Heralded multiphoton states with coherent spin interactions in waveguide QED

WaveguideQEDoffers the possibility of generating strong coherent atomic interactions either through appropriate atomic configurations in the dissipative regime or in the bandgap regime. In this work, we show how to harness these interactions in order to herald the generation of highly entangled atomic states, which afterwards can be mapped to generate single mode multi-photonic states with high fidelities.Weintroduce two protocols for the preparation of the atomic states, we discuss their performance and compare them to previous proposals. In particular, we show that one of them reaches high probability of success for systems with many atoms but low Purcell factors.


Introduction
Non-classical states of few photons can be generated in a variety of physical systems. Triggered single photonsources [1] can be found in solid state systems [2][3][4], in neutral atoms or ions coupled to optical cavities [5][6][7][8][9][10] and in collective atomic ensembles [11][12][13][14][15][16][17][18]. By combining these single-photon sources with linear optics tools and post-selection, it is possible to achieve higher photon number states but with an exponentially small probability, which precludes the generation of larger photon numbers (see [19] and references therein) and, typically, destroying the state after heralding.
The enhanced light-matter interactions provided by waveguide QED [20][21][22][23][24][25][26][27][28][29][30][31] presents an excellent arena for the generation of multi-photon states. One possibility is to use atom-like metastable states of atom-like systems as quantum memories that can afterwards be triggered to generate photonic states with controllable temporal shape [32,33] and with a very favorable scaling of the infidelity I m NP d phot 2 1 µ ( ), with N being the number of quantum emitters and P d 1 the (single atom) Purcell factor of the system which characterizes how much emission goes into the waveguide with respect to free space emission. Hence, the generation of arbitrary photonic states reduces to the preparation of arbitrary symmetric excitations in an ensemble of quantum emitters, which is the main focus of this work.
In recent proposals [33,34], we designed both deterministic and probabilistic methods to generate collective atomic states using equally spaced atoms within the purely dissipative waveguide regime. It was shown that this simple atomic configuration allows one to prepare collective atomic states of m excitations with either the fidelity [33] or the heralding probability [34] deviating from unity only by a factor scaling with P d 1 1 2 -. The key resource of these protocols is the long-range dissipative coupling induced by the waveguide which enforces effective unitary dynamics through the quantum Zeno effect.
In this work, we present two protocols that harness long-range coherent interactions induced by the guided modes to generate collective atomic excitations within an ensemble of atoms. atomic configuration determines the coherent and the dissipative interaction between the emitters. The atomic configuration in our protocol was inspired by [35], in which a pair of atomic mirrors are placed next to a single emitter and where the analogy to cavity QED with g N µ was shown. By placing another set of mirrors around this atomic cavity, we obtain the tools necessary for the heralded generation of collective atomic excitations in the first pair of mirrors.
• The second protocol ('dipole-dipole') is designed for emitters, whose resonance frequency is in the bandgap, see figure 1(b), and which is, e.g., well suited for engineered dielectrics. In this regime, dipole-dipole interactions mediated by an atom-photon bound state formed in the bandgap emerge [36][37][38][39][40]. Here, the advantage is that dissipation (through the waveguide modes) is strongly suppressed.
We analyze in detail the performance of both protocols, and compare it to previous ones. The rest of the manuscript is divided as follows: in section 2, we introduce the two configurations and explain how the coherent coupling emerges in each case together with the common ideas of the protocol. In section 3, we discuss the situation for the double mirrors setup, explaining one main protocol, together with different variations of it. In section 4, we discuss how to adapt the protocol in the dissipative regime for the situation where dipole-dipole interactions are mediated by atom-photon bound states. Finally, in section 5 we summarize the figures of merit and scaling of the different protocols and make a comparison with previous proposals [33,34].

System and general protocol
As shown in figure 1, the common ingredient for both the dissipative and bandgap regime is to have three individually addressable ensembles, namely, a source atom, used to transfer single excitations to the target ensemble, which we herald with a change of state in the detector ensemble. Moreover, the emitters must have two dipole transitions g e ñ « ñ | | and s e ñ « ñ | | coupled to two waveguide modes 6 as shown in figures 2(a) and (b) in which either coupling can be controlled, e.g., by using a M-type level scheme (see figure 2(b) or by Stark-shifting the respective levels out of resonance. The two guided modes are required so that the source emitter and the detector ensemble can couple to different modes. This ensures that no direct excitation transfer between them can take place. An excitation in the source emitter can only excite the detector atoms when a collective excitation is created in the target ensemble. Furthermore, we require in both cases that the source/target/detector ensembles are individually addressable.    Apart from these common ingredients, the two protocols in the dissipative and bandgap regime require additional conditions. For example, in the dissipative regime we demand that: (i) The two guided modes mediating the interaction have equal wavelength 0 l , defined by the characteristic atomic frequency: q q q 2 . This can be achieved, e.g., by the use of two different polarization modes.
(ii) The emitters inside the target/detector ensemble are placed at distances commensurate with 0 l , whereas the source/target and target/detector are placed at a distance n 4 0 0 (iii) We can neglect finite propagation lengths and non-Markovian effects and thus use a Markovian description to analyze the performance. The impact of these effects has been discussed in [41][42][43]. This requires that the maximal distance between atoms is small compared to the propagation length, v T g , during the time of operations, where v g is the group velocity in the waveguide.
For the configuration within the bandgap regime, the only additional requirement is the existence of a bandgap for the two guided modes, such that their interactions can be mediated by virtual guided modes.

Theoretical description in the dissipative regime
In the dissipative regime we use what we denote as double mirrors configuration as sketched in figure 1(a), in which the source atom is embedded by two atomic cavity mirrors with N atoms each which play the role of target atoms. Moreover, we embed the source/target system within two other atomic mirrors with N atoms each, that altogether form the detector ensemble. When tracing out the reservoir degrees of freedom, we obtain an effective master equation which describes the atomic dynamics. In this configuration, the waveguide induces strong and long-range coherent spin interactions between the different ensembles described by the  w n ) and guided mode profile (contained in g q h | |) [44]. Together with the coherent spin interactions also collective decay terms emerge, which are given by † † . Obviously, apart from the decay into the desired waveguide modes, the excited states e n ñ | may also emit photons into free space, or even to other non-guided waveguide modes. We include all these processes into a rate, * G , typically of the order of the natural linewidth a G , which can be described through an additional Lindblad term in the Liouvillian as  G G which may be tunable or not, depending on the particular setup. The source/target configuration is inspired by [35], where it was shown that this model can be mapped to a cavity QED configuration, where the source atom plays the role of the two-level system with effective decay . In standard cavity QED, one could improve the scaling to C 1 e~, using off-resonant Raman transitions [46], where C g 2 kg = ( )is the so-called cooperativity. In the effective model within the waveguide setup, however, both G and * G (and consequently the effective cavity QED parameters g, , k g) renormalize in the same way when using off-resonant transitions such that the optimization is not possible. 7 In fact, the protocol also works for distances within an ensemble of 2 0 l (or multiples thereof) and/or distances between source/target or target/detector of odd multiples of 4 0 l if the collective external drivings are adjusted accordingly. 8 See appendix for (i) a discussion on the atomic dynamics and (ii) details of the protocol in thedissipative regime.

Theoretical description in the bandgap regime
If we assume that both the e−g and the e−s transitions are within the bandgap regime, it can be shown [39,40] that the excited atomic states are dressed by a photon cloud of size d x which allows to exchange interaction between the emitters, which are assumed to have an equidistant spacing d a 2p = , where a is the period of the photonic crystal. This distance is well-suited for the generation of photonic states in a later step and also avoids the sign alternation due to the phase acquired by the Bloch mode at the cut-off frequency [47]. The photon cloud can be seen as an off-resonant atom-induced cavity of length d x , that allows to exchange interactions between the emitters described by the Hamiltonian G are the decay rates at the bandgap frequency cutoff, e z z d is the overlap between the effective cavity of the nth atom with the mth atom, and the 1 x dependence is the decrease of the coupling strength to the cavity due to the increase of the mode length. Notice that apart from H bg , the photon cloud also induces dipoledipole couplings within the target (and detector ensemble). We did not write them here explicitly, because their effect on our protocols can be compensated with appropriate laser detunings as will be explained in section 4. In the limit of L d N x   , where L is the length of the photonic crystal, the Hamiltonian of equation (4) converges to the one of equation (1) with a renormalized d g s 1 , G by the factor 1 x but with the advantage of eliminating the collective quantum jumps of equation (2).

Protocol
The basic principle of our protocol for both the dissipative and the bandgap regimes is depicted in figure 2(c). In order to add one (symmetric) excitation to the target ensemble already containing m 1 excitations, (i) the source atom is excited, (ii) by dipole-dipole coupling the excitation is collectively transfered to the target ensemble through the d g 1 G -mode, and further to the detector ensemble through the d s 1 G -mode, and finally (iii) a fast π-pulse on the detector ensemble's g e ñ « ñ | | -transition terminates the dynamics. If a heralding measurement on the gñ | state on the emitters of the detector ensemble is successful, a symmetric excitation in the target ensemble must have been generated with a heralding probability that we denote as p m m 1 - and an error or infidelity I m m 1 - . If a collective excitation has been added, the source emitter and the detector ensemble are reinitialized and the process is repeated. Hence, to reach any state with m (symmetric) excitations in the target ensemble, one will have to repeat the above procedure successfully m times.
If at some point a heralding measurement fails, either a quantum jump in one of the ensembles has occurred or the excitation has not been transfered to the detector ensemble yet. Some of these processes do not spoil the coherence of the target ensemble state and would be correctable. However, because all these processes are indistinguishable, the whole protocol has to be repeated from the very beginning to avoid a low fidelity of the final state, that is, a low overlap with the target state. The final protocol to accumulate m excitations will be characterized by the average number of operations R p m k ) , which is in general exponential in m, and has a total infidelity where t y ñ | is the target state and ρ is the actual final state. In section 3, we first discuss in detail the protocol in the dissipative regime and will find an ultimate limit that is imposed by the collective quantum jumps of equation (2). Then, in section 4 we discuss how to adapt the protocol to the case of finite range of the dipole-dipole couplings d x and the limitations imposed by it.

Detailed protocol in the dissipative regime
The practicality of the outlined protocol is gauged by the heralding probability and the fidelity of the final state with respect to the target state, c t m m Because the Hamiltonian H wg of equation (1) leaves the excitation number m invariant and because the state is heralded at the end of every cycle, we only need to treat the case in which one excitation is added to the state of the target ensemble m 1 y ñ -| . If the heralding measurement is successful, the state is then m y ñ | . The full initial state of the system is denoted by s s . In the following analysis, we will skip most of the technical details and refer the interested reader to the appendix (see footnote 6).

Holstein-Primakoff-approximation: calculation of probability
For large ensemble sizes, which are necessary for the photon generation step, the low excitation regime, i.e., m N  , is approximately bosonic. In this case, the multilevel Holstein-Primakoff approximation (see [48,49]) can be applied 9 . Then, the spin operators in the ensembles t 1 and t 2 are approximated by bosonic operators Then, the Hamiltonian H wg of equation (1) couples the initial state to two other normalized states, that is, Other non-excited states that are reached by quantum jumps can be neglected because of the heralding step. G = G , may also lead to a sufficiently high success probability and fidelity. These variations are discussed at the end of this section. In the optimal case, the success probability of the heralding measurement (figure 3(a)) is (see footnote 6) The scaling originates from the fact that the process is very fast, T N 1 2 µ -, and that the non-Hermitian terms, which lead to the reduction of the success probability, scale with arise from the population of the specific states which are subject to the respective quantum jumps, that . The dependence on the Purcell Factor is exact because the evolution takes place in the subspace of a single excitation and every state is affected in the same way by spontaneous emission.
By repeatedly adding heralded single excitations, the state with m collective excitations m y ñ | can be reached. Clearly, the average number of repetitions is exponential in the number of excitations, i.e.,  3.2. Beyond Holstein-Primakoff approximation: calculation of fidelities For the generation of a single excitation the Holstein-Primakoff Approximation is exact. However, for higher excitations, the non-Hermitian part of the Hamiltonian leads to a coupling to additional states, that are linearly independent of the three states treated above. Also, the mapping of the source and target ensemble to cavity QED is no longer perfect and will suffer from a similar loss in fidelity as our protocol. The deviations from the approximation can be investigated numerically by using the exact Holstein-Primakoff Transformation (see appendix). Instead of three orthonormal states as above, one then has to consider m 4 1 + orthonormal states, which are symmetric in each ensemble. For obtaining the results, the bosonic operators b e i , and b s i , are cut-off at 2 and m 1 + , respectively.
The multitude of additional states that the non-Hermitian Hamiltonian couples to may lead to a non-unit overlap with the target state m y ñ | , which should go to unity in the limit of large ensemble size N 1  . Therefore, also the new initial state of the target deviates from m 1 y ñ -| and has to be obtained from the final state of the preceding step. The results from the full numerical analysis (figure 3) agree very well with the results obtained by applying the Holstein-Primakoff-approximation for N m  . Furthermore, the average accumulated infidelity as defined in equation (5) is very close to unity and scales (for m N  ) as where the prefactor was obtained by a fit of the results from numerical integration of the master equation. The fidelity is independent of the Purcell Factor because every state is affected in the same way by spontaneous emission and the transitions to unwanted states only happens through collective operators.

Variations of the protocols
The protocol described in the previous section used several requirements, e.g., tunable coupling to guided modes or fast π-pulses, to maximize the heralding probability while keeping the infidelity minimal. If some of these ingredients are not available there exist several alternatives to obtain still high heralding probabilities. For example 10 , • Fixed coupling to waveguide modes. Typically, the Purcell factor P d see appendix for details. Interestingly, the infidelity of equation (9) is unchanged.
• Replacing fast π-pulses. The fast π-pulse on the source atom at the beginning and on the detector ensemble at the end of each step can be avoided by applying a continuous external field with the same Rabi coupling strength Ω to the respective transitions. These are the s e ñ -ñ | | -transition of the source atom and the g e ñ -ñ | | -transition of the detector ensemble. The success probability is then maximized for the same ratio of the decay rates, i.e., m d g d   G , which implies that the probability would still scale exponentially with m N . In addition, one requires a measurement device which can resolve the excitation number of the detector ensemble to guarantee the transfer of m excitations to the target ensemble. Even if that is possible, e.g., the probability for generating two excitations at once is lower than the probability, p p 0 1 1 2   , obtained through the original protocol (see appendix).
In all of these variations, the final goal is to accumulate several excitations within the same hyperfine level sñ | . When the heralding fails, we reinitialize the process all over again, which yields an exponential number of operations R m with the number of excitations we want to create. Moreover, the existence of m excitations already in the state s causes the enhanced decay m d s 1 G of the target ensemble, which leads to a scaling of the success probability with m and to the necessity of a tunable ratio d G G for maximizing the success probability. The former can be avoided if after each heralding of a single collective excitation, it is stored in other hyperfine levels available s n ñ | to combine them a posteriori using Raman two-photon and microwave transitions plus atomic detection. It can then be shown [34,53,54] that by using only one additional hyperfine level, s 1 , the number of operations is still exponential R e m m µ , whereas, if we use m log 2 levels a subexponential scaling of R m can be achieved. In these cases, carefully constructed repumping schemes have to be used to avoid introducing additional errors during the repumping step [34].

Protocol in the bandgap regime
In the previous section, we showed how the success probability of the protocol in the dissipative regime is limited by quantum jumps into the waveguide, which leads to the scaling with . As we showed in section 2, a possibility to get rid of the quantum jumps while maintaining the dipole-dipole interactions is to use interactions mediated by the bandgap [36][37][38][39][40]. This can be interpreted as the formation of an atom-photon bound state mediating the exchange of interactions between emitters. By using this configuration we eliminate the quantum jumps into the waveguide at the price of reducing the dipole-dipole couplings due to their finite range d x . In principle, one can make d x much larger than the characteristic length of the system, Nd, so that all the emitters couple homogeneously as in the dissipative regime. However, this comes at a price of enlarging the length of the atom-induced cavity and therefore the subsequent reduction of the dipole-dipole coupling. In this section, we first discuss the scaling of the success probability and infidelity in the ideal limit N x  . In realistic cases, d x is limited by the length of the photonic crystal L, that is we require a finite d L x  . Thus, we also explore the limitations imposed by this trade-off to generate multipartite entangled states.

Ideal case
The idea of the protocol is analogous to the one in the dissipative regime: transfer a single excitation from the source atom to the target ensemble through the d g 1 G mode, and then from the target to the detector through d s 1 G . In the limit N x  the Hamiltonian H bg of equation (4) where the dipole-dipole couplings within the ensembles have been included in the Hamiltonian H LS . The dynamics of the system can be again best analyzed by using the Holstein-Primakoff-transformation. The main differences to the dissipative regime are the following: (i) The emitters within each ensemble suffer dipole-dipole interactions irrespective of their position, whereas in the dissipative regime these can be canceled by choosing the 2p (or π) distances. These dipole-dipole interactions for equidistantly spaced atoms lead to a collective Lamb-shift, which is, e.g., for the S eg  x µ G ( ), whereas the only process that makes the norm decay is the spontaneous emission probability with rate * G .
In general, this scaling is not better than in the dissipative regime. But as we showed in the previous section, the imperfect fidelity was arising from the collective quantum jump terms, which are vanishing in this case such that the infidelity with the final state satisfies I 0 m = .

Realistic case
For finite ξ one needs to take into account the changes in the collective Lamb-shifts and the coherent couplings. When taking this into account, the dynamics still lead to a (almost full) depletion of the population in the excited state of the source atom, see figure 4(a) for N=100 and 100 x = . At the point of maximal population transfer we plotted the phase c arg n ( ) and the intensity distribution c n 2 | | for the coefficients c n of g n eg in the target ensemble in figure 4(b). We see that in spite of the limited range N x~, the collective mode is approximately homogeneous. For smaller ξ, the phase and intensity distributions become inhomogeneous and therefore cannot be used to transfer the excitations coherently. In figure 4(c) we show the scaling of the infidelity of the intermediate state with respect to the completely symmetric state and see that the infidelity scales favorably with the cavity length, i.e., as I 2 x µ -.

Comparison between different protocols
The protocols presented in this manuscript together with the ones presented in [33,34] constitute a set of methods for quantum state preparation using different resources present in waveguide setups. To give a full understanding, we summarize the conditions and figures of merit for each protocol, identifying which ones are more suitable depending on the available resources (see Table 1): • In [33], we use atomic Λ-systems with equally spaced atoms to build up arbitrary superpositions of atomic/ photonic states. The protocol requires P 1  same conditions hold, i.e., if one works in the bad-cavity limit and has an ancilla atom which can be addressed individually.
• The first protocol discussed in [34] also uses Λ-systems, requires N P 1 d 1  and the use of an external single photon detector. The protocol heralds (by a photodetector with efficiency η) the transfer of single collective excitations with probability p, which can be controlled at will, but with a trade-off with the infidelities, which scales as I p • The other protocols discussed in [34], also exploit the long-range dissipative coupling for equally spaced atoms, and require P 1 d 1  . The advantage is that the probability of heralding a single collective excitation p e P d 1 µcan be made close to 1 for systems with P 1 d 1  . Moreover, the infidelity of accumulating m excitations is strictly I m = 0. This is certainly the best suited method in terms of fidelities but to obtain high probabilities we require systems with P 1 µ . This is probably the best method for optical fiber setups [26][27][28][29][30][31].
• The protocol within the bandgap regime is only suited for engineered dielectrics where the existence of bandgaps is possible. Though it has the advantage of eliminating quantum jumps into the waveguide, the finite range of the interactions ξ, leads to a worse heralding probability scaling with p e

Conclusions
In conclusion, we have proposed several methods for the heralded preparation of symmetric states in ensembles of emitters using coherent atom-atom interactions induced by their coupling to two guided modes in waveguide QED setups in both the dissipative and the bandgap regime. In the dissipative regime, we showed how the collective quantum jumps into the waveguide limit the single excitation heralding probability p e m m m N 1  p - which can still be close to 1 for systems even with P 1 d 1 < , which is very relevant for the experiments with optical fiber setups [26][27][28][29]. We also consider the situation of enginereed dielectrics within the bandgap regime in which the finite range of atom-induced cavities gives a more limited scaling of the probabilities. In all cases by using atomic detection and post-selection we rule out most of the errors, giving rise to very low global infidelities in both the dissipative (I m N m 2 2 µ ) and bandgap regimes (I m = 0, for N x  ) for the preparation of atomic states.
These prepared states can then be mapped to a photonic state of the waveguide with controllable temporal shape [32,33]. This mapping scales favorably with the system parameters, in particular the emitter number N and the Purcell factor P d 1 , that is the infidelity (or error) of this process scales as I m NP d ph 2 1 µ ( ). Therefore, this protocol can be used for the efficient preparation of triggered multiphoton states.  Typically, the relaxation timescales of the reservoir are much faster than the atomic timescales. This separation of timescales justifies the so-called Born-Markov approximation that allows to calculate the evolution of the atoms, through their reduced density matrix ρ, after tracing out the reservoir degrees of freedom. This approximation requires that one can neglect finite propagation lengths and non-Markovian effects, which requires that the maximal distance between atoms is small compared to the propagation length, v T g , during the time of operations, where v g is the group velocity in the waveguide. In the case of a onedimensional reservoir the evolution is then governed by the master equation [ depends on both the energy dispersion ( q, w n ) and guided mode profile (contained in g q h | |) [44]. Note, that the master equation above allows for two very distinct regimes for different interatomic spacings: • If the distance between two emitters is a multiple of q 2 0 0 p l = the coherent terms vanish and the evolution is purely dissipative and the atoms decay through a collective operator.
• If the distance between two emitters is an odd multiple of q 2 4 0 0 , the dipole-dipole interactions are at their maximum.
In this work, we exploit coherent dipole-dipole interactions such that a beneficial configuration is based on two atomic mirrors (see figure 1(a)) surrounding an atom acting as a source for distributing atomic excitations symmetrically to the inner mirrors (target ensembles t 1 and t 2 ). Collective and individual quantum jumps and other experimental imperfections cause transitions to undesired states. These transitions can be corrected by using the second guided mode and heralding measurements on the outer mirror emitters, the detector ensembles d 1 and d 2 .
In particular, the coherent dynamics in the double mirrors configuration (in the frame rotating at the frequency of the atomic frequencies) is then described by † † . Apart from interacting with the waveguide modes, the excited states e n ñ | may also emit photons into free space, or even to other polarizations that do no create collective coupling between emitters. We embedded all these processes into a rate, * G , typically of the order of the natural linewidth a G , which can be described through an additional Lindblad term in the Liouvillian as      ( ) . In the optimal case, the success probability of the heralding measurement ( figure 3(a) The scaling originates in the fact that the process is very fast, T N 1 2 µ -, and that the non-Hermitian terms after normalization is unity within the Holstein-Primakoff approximation. We neglect errors originating from finite detection efficiencies and dark counts, because the detection via the detector ensemble can be repeated as many times as necessary.
By repeatedly adding excitations, we can accumulate several excitations within the same mode  » for m N 1   , where the prefactor was found by a numerical fit. The fidelity is independent of the Purcell Factor because the error stems from nonlinear corrections to the Holstein-Primakoff picture that enter through collective rather than spontaneous emission events which affect every state in the the same way.

B.5. Variations of the protocol
The scheme previously proposed can be modified if some of the demanded ingredients are not available. The goal is to maximize the success probability (equation (B.18)) constrained by the parameters that are experimentally achievable:  figure B1. The prefactor was obtained from a numerical fit. The overlap with the goal state is 1 within the Holstein-Primakoff-approximation and for m=0 and slightly deviates from unity by N 1 2 for the full solution. As expected, they agree in the limit of large ensemble sizes N 1  . Figure B2. Moreover, it is important to highlight another difference with respect to the previous protocol. Previously, when a heralding measurement failed we reinitialize the target ensemble back to gñ | destroying the stored m excitations. However, in this case the excitations stored in s i ñ | can be salvaged with minimal error by using an appropriate repumping scheme in which the symmetry of the state is unaffected. This is achieved by switching 0 G -mode due to the symmetry of the state of the target ensemble. There are two types of errors: (i) a spontaneous jump in the target ensemble can already have occured before the repumping step, in which case the error probabiliy scales as ; Concerning the methods to accumulate excitations into a single level, there are multiple approaches that sketch briefly: Figure B3. (a) If only a single guided mode is available, the protocol can still be applied if another metastable state with strongly suppressed spontaneous emission is available. (b) Beam-splitter-like transformations between metastable states are obtained by applying a corresponding external driving field. (c) Excitations can also be added by using the decay through the guided mode. (d) The heralding probability for generating two excitations at once is lower than the heralding probability of the original protocol.