atoms or resonators), each generally interacting with the field at many coupling points (``giant'' emitter). In the regime of negligible delays, the unitary describing each collision in particular features a contribution of a chiral origin resulting in an effective Hamiltonian. The picture is applied to derive a Lindblad master equation (ME) of a set of giant atoms coupled to a (generally chiral) waveguide field in an arbitrary white-noise Gaussian state, which condenses into a single equation and extends a variety of quantum optics and waveguide-QED MEs. The effective Hamiltonian and jump operators corresponding to a selected photodetection scheme are also worked out.


I. INTRODUCTION
A major focus of quantum optics is the interaction of quantum emitters, such as (artifical) atoms or resonators, with a field modeled as a continuum of bosonic modes. Accordingly, describing the dynamics generally requires to keep track of all the field modes, a task which at times can be circumvented when the focus is the open dynamics of the emitters, provided that a master equation is preliminarily derived and ensured to be completely positive. This tool is yet insufficient and must be complemented with appropriate field equations whenever one is interested in the dynamics of photons.
The CM-based description has a number of interesting features such as: 1. A simple and intuitive picture of the joint dynam- The field is decomposed into non-interacting time bins traveling at constant speed. One at a time, these undergo a short two-body interaction with the emitter (collision). In the regime of negligible time delays, a similar conveyor-belt picture holds for many emitters each of which can be giant (i.e., interacting with the field at many coupling points).
ics, helpful to get insight into the problem at hand.

A direct,
Born-Markov-approximation-free, derivation of Lindblad MEs guaranteed to be completely positive. 3. The time-bin evolution is easily worked out, thus enabling to keep track of a relevant part of the field dynamics. 4. CMs are the natural microscopic framework to describe continuous weak measurements, which can be applied to photon detection [8,16,17,36]. 5. When formulated as a CM, the dynamics turns into an equivalent quantum circuit, allowing in particular for Matrix Product States simulations [1, 8,11,[37][38][39].
In the framework of quantum optics, so far only CMs for pointlike quantum emitters were fully developed (only one coupling point). While CMs featuring two coupling points were considered in the regime of long time delays [1, 2,5], a comprehensive formulation of the negligible-delay regime (occurring in most experiments) is still missing.
In this work, we present a general theory of the CM-based description of quantum optics in the case of many emitters. We allow each of these to generally couple to the field at many coupling points so as to encompass systems such as the so called "giant" atoms [40,41], which can now be experimentally implemented and operated [42,43], or bosonic oscillators/atomic ensembles coupled to 1D fields in looped geometries [44,45] as explicitly discussed in Ref. [46]. The framework is first formulated by considering a unidirectional field (just like in standard input-output formalism [47]) and then extended to a bidirectional field. While both the regimes of negligible and long time delays are discussed, our main focus is the former. In which case, it will be proven that each collision can be effectively represented as a collective coupling of all the emitters with one field time bin plus an internal coherent dipole-dipole interaction between the emitters described by a Hamiltonian originating from the intrinsic system's chirality.
While the presented collisional framework has many potential uses, here we apply it to derive the Lindblad master equation of a set of giant emitters coupled to a, generally chiral, one-dimensional waveguide when the field starts in an arbitrary Gaussian state. This condenses in a single equation and extends a variety of master equations used in waveguide QED [40,[48][49][50], as will be illustrated in detail. Moreover, we show that the recently discovered possibility to realize decoherence-free Hamiltonians with giant emitters [46,51] is naturally predicted in the collisional picture, without the need to resort to the master equation, thus highlighting its independence of the field state. Additionally, for an arbitrary photodetection scheme, we calculate the Kraus operators corresponding to a measurement outcome and use these to derive the effective Hamiltonian and jump operators generating the quantum trajectories.
The present paper in fact comprises two parts. The first of which presents the general emitters-field microscopic model (Section II), outlines the main collision model features without proof, the aforementioned general master equation and the description of photodetection and related quantum trajectories (Section III). Special cases of the master equation are illustrated in a separate section (Section IV), which ends with a discussion of decoherence-free Hamiltonians (Section IV A).
The second (more technical) part derives in detail the collision model for a unidirectional field (Section V), works out the ensuing master equation (when existing) in the negligible-delays regime (Section VI), ex-tend these tasks to a bidirectional field (Sections VII and VIII) and finally addresses in detail photodetection and quantum trajectories (Section IX).

II. MICROSCOPIC MODEL
The general emitters-field microscopic model we consider is essentially the same as that underpinning the standard input-output formalism of quantum optics [47] and related theories such as SLH [52].
Let S be a system made out of N e quantum "emitters" of frequency ω 0 and associated ladder operatorŝ A j ,Â † j for j = 1, ..., N e . The statistical nature of these operators is left unspecified, hence in particular each emitter could be a harmonic oscillator or a pseudospin (linear and non-linear, respectively). The emitters are weakly coupled to a unidirectional bosonic field with normal-mode ladder operators . The jth emitter interacts with the field at N j distinct coupling points. For N j = 1 we retrieve the standard local coupling and the emitter is called "normal" [see Fig. 2(a)]. Instead, if N j ≥ 2, the coupling is multi-local and the emitter is dubbed "giant" [see Fig. 2(a)]. The spatial coordinate of the th coupling point of the jth emitter is x j (the field is along the x-axis). Under the usual rotating-wave approximation (RWA) and assuming white coupling, the total Hamiltonian reads (we set where all integrals run over the entire real axis compatibly with the RWA. Here, τ j = x j /v is the coordinate in the time domain of each coupling point (the field dispersion law is ω = vk). Note that here ω are frequencies measured from the emitters' energy ω 0 (i.e., detunings in fact). We also point out that each coupling point has an associated position-dependent phase factor e iω 0 τ j , which can be equally written in the space domain as e ik 0 x j with k 0 = ω 0 /v. Instead of ω-dependent normal modes, the field can be equivalently represented in terms of time modes with ladder operatorŝ fulfilling bosonic commutation rules (a): A normal emitter (such as 1) interacts with the field at a single coupling point, while a giant emitter has two or more coupling points (like emitters 2 and 3 here). (b): Instead of a double index as in (a), we can use a single index ν to label coupling points from left to right, defining for each a ladder operatorÂ ν incorporating the coordinatedependent phase factor (e.g.,Â 4 = e −ik 0 x 22Â 2 ). Thus, formally, the system is equivalent to a set of normal but not independent emitters, i.e., [Â ν ,Â † ν ] for ν = ν is generally non-zero (e.g., [Â 1 , Transformation from indexing (b) to (a) is described by the pair of index functions j = J ν and = L ν . These and the inverse function can be represented through the plotted diagram, where values of ν (in red) label the black dots. The Cartesian coordinates of each dot indicate the corresponding pair (j, ). The diagram thus encodes the coupling points topology.

A. Interaction picture and relabeling
Passing to the interaction picture with respect toĤ 0 =Ĥ S +Ĥ f transforms ladder operators aŝ A j →Â j e −iω 0 t andb ω →b ω e −i(ω 0 +ω)t so that the joint emitter-field state σ now evolves asσ Now, following Ref. [46], it is convenient to introduce an index ν = 1, ..., N labeling all the coupling points from left to right, i.e., x 1 < x 2 < ... < x N [see Fig. 2 withÂ j the ladder operator of the corresponding atom and e −ik 0 x j the corresponding phase shift. For instance, in the case of Fig. 2 Formally, the mapping between (j, ) and ν is a expressed by a pair of discrete functions j = J ν and = L ν , a diagrammatic representation of which is shown in Fig. 2(c). Note that ladder operators {Â ν } with different indexes do not necessarily commute, that is [Â ν ,Â † ν =ν ] is generally non-zero [e.g., in Fig. 2 . This way the system could be thought as a set of N normal emitters (as many as the coupling points), which yet are not independent. Their dynamics is governed by the Hamiltonian [cf. Eq. (6)]

B. Bidirectional field
For a bidirectional field, each normal frequency ω now has associated right-going and left-going modes with ladder operatorsb ω andb ω , respectively (b ω fulfill commutation rules analogous tob ω ). In the total Hamiltonian (1), the field and coupling Hamitonians are replaced bŷ where we allowed generally different coupling strengths to right-and left-going modes so as to encompass chiral dynamics [53] (the previous unidirectional case is retrieved for γ = 0). Note the different phase factors in right-going terms compared to leftgoing ones. A detailed derivation of the microscopic Hamiltonian is reviewed in Appendix A. Left-going time modes are defined analogously to (4) asb fulfilling commutation rules analogous to (5).
Proceeding similarly to the unidirectional case leads to the interaction-picture coupling Hamiltonian (12) withÂ ν defined as in (7) andÂ ν where, just like in Eq. (13), j = J ν and = L ν (note however the change of phase compared toÂ ν ).

III. SUMMARY OF MAIN RESULTS
In this section, we sum up some of the main results of this work.

A. Unidirectional field
Let t n = n∆t, with n integer and t 0 the initial time, be a mesh of the time axis. In the regime of negligible time delays defined by t N − t 1 ∆t γ −1 , the propagator of the joint dynamics (in the interaction picture) is wellapproximated aŝ whereĤ (note the characteristic 1/ √ ∆t dependence of the coupling strength). Here,Â is the collective emitters' oper-atorÂ = ∑ νÂν , whilê is the annihilation operator associated with the nth time bin of the field. Time-bin ladder operators fulfill standard bosonic commutation rules [b n ,b n ] = [b † n ,b † n ] = 0 and [b n ,b † n ] = δ n,n . Thus the dynamics effectively consists of a sequence of pairwise collisions (short interactions). During the nth collision, the emitters collectively couple to the nth field's time bin (interactionV n ) and at the same time undergo an effective dipole-dipole interaction described by HamiltonianĤ vac . Note that time bins are noninteracting with each other and that the nth time bin interacts with the emitters only during the time interval [t n−1 , t n ] in a conveyor-belt fashion, in this respect just like the standard case of one normal emitter (see Fig. 1).
Let σ n = σ t=t n be the joint state of the emitters and all time bins with σ 0 = ρ 0 ⊗ ρ f , where ρ 0 (ρ f ) is the initial state of emitters (field). At each collision, σ n evolves according to where ∆σ n = σ n − σ n−1 and [..., ...] + stands for the anticommutator (the ∆t-dependence of 2nd-order terms is only apparent sinceV n ∼ 1/ √ ∆t). If the initial state of the time bins corresponding to the field state ρ f is of the form n η n (no correlations), then tracing off the field in Eq. (18) yields that the emitters undergo a Markovian dynamics described by with ρ n the state of the emitters at time t n , ∆ρ n = ρ n − ρ n−1 and .
where {Ĵ m } is a suitable collection of jump operators. The Lindblad form is guaranteed because at each collision the emitters evolve according to a completely-positive and trace-preserving (CPT) map, ρ n = Û n ρ n−1Û † n . The most general white-noise Gaussian state of the field is fully specified by the 1st and 2nd moments [54] with db t = t+dt t dsb s the well-known quantum noise increment. Correspondingly, the most general Gaussian, uncorrelated state of the time bins is fully specified by the moments with α n = α t=t n , N ≥ 0 and |M| 2 ≤ N(N + 1).
Replacing the explicit expression ofV n in Eq. (19) using (22) and carrying out the continuous-time limit γ∆t → 0, the discrete master equation (19) is turned into the general continuous-time master equation This can be expressed in terms of original ladder oper-atorsÂ j using (7) and recallingÂ = ∑ νÂν .

B. Bidirectional field
In the case of a bidirectional field, the time bin is now bipartite (see Fig. 4) having associated ladder operatorŝ b n [cf. Eq. (17)] andb n , the latter given bŷ whileĤ vac andV n are now generalized aŝ Inside brackets of (25), note that the second term has swapped subscripts compared to the first. This is due to the opposite interaction time ordering for left-and right-going modes. Accordingly, the dissipator in the Lindblad master equation (23) now naturally splits into a pair of analogous contributions: one featuring operatorsÂ ν 's and moments of right-going field modes (α t , N, M) and another one involvingÂ ν 's and left-going-mode moments (α t , N , M ). The latter moments are defined analogously to (21) withb t →b t . This leads to the master equation where we recall Eqs. (20) and (25). This can be expressed in terms of original ladder operatorsÂ j through (7), (13) and (27). Master equation (23) for a unidirectional field is retrieved for γ = 0.

C. Photodetection and quantum trajectories
For a unidirectional field, photodetection translates into measuring each time bin right after its collision with S in a selected basis {|k n } (defining the photodetection scheme). For time bins initially in state n |χ n (thus η n = |χ n χ n |) and negligible time delays, the (unnormalized) state of S after a specific sequence of measurement outcomes {k 1 , ..., k n } is the associated probability being p k 1 ··· k n = Tr S {ρ n } and with each Kraus operator given bŷ A measurement on time bin n with outcome k (at the end of the nth collision) thus projects S into the (unnormalized) stateρ n =K k ρ n−1K † k , defining the conditional dynamics. Summing over all possible outcomes yields the CPT map ρ n = E [ρ n−1 ] = ∑ kKk ρ n−1K † k , defining the unconditional dynamics.
A (pure) time-bin state generally depends itself on √ ∆t. We consider those states such that to the lowest order in √ ∆t read with |χ (j) n such that χ n |χ n = 1 + O(∆t) (|κ n with κ n = 0, 1, ... denote the time-bin Fock states). Plugging this andÛ n = e −i(Ĥ vac +V n ) into ρ n = E [ρ n−1 ] and dropping high-order terms eventually leads to the master equation with the effective HamiltonianĤ eff and jump operatorŝ J k given bŷ This in fact defines an unraveling of master equation (23) [which is indeed equivalent to (32)] corresponding to the photodetection scheme {|k n } in the case of a unidirectional field.
For a coherent-state wavepacket of amplitude ξ t (in the time domain) [55], |ξ = e dt (ξtb † t −ξ * tb t ) |0 (with |0 the field vacuum), the corresponding time-bin state is with ξ n = ξ t=t n . Hence, |χ (1) n = ξ n |1 n . In the case of photon counting, {|k n } are the Fock states. The effective Hamiltonian and the only surviving jump operator are thus immediately calculated aŝ The continuous-time limit expressions are simply obtained by replacing ξ n → ξ t .
For a bidirectional field, photodetection consists in measuring both the right-and left-going time bins (see Fig. 4) in a basis |k, k . A measurement outcome k, k is now described by the Kraus operator [cf. Eq. (30)] with |χ m (|χ m ) the initial state of the right-going (leftgoing) time bin andÛ n = e −i(Ĥ vac +V n ) withĤ vac and V n now given by (25) and (26). The effective Hamiltonian and jump operators are given by [cf. Eqs. (33) and (34)]

IV. EXAMPLES OF MASTER EQUATIONS AND DECOHERENCE-FREE HAMILTONIANS
The aim of this section is to illustrate how (28) encompasses and generalizes various quantum optics and waveguide QED master equations with a special focus on giant atoms and decoherence-free Hamiltonians. As such, it could be skipped by a reader solely interested in the collision-model derivation.
For a pair of normal emitters coupled to a unidirectional field, we have: (23) [or (28) for γ = 0] reduces to the well-known ME of a pair of cascaded emitters in vacuum [56,57].
Thus Eq. (28) generalizes the squeezed-bath master equation to giant emitters.

thus read
Plugging these into Eq. (28), for α t = α t = N = N = M = M = 0 we retrieve the vacuum master equation [51,61] For a pair of giant emitters with two coupling points each and a bidirectional waveguide: γ = γ = Γ/2, N e = 2, N = 4. TheÂ ν 's andÂ ν 's depend on the pattern of coupling points, for which three different topologies are possible: serial, nested and braided (see Fig. 5). Setting ϕ ν = k 0 x ν = ω 0 τ ν and as usual whileÂ has an analogous expression with ϕ ν → −ϕ ν . Plugging these into (28), for ϕ ν = νϕ (uniform spac-ings) and the field vacuum state one getṡ which was derived through the SLH formalism in Ref. [51] alongside other master equations for different configurations and number of atoms [these can all be retrieved from (28) likewise].

A. Decoherence-free Hamiltonians with giant atoms
A major appeal of giant emitters is that they allow to implement decoherence-free many-body Hamiltonians. A paradigmatic instance is the braided configuration in Fig. 5. By adjusting a π-phase shift between the coupling points of the same emitter, e.g., setting ϕ = π/2, all the dissipative terms in Eq. (44) vanish but the HamiltonianĤ vac , which effectively seeds a dissipationless coherent interaction [51].
In the collisional picture this phenomenon can be predicted without working out the master equation, making clear at once that it occurs regardless of the field state [thus being not limited to the vacuum state assumed in the derivation of Eq. (44)]. Indeed, the condition that collective operators (27) vanish, (or justÂ = 0 with a unidirectional field), guarantees that the joint emitters-field propagator reduces tô U t = exp(−iĤ vac t). This is because (45) effectively decouples the emitters from the field time bins in light of Eqs. (14), (25) and (26), thus inhibiting dissipation.
Having giant emitters is clearly indispensable since for normal emitters there is no way forÂ andÂ to identically vanish in the entire Hilbert space. The question is now whether or not (45) yields in addition a nullĤ vac (if so no evolution takes place). For a giant atom [cf. Eq. (41)], the conditionÂ =Â = 0 holds for ϕ = (2n+1)π which will also entailĤ vac = 0. For two giant atoms, the collective operators vanish for any π-phase shift between the coupling points of the same emitter [cf. Eq. (43)]. Using (25), one can check that this always yieldsĤ vac = 0 in the serial and nested topologies (see Fig. 5) whereas in the braided oneĤ vac can be non-zero (for a comprehensive analysis we refer the reader to Ref. [62]).
In the collisional picture, occurrence ofĤ vac = 0 with zero decoherence means that each time bin ends up uncorrelated with the emitters as the collision is complete. Notwithstanding, during the collision, it mediates a crosstalk between the emitters which thus get correlated with one another.

V. COLLISION MODEL DERIVATION
In this section, we address the derivation of the collision model for a unidirectional field (the generalization to the bidirectional case in presented in Section VII).
Note that regime (1) is often dubbed "Markovian". Strictly speaking, this is an abuse of language relying on the fact that for many typical field states (such as vacuum, thermal, coherent or broadband squeezed states) dynamics in regime (1) are Markovian and described by a Lindblad master equation. This is not necessarily the case, though, with more general field states such as single-photon wavepackets, even for a single coupling point [63]. Intermediate regimes between (1) and (2) are of course possible, but these can be described as a combination of (1) and (2). Most of the present section concerns the regime of negligible time delays (1) (our main focus in this work), which still occurs in the vast majority of experimental setups (see e.g. Ref. [64] for a discussion on circuit-QED systems). Nevertheless, we begin with some general considerations and properties common to both regimes.
Consider a time mesh defined by t n = n∆t with n = 0, 1, ... integer and ∆t the time step (later on this will be interpreted as the collision time). In the interaction picture (see Section II A), the propagatorÛ t can be decomposed as [65] withV(s) given in Eq. (6) andT the usual timeordering operator, and where each unitaryÛ n describes the evolution in the time interval t ∈ [t n−1 , t n ] This discretization of the joint dynamics underpins the collision-model description (in any regime). Throughout, we will consider a time step much shorter than the characteristic interaction time, i.e., ∆t γ −1 . Accordingly, we apply Magnus expansion [66] and approximate (47) up to second order in ∆t aŝ with 1 1 the identity operator and whileĤ (1) n is the sum of three termŝ

A. Negligible time delays
When time delays are negligible we can coarse grain the dynamics over a time scale defined by ∆t such that meaning that the overall length of the coupling points array (hence the distance between any pair τ ν − τ ν ) is negligible compared to the time step defining the time scale [see Fig. 3(a)]. We can take advantage of (56) and obtain approximated expressions ofĤ  (51) can be approximated as t n−1 − τ ν t n−1 and t n − τ ν t n so that (we set τ 1 = 0 throughout) where we defined theb n 's as (17). It is easily checked that the commutation rules for theb t 's [cf. Eq. (5) Thus thê b n 's define a discrete collection of bosonic modes, which we will usually refer to in the remainder as "time-bin modes" or at times simply as "time bins". Thus (51) in the present regime reduces tô whereÂ = ∑ νÂν is a collective operator of the emitters. Note the characteristic scaling ∼ ∆t −1/2 of the emitter-(time bin) coupling strength, which is a hallmark of CMs [4]. (1) sq can be neglected (note that Appendix B refers to Section V B to be discussed shortly).
Thus we are left only with the vacuum contribution H (1) vac . To work this out, we first note that the each double integral in Eq. (53) runs over the shaded triangle sketched in Fig. 6. For a given pair (ν, ν ), the twovariable δ function is peaked on the line s = s − (τ ν − τ ν ). As shown in Fig. 6, this line falls within the triangle for ν > ν and outside of it for ν < ν (since τ ν − τ ν > 0 for ν > ν ). Hence, only terms ν > ν contribute toĤ The above shows that, for delays negligible with respect to ∆t this being in turn much shorter than the interaction characteristic time scale γ −1 , in Eq. (48) we can approximateĤ (0) n V n andĤ (1) n Ĥ vac . Thereby, showing that in this regime the joint emitter-field dynamics can be effectively pictured as a sequence of short pairwise interactions (collisions) of duration ∆t (collision time), as sketched in Figs. 1 and 3(a). In each of which the emitters collectively couple to a fresh time bin (only one) according to the coupling Hamiltonian V n and at the same time coherently interact with one another through the second-order many-body Hamil-tonianĤ vac . Note that time bins are uncoupled from one another and that each collides with the emitter only once in a "conveyor-belt" fashion (see Fig. 1).
As said, to arrive at Eq. (60), all time delays τ ν − τ ν were neglected. We point out that this is different from setting τ ν − τ ν = 0. Instead, it corresponds to performing the limit τ ν − τ ν → 0 + for all pairs (ν, ν ) with ν > ν . Indeed, it is easily checked that setting τ ν − τ ν = 0 entailsĤ (1) vac = 0 since in this case both terms ν > ν and ν < ν must be accounted for but exactly cancel out (the two dashed lines in Fig. 6 now both reduce to s = s). Physically, this means that the effective HamiltonianĤ vac stems from the fact that, while traveling from left to right [see Fig. 3(a)], the nth time bin interacts first with coupling point ν and only afterwards with ν + 1, no matter how short the delay τ ν+1 − τ ν . This is in line with similar observations made in derivations of cascaded MEs through other methods (see e.g. [56]). Interestingly, the collisional picture allows for a complementary interpretation of this phenomenon in terms of far-detuned time-bin modesb n,k , which we introduce next.

B. Time-bin modesb n,k
It should be clear from their definition (17) that, for a finite ∆t, modesb n generally capture only part of the field degrees of freedom. Formally, this can be seen by expanding the continuous time modes as [8] with Θ n (t) = 1 for t ∈ [t n−1 , t n ] and 0 otherwise, and whereb n,k = 1 √ ∆t t n t n−1 dt e i2πkt/∆t b t .
Ladder operatorsb n,k fulfill [b n,k ,b † n ,k ] = δ n,n δ k,k , [b n,k ,b n ,k ] = [b † n,k ,b † n ,k ] = 0. Moreover, for k = 0 we retrieve modesb n [cf. Eq. (17)], i.e.,b n,0 ≡b n . A straightforward Fourier analysis shows that time-bin modeŝ b n,k =0 are dominated by field normal modes whose detunings from the emitter grow as ∼ |k|/∆t, (while modesb n,0 contain field frequencies quasi-resonant with the emitter) [8]. For ∆t → 0 [still fulfilling (56)], corresponding to the continuous-time limit of the dynamics, these frequencies become divergent. Accordingly, it is reasonable to assume there are no photons populating modesb n,k =0 . This is equivalent to stating that the most general field state is of the form with η bins the (generally mixed) state of modesb n ≡b n,0 and |0 n,k the vacuum state of modeb n,k .

C. Differences with the single-coupling-point case
For a single coupling point (N = 1)Ĥ vac of course does not arise and we are only left withV n (containing onlyb n ≡b n,0 ), meaning that the coupling to time-bin modes k = 0 is negligible. Yet, for two or more coupling points (N ≥ 2), these off-resonant modes yield non-negligible effects despite they do not explicitly appear inĤ vac (not even inV n , of course). Indeed, they are in fact responsible for the emergence ofĤ vac . This can be seen from Eq. (53) featuring a singularity in the integrand function due to the field commutator. Such a singular behavior forbids to retaining only k = 0 terms in expansion (61) no matter how small ∆t (indeed it is easily checked that expanding each field operator entering Eq. (53) and retaining only modesb n,0 =b n would yield a vanishingĤ vac ).
Thus all time-bin modesb n,k in fact contribute to the dynamics for N > 1. However, unlike k = 0 modes, off-resonant modes k = 0 are only virtually excited, explaining why they do not explicitly appear inĤ vac .

D. Non-negligible time delays
A comprehensive treatment of the regime of nonnegligible delays is beyond the scope of the present paper. Yet, we wish to highlight a major difference from the negligible delays regime, this being that at each time step the emitters collide with as many time bins as the number of coupling points (instead of only one). To illustrate this, we work out nextĤ In contrast with the negligible delays regime, now one can take a time step negligible compared with all the system's time delay, i.e., ∆t τ ν − τ ν−1 for all ν (note that this is compatible with condition ∆t γ −1 that we assume throughout). For sufficiently short ∆t, the coupling points coordinates can be discretized as τ ν = m ν ∆t, where {m ν } are N integers such that m 1 < m 2 < ... < m N , and set τ 1 = m 1 = 0. Accordingly, (51) becomes [recall that t n = n∆t] showing that, during a given time interval [t n−1 , t n ], each coupling point ν interacts with a different time bin n − m ν [see Fig. 3(b)].
In the presence of giant emitters (even a single one), this dynamics is tough to tackle analytically. Through an elegant diagrammatic technique, Grimsmo found an analytical solution for the open dynamics of a driven giant atom with two coupling points [2], while Pichler and Zoller found an efficient matrix-product-state approach which they applied to a pair of driven normal atoms coupled to a bidirectional field [1] (the collisional picture for a bidirectional field is addressed in Section VII). A major reason behind the complexity of this dynamics lies in its generally non-Markovian nature (conditions for Markovian behaviour are discussed in Section VI).

VI. MASTER EQUATION FOR NEGLIGIBLE TIME DELAYS
In section V A, we focused on the total propagator showing that for negligible time delays it can be decomposed as a sequence of collisions between the emitters (jointly) and a field time bin, each described by the two-body elementary unitaryÛ n in Eq. (60), which is fully specified byĤ vac andV n . In this section, we derive master equations for the emitters and time bin in the regime of negligible time delays.

A. Conditions for Markovian dynamics
Based on (63) and related discussion, from now on time-bin modesb n,k =0 will be ignored. The joint state of the emitters and all time bins (modesb n ≡b n,0 ) evolves at each time step as σ n =Û n σ n−1Û † n with σ n = σ(t n ). A corresponding finite-difference equation of motion is worked out by replacingÛ n with (60) and retaining only terms up to second order in ∆t where ∆σ n = σ n − σ n−1 (recall thatV n ∼ 1/ √ ∆t). Under the usual assumption of zero initial correlations between the emitters and the field, the initial condition reads σ 0 = ρ 0 ⊗ η bins , where ρ 0 and η bins are the initial states of all emitters and all time bins, respectively. We next ask whether or not the reduced dynamics of the emitters ρ n = Tr bins {σ n } is Markovian and describable by a Lindblad master equation. We note that this is generally not the case when time bins are initially correlated, namely η bins is not a product state, since in these conditions the emitters can get correlated with a time bin even before colliding with it [4,27]. This indeed rules out that that the evolution of the emitters (open system) at each elementary collision be described by a completely positive and trace preserving (CPT) quantum map [67], which is the key requirement in order for a Lindblad master equation to hold. A typical instance is a single-photon wavepacket of bandwidth comparable with γ [10][11][12][13]68].
We thus consider the case that time bins are initially uncorrelated, that is with η n the reduced state of the nth time bin (mode with latter operatorb n =b n.0 ). This entails ρ n = Tr bins Û n σ n−1Û † n = Tr n Û n ρ n−1 η nÛ † n , (67) where Tr n is the partial trace over the time bin n (modê b n ≡b n,0 ). This defines a CPT map describing how the emitters' state ρ n is changed by the nth collision. Likewise, the nth time bin evolves according to η n = Tr S Û n ρ n−1 η nÛ † n (68) with Tr S the partial trace over the emitters. This is a CPT map describing the change of the single time bin state due to collision with the emitters (after the collision this will no longer change since time bins are noninteracting). Note that map (68) depends parametrically on the current reduced state of emitters (updated at each collision).
Although not explicit, this equation is in Lindblad form as is easily checked by spectrally decomposing η n [69]. The Linbdlad form is a guaranteed by the fact that the emitters evolution at each collision is described by a CPT map [last identity in Eq. (67)]. Using (16) the first-order Hamiltonian and secondorder dissipator can be put in the more explicit form with µ, µ = 1, 2 and where we set Now Eq. (69) is expressed fully in terms of the time-bin moments b n , b † nbn and b 2 n , which depend on η n in turn dependent on the initial field state [cf. Eqs. (??) and (66)].
The time-bin moments can be determined for the most general white-noise Gaussian state of the field. Such a state is fully specified by [54] (74) with db t = t+dt t dsb s the quantum noise increment (it is understood that 2nd moments for t = t are zero). Parameters N and M fulfill the constraints N ≥ 0 and |M| 2 ≤ N(N + 1). Noting thatb n = t n t n−1 dr t / √ ∆t, it is evident that for such a field state, b † nbn = b nbn = 0 for n = n . This, because of the Gaussianity hypothesis, is equivalent to Eq. (66). Thus time bins are initially uncorrelated. Moreover, we find Here, α n = t n t n−1 dt α t /∆t, which for ∆t short enough reduces to α n α t n . Note that b n ∝ √ ∆t, which cancels the 1/ √ ∆t factor in Eq. (71). Plugging moments (75) into the finite-difference Eq. (69) and taking the continuous-time limit such that γ∆t → 0, t n → t, ρ n−1 → ρ t , ∆ρ n /∆t → dρ/dt we end up with the general master equation (23).

Time bin master equation
An equation for the rate of change of the single time bin state, ∆η n /∆t with ∆η n = η n − η n , can be similarly worked out. We again start from Eq. (65) but now trace over all emitters and all time bins n = n, obtaining with ... ρ = Tr S {... ρ} and where Tr S {...} is the partial trace over the system. Note that this equation parametrically depends on the state of the emitters, ρ n−1 , which changes at each time step. Eq. (76) expresses map (68) in the short-collision-time limit.

VII. COLLISION MODEL FOR A BIDIRECTIONAL FIELD
For a bidirectional field (see Section II B), unitariesÛ t andÛ n are formally the same as (46) and (47), respectively, butV t is now given by Eq. (12). TheÛ n 's lowestorder expansion (48) is formally unchanged. Through a reasoning analogous to that in Section V, in light of (12), Eqs. (51) and (52)

A. Negligible time delays
RegardingĤ (0) n , an argument analogous to that leading to (58) now yields that in the present regimeĤ  s+τ ν ]. This peaks on the line s = s − (τ ν − τ ν ), which differs from the δ function coming from right-going modes [cf. Eq. (59)] for the exchange ν ↔ ν . Accordingly, in Fig. 6, the lines corresponding to ν < ν and ν > ν are swapped, hence now only terms ν < ν (instead of ν > ν ) contribute toĤ Thereby, for τ N − τ 1 ∆t (1/γ, 1/γ ), the joint dynamics can be be represented by an effective collision model (see Fig. 4), where at each collision the emitters jointly collide with a right-going and a left-going time bin, at once being subject to an internal coherent dynamics governed by the second-order Hamiltonian (25). Note that, formally, this can still be thought as a collision model featuring a single stream of time bins [like Fig. 1] provided that one defines a two-mode time bin (b n ,b n ).

B. Non-negligible time delays
An argument analogous to that in Section generalizes Eq. (64)] aŝ (the former commuting with the latter). Note the different subscripts inb n−m ν andb n+m ν , reflecting that rightand left-going time bins travel in opposite directions as sketched in Fig. 7.
Similarly to the unidirectional case discussed in Section V D, analytical descriptions of this dynamics are demanding [1, 70,71].

VIII. MASTER EQUATION FOR A BIDIRECTIONAL FIELD
With the extended definitions ofĤ vac andV n for a bidirectional field discussed in the previous section (regime of negligible time delays), the finite-difference equation of motion (65) for the joint dynamics still holds. The initial state of the time bins η bins is obtained from the initial field state by using (61) and tracing off time-bin modes k = 0 (with left-going time-bin modeŝ b n,k also accounted for).
Likewise, a Markovian open dynamics will arise with field states which in the collisional picture turn into uncorrelated states of the time bins η bins = n (η r,n ⊗ η l,n ) with η r,n (η l,n ) the reduced state of the nth right-going   75)]. Plugging these into the finite-difference Eq. (69) and taking next the continuous-time limit as done in the unidirectional case, we end up with master equation (28).

IX. PHOTODETECTION AND QUANTUM TRAJECTORIES
As anticipated in the Introduction, a major advantage of collision models is that they naturally accommodate quantum weak measurements [17]. In the present quantum optics framework, this translates into a convenient description of photodetection [54,72].
Let {|k n } be an orthonormal basis of the nth time bin (henceforth we will mostly prefer the compact notation |k n ). In the collisional picture, photodetection consists in measuring each time bin right after its collision with S. The photodetection scheme is defined by the measurement basis {|k }. Assuming an uncorrelated initial state of the time bins [cf. Eq. (66)], the (unnormalized) evolved state of the joint system after a specific sequence of measurement outcomes {k 1 , ..., k n } is given bỹ σ n = |k n k n |Û n · · · |k 1 k 1 |Û 1 ρ 0 m η m Û † 1 |k 1 k 1 | · · ·Û † n |k n k n | (87) withÛ n = e −i(Ĥ vac +V n )∆t . The probability p k 1 ··· k n of getting this sequence of measurement outcomes is the norm ofσ n , hence the normalized state is σ n =σ n /p k 1 ··· k n .
Let each time bin be initially in a pure state η m = |χ m χ m | (the mixed case is commented later). Plugging this into (87) and tracing off all the time bins yields the unnormalized state of the emitters S at step ñ ρ n =K k n · · ·K k 1 ρ 0K † while the time bins are of course in state |k 1 · · · |k n (uncorrelated with S). Thus, since Tr n {σ} = Tr S {ρ n }, p k 1 ··· k n can be expressed as [cf. Eq. (88)] A time-bin measurement with outcome k (at the end of the nth collision) thus projects the emitters into the (unnormalized) statẽ (ρ n−1 denotes as usual the normalized state of S right before collision with time bin n). This map defines the conditional dynamics. Summing next over all possible k's yields ρ n = ∑ kKk ρ n−1K † k , this CPT map defining the unconditional open dynamics.
We derive next the lowest-order expansion of each Kraus operatorK k . Let us first arrange (60) in the form where we used (16). Moreover, we allow the time-bin state |χ n to generally depend on √ ∆t (this is for instance the case of coherent states as illustrated later). Hence, to lowest order |χ n |0 n + |χ Note that the 0th-order term was set equal to the timebin vacuum state to ensure that the field energy density b † nbn /∆t does not diverge in the limit ∆t → 0, which would lead to nonsensical photon-counting evolution [8,54] [we come back to this issue shortly after Eq. (101)]. Also, |χ (1) n and |χ (2) n are subject to the constraints Re 0|χ which follow from the normalization condition of |χ n to the 1st-order in ∆t. Henceforth, subscript n will be dropped in |0 n , |χ (1) n and |χ (2) n . Plugging (93) and (94) into (90) and grouping together terms of the same order in √ ∆t, to leading order we getK wherê K (1) with time-bin operatorsĉ µ and emitter operatorsĈ µ (µ, µ = 1, 2) given by (73). Replacing (96) into the conditional map (92) yields (to leading order) Summing the right-hand side over k, we end up with the Lindblad master equation (see Appendix C for more details) [recall definition (20)], where the effective Hamiltonian and jump operators are given bŷ (101) This equation is equivalent to the (white-noise, Gaussian) master equation (23) [or (28) for γ = 0], but at variance with this is not expressed in terms of field moments (requiring instead a more detailed knowledge of the time-bin state). Eqs. (100) and (101) in fact define an unraveling of the master equation corresponding to a desired photodetection scheme. Two comments follow.
First, there are white-noise Gaussian field states, such as squeezed and thermal states, for which the 0th-order term of expansion (94) differs from |0 n since their infinite bandwidth corresponds to an infinite photon flux (see also Ref. [8]). Eqs. (100) and (101) thus do not apply to such states.
Second, for most physically relevant field states, in a single time bin the single-photon and two-photon amplitudes are at most of order √ ∆t and ∆t, respectively. This means that only the outcomes k = 0 and k = 1 occur with meaningful probability, corresponding respectively to "click" and "no-click" outcomes (in line with most treatments of photodetection which indeed limit themselves to "click"/"no-click" outcomes at each infinitesimal time increment).
As an illustration, in the next subsection we will show how the above applies to photon counting in the case of a coherent-state wavepacket.
In the most general case of a mixed time-bin initial state, whose spectral decomposition reads η n = ∑ κ q κ |χ κ n χ κ | (with probabilities q κ fulfilling ∑ κ q κ = 1), the Kraus operators will be indexed not only by the measurement outcome but also by the eigenstate |χ κ ,K and (92) turns into a sum over κ,

A. Photon counting for a coherent-state wavepacket
In the case of photon counting, the time-bin measurement basis are the Fock states {|k n } with k = 0, 1, ... . A coherent-state wavepacket in terms of time modes (4) reads [73] |ξ = e dt (ξtb † with |0 the field vacuum state and ξ t the wavepacket amplitude in the time domain [55]. One can decompose the time integrals into a sum over intervals {[t n−1 , t n ]} and, if ∆t is small enough, in each interval replace ξ t → ξ n = 1 ∆t t n t n−1 dt ξ t . This yields hence (66) holds for η n = |χ n χ n | with [74] |χ n = e ξ n √ ∆tb † n −ξ * n √ ∆tb n |0 n .
Plugging this in (101), in the continuous-time limit such that ξ n → ξ t , we get the effective Hamiltonian and jump operator In addition toĤ vac ,Ĥ eff features the standard drive Hamiltonian, arising from the first-order termV n in the collision unitary (60) [75]. Also, note the c-number shift inĴ 1 . Analogous shifted jump operators, physically due to the coherent superposition of light emitted from S and the incoming beam, were previously derived (for normal emitters) via the input-output formalism (see, e.g, Refs. [54,[76][77][78]).

X. CONCLUSIONS
In this paper, we formulated the collision-modelbased description of quantum optics dynamics in the presence of many quantum emitters, each able to interact with a generally chiral field at many coupling points. The collisional picture maps the field into a stream of discrete time-bin modes interacting with the emitters in a conveyor-belt-like fashion. In the regime of negligible time delays (usual in most experiments) the dynamics is effectively represented as a sequence of pairwise collisions each between a field time bin and all the emitters collectively. These at once undergo an internal dynamics ruled by an effective second-order Hamiltonian describing dipole-dipole interactions. This Hamiltonian origins from the fact that the traveling time bin reaches the system's coupling points in sequence, no matter how short the delays. As such, the effective Hamiltonian depends on the coupling points topology. We applied the collisional picture to derive a general Lindblad master equation of a set of (generally) giant emitters coupled to a chiral waveguide for an arbitrary white-noise Gaussian state of the field. This combines into a single equation and extends a variety of master equations used in quantum optics and waveguide QED. In addition, building on previous work [8], we worked out a general recipe that, for a given photodetection scheme, returns the effective Hamiltonian and jump operators generating the ensuing quantum trajectories.
For the sake of argument and in order to keep the number of parameters at a reasonable level, throughout we considered identical quantum emitters each featuring a single transition coupled to the field and no external drive. Extending the theory so as to relax these assumptions is straightforward.
It is natural to compare the collision-model picture with the longstanding input-output formalism (IOF) of quantum optics (and methodologies underpinned by the latter such as the SLH approach [52]). As anticipated, these are both grounded on the same microscopic model, in particular the white-noise coupling assumption, and share the field representation in term of temporal modes. On the other hand, some significant differences stand out. One especially evident is that in the IOF time is a continuous variable, while in the collisional picture one in fact works with a discrete (coarse-grained) dynamics taking the continuous-time limit only at the end.
Notably, while in the IOF one evolves the operators in a Heisenberg-picture-like fashion, the collisional framework essentially deals with evolutions of states in the spirit of the Schrödinger picture. The role of input and output field operators in the IOF is played by the initial and final state of the time bins η n and η n , while the central equation of IOF that connects the output field to the input field and system operator is replaced by the pairwise unitaryÛ n describing each collision. The collision unitary concept makes the collisional picture particularly advantageous to carry out tasks such as deriving in a natural way CPT master equations, jump operators or effective decoherence-free Hamiltonians [62]. Moreover, the joint emitters-field dynamics is in fact mapped into an effective quantum circuit, which can help quantum simulations and allows to potentially take advantage of already developed quantum information/computing techniques. In this respect, it was recently proved [8] that all quantum optics master equations and photon detection schemes for a single (normal) emitter can be simulated through a collision model with time bins replaced by qubits. The present work in fact extends the same property to many giant emitters.
Finally, although in this paper the illustrations of the general theory mostly targeted the open dynamics of the emitters, we stress that the collisional picture captures the joint dynamics including the field. The framework is thus potentially as useful in problems such as multi-photon scattering from atoms [? ] or nonequilibrium thermodynamics of quantum optics systems (or generally bosonic baths) [79].