Reliable multiphoton generation in waveguide QED

On-demand generation of optical propagating photons is at the basis of many applications in quantum information science, including multipartite teleportation murao99a , quantum repeaters briegel98a , quantum cryptography gisin02a ; durkin02a , and quantum metrology giovannetti04a . While single photons are routinely produced in different experimental setups lounis05a , single–mode multiphoton states are much harder to generate dellanno06a . Current methods are limited by either an exponentially small success probabilities or by low fidelities. Here we show that reliable single-mode multiphoton sources can be constructed with emitters close to nano-photonic waveguides by combining both the strong coupling present in those systems and collective phenomena. In particular, we show how collective excitations can be efficiently loaded in a collection of atoms, which can then be released with a fast laser pulse to produce the desired multi-photon state porras08a ; gonzaleztudela15a . In order to boost the success probability and fidelity of each excitation process, our method utilizes atoms to both generate the excitations in the rest, as well as to herald the successful generation. Furthermore, to overcome the exponential scaling of the probability of success with the number of excitations, we design a protocol to merge excitations that are present in different internal atomic levels with a polynomial scaling.

The enhancement of light-matter interactions provided by quantum nanophotonics opens up new avenues to create high-fidelity multiphoton states. For example, $m$ quantum emitters can be strongly coupled to structured waveguides, which show large Purcell factors, $P_{\mathrm{1d}}$, so that $m$ atomic excitations can be mapped to a waveguide mode with an error (or infidelity, $I_{m}$) scaling as $m/P_{\mathrm{1d}}$. However, the resulting state is not a single mode, but a complex entangled state of several modes gonzaleztudela15a , so that it cannot be directly used for quantum information purposes. Single-mode multiphoton states can be created by storing $m$ collective excitations in $N\gg m$ atoms, which are then mapped to a photonic state of the waveguide. While the latter process can be achieved with very low infidelity, scaling as $m^{2}/(NP_{\mathrm{1d}})$ porras08a ; gonzaleztudela15a , present schemes for the first part scale like $I_{m}\propto m/\sqrt{P_{\mathrm{1d}}}$ gonzaleztudela15a , as they still do not fully exploit the strong coupling to the waveguide nor collective effects. This ultimately limits the fidelity of the whole procedure.

In this work we show how this limitation can be overcome with new schemes for the heralded generation of $m$ collective excitations in $N\gg m$ atoms coupled to a waveguide. The basic idea is to use the atoms themselves to both create the excitations one-by-one, and to herald the success of the process. In this way, arbitrarily small infidelities, $I_{m}$, can be obtained at the expense of making the process non-deterministic. Depending on the scheme, we find that the global probability of success (or, inversely, the average number of operations, $R_{m}$) decreases (increases) exponentially with $m$, and thus it cannot be scaled to arbitrarily large photon production. Finally, we also show how to overcome this exponential law by using additional atomic states, atom number resolved detection, and a specific protocol to merge excitations, while keeping a very low global infidelity, $I_{m}\sim{\rm poly}(m)/(NP_{\mathrm{1d}})$.

Structured waveguide setups with trapped atoms offer several interesting characteristics that we exploit to design our protocols, namely, i) regions of large Purcell Factor $P_{\mathrm{1d}}\gg 1$, e.g., due to slow light in engineered dielectrics laucht12a ; lodahl15a ; yu14a ; goban15a , while keeping at the same time ii) long-propagation lengths of the guided modes compared to the characteristic wavelength ($\lambda_{\rm a}$) that give rise to long-range dissipative couplings chang12a . Moreover, as shown in, e.g. Ref. chang12a , in order to avoid dipole-dipole interactions and fully exploit superradiance effects we assume iii) the atoms to be placed at distances $z_{n}=n\lambda_{\rm a}$, with $n\in\mathbb{N}$. ¹¹1In fact all results can be extended to separations of $\lambda_{\rm a}/2$. Finally, iv) it is possible to read the collective atomic state very efficiently through the waveguide due to the naturally short timescales and large collection efficiencies. We use atomic detection for heralding, which has already reached accuracies of $10^{-4}$ myerson08a in trapped ion setups, where the collection efficiency is not enhanced by the presence of a waveguide, as in our scheme. Thus, here we assume the atomic detection to be perfect, and consider $P_{\mathrm{1d}}$ and $N$ as the main resources to analyze the figures of merit of the protocols. For the analysis of three different schemes, we adopt the following strategy for each one: we analyze the process of generating a single collective excitation, assuming the atoms already store $m$ excitations. We denote by $p$ the probability of success, and $I_{m\rightarrow m+1}$ the corresponding infidelity. We finally analyze the average number of operations $R_{m}$ and final infidelities $I_{m}$ for accumulating all $m$ excitations. To simplify the expressions along the main text, we assume to work in a regime with $N\gg m$ and $P_{\mathrm{1d}}\gg 1$, though complete expressions can be found in Supplementary Material supmat .

Let us first analyze a protocol inspired in a method originally devised to create long-distant entangled states in atomic ensembles duan01a , that requires $N$ atoms placed close to a 1d waveguide as depicted in Fig. 1(a). The atoms must have a level structure as depicted in Fig. 1(b), where the atomic transition $e\leftrightarrow s$ is coupled to the guided modes at a rate $\Gamma_{\mathrm{1d}}$ and the transition $g\leftrightarrow e$ is driven equally by a laser with Rabi frequency $\Omega$. The excited states $e$ also radiate into leaky modes (of the waveguide or outside) other than the relevant one, that give rise to a decay rate $\Gamma^{*}$, leading to the Purcell factor $P_{\mathrm{1d}}=\Gamma_{\mathrm{1d}}/\Gamma^{*}$. The excitations are stored in the states $g$ and $s$, which are assumed to be decoherence-free like any other hyperfine ground state. The procedure assumes that we start with $m$ collective excitations in level $s$, and the goal is to add another one. Thus, the initial state is $|\Phi_{m}^{s}\rangle\propto\mathrm{sym}\{|s\rangle^{\otimes m}\otimes|g\rangle^{N-m}\}$, where ‘sym’ denotes the symmetrizing operator. The idea is to weakly excite the atoms in $g$ to level $e$ with a short laser pulse of duration $T\ll 1/(N\Gamma_{\mathrm{1d}})$ and if a photon in the waveguide is detected, it heralds the addition of an excitation in state $s$. As all the atoms are equally coupled to the waveguide, the excitation will be collective.

Let us denote by $x=\Omega\sqrt{N}T/2\ll 1$, so that right after the pulse we have the state (up to a normalization constant), $\big{[}\mathbf{1}+x\,S_{eg}+x^{2}\,S_{eg}^{2}+O(x^{3})\big{]}|\Phi_{m}^{s}\rangle$, where we used the notation $S_{\alpha\beta}=(N)^{-1/2}\sum_{n=1}^{N}\sigma_{\alpha\beta}^{n}$ for the collective spin operators and $\sigma^{n}_{\alpha\beta}=|\alpha\rangle_{n}\langle\beta|$ with atom number $n=1,\dots,N$. After the pulse, we leave the system free to evolve under the interaction with the waveguide for a time $t\gg 1/\Gamma^{*})$, in which case the wavefunction terms with excitations in $e$ decay either to waveguide/leaky photons. If a waveguide photon is detected, either it comes from the lowest order term, $O(x)$, in which case the atomic state will be the desired one, i.e., $S_{sg}|\Phi_{m}^{s}\rangle$, or from the double excited state, in which case we will prepare the wrong state introducing an error. Emission of leaky photons also produces errors, but of smaller order supmat . Denoting by $\eta$ the detection efficiency, the success probability in generating the desired state is $p\approx\eta x^{2}$ and the infidelity is $I_{m\rightarrow m+1}\propto(1-\eta)x^{2}$ [to lowest order in $x$]. To create $m$ excitations, one has to detect $m$ consecutive photons, leading to $R_{m}=p^{-m}$ and $I_{m}\propto m(1-\eta)x^{2}$. As in Ref. duan01a , the error can be made arbitrarily small at the expense of decreasing the success probability. If a high fidelity is required (e.g., $0.999$) the method is practicable only for few excitations. One way of reducing $R_{m}$ is by using an additional metastable state $s_{1}$ in which the heralded excitation is stored after each successful addition. This can be done by combining, e.g., a two-photon Raman transition, to make a beam splitter transformation between levels $s$ and $s_{1}$, and post-selection conditioned on no atomic detection in $s$. Then, assuming that we have $m$ excitations already stored in $s_{1}$, this procedure generates $m+1$ within the same level. It can be shown supmat ; dakna99a that this strategy leads to an average number of operations $R_{m}\propto e^{m}/p$.

To overcome the trade-off between probabilities and fidelities coming from zero and double excitations, we propose a protocol relying on a configuration as depicted in Fig. 2(a): we replace the write field of the previous scheme by a single source atom that guarantees the transfer of at most a single excitation to the target ensemble. Furthermore, in a second step, $N_{d}$ detector atoms act as a detector to herald the transfer of excitations, replacing the photon detector. Both the source and detector atoms should be separated from the target ensemble by a multiple of $\lambda_{\rm a}/2$ and sufficiently separated for independent addressing with external control fields. The protocol requires a level structure as shown in Fig. 2(b) where two dipolar transitions $e_{1}\leftrightarrow g$ and $e_{2}\leftrightarrow s$ are coupled to the same waveguide mode with rates that we set to be equal for simplicity and with a corresponding Purcell Factor $P_{\mathrm{1d}}$. We require the use of other hyperfine, auxiliary levels, $a_{1},a_{2},s_{1}$. The transition $a_{1}\leftrightarrow e_{2}$ is connected by a laser, whereas the $a_{1}\leftrightarrow e_{1}$ is a two-photon transition mediated by $a_{2}$, with effective Rabi frequency $\Omega_{a}$, so that direct spontaneous emission from $e_{1}\to a_{1}$ is forbidden vanenk97a ; porras08a ; borregaard15a ; supmat . The level $s_{1}$ is used to store excitations and decouple them from the dynamics induced by the waveguide. We require that $s_{1}$ is not connected to neither $e_{1}$ nor $e_{2}$ by a dipole transition, so that it will only be connected to $s$ coherently through microwave fields.

As before, this protocol starts with a superposition in the target ensemble $|\Phi_{m}^{s}\rangle_{t}$, and with the source/detector atoms in $a_{1}/s$ respectively. The heralded transfer of a single collective excitation consists of several steps [Fig. 2(c)]. The first one, coherently transfers the excitations from $s\to s_{1}$ in the target and detector ensemble to protect them from the waveguide dynamics. The second step uses a short laser pulse in the source atom to excite it to state $e_{1}$, and then switches on the lasers driving $e_{1}\leftrightarrow a_{1}$ via a two-photon transition in the target ensemble with (effective) Rabi frequency $\Omega_{a}$ for a time $T_{a}$. Ideally, the source atom exchanges the excitation with the target ensemble by a virtual photon in the waveguide, thus generating a collective excitation in $a_{1}$-state of the target ensemble. After that, the laser $\Omega_{a}$ is turned off and one waits for a time $t\gg\left(\Gamma^{*}\right)^{-1}$ such that any (non-ideal) remaining population in the excited state decays. Thirdly, we apply $\pi$-pulses to decouple the source atom and couple the target and detector ensemble. Another short pulse is applied to move the collective excitation in the target ensemble from $a_{1}$ to $e_{2}$ and we switch on the laser $\Omega_{b}$ driving the $e_{2}\leftrightarrow a_{1}$ transition in the detector ensemble for a time $T_{b}$ with Rabi frequency $\Omega_{b}$. This transfers the collective excitation in the target ensemble to $s$ and creates a collective excitation in $a_{1}$ in the detector ensemble. At the end, a measurement of that internal state, $a_{1}$, of the detector atoms (through fluorescence in the waveguide) heralds the successful preparation of a collective excitation in the target ensemble, i.e., $|\Phi_{m+1}^{s}\rangle_{t}$.

Let us now analyze the protocol in detail. In the second step, the evolution of the source/target atoms is described by a master equation supmat , which can be analytically solved in the limit $NP_{\mathrm{1d}}\gg 1$. By choosing the Rabi frequency $\Omega_{a}\approx\sqrt{N\Gamma_{\mathrm{1d}}\Gamma^{*}}$, and $T_{a}=\pi/\sqrt{\Gamma_{\mathrm{1d}}\Gamma^{*}}$, the probability for the ensemble to end up in the desired state after the second step, $|g\rangle_{s}\otimes S_{a_{1}g}|\Phi_{m}^{s_{1}}\rangle_{t}$, is maximized $p_{\rm a}\approx e^{-\pi/\sqrt{P_{\mathrm{1d}}}}$, which can be improved by appropriate pulse shaping supmat . Similarly, in the third step, the evolution of the source/detector atoms can be analytically solved in the limit $N_{d}P_{\mathrm{1d}}\gg 1$, obtaining a probability to end up in the desired state $S_{sg}|\Phi_{m}^{s}\rangle_{t}\otimes S_{a_{1}s}|s\rangle_{d}^{\otimes N_{d}}$ given also by $p_{\mathrm{b}}\geq p_{\rm a}$. Thus, the total probability of success of the protocol is lower bounded by $p\gtrsim p_{\rm a}^{2}$, with a heralding infidelity $I_{m\to m+1}=0$, as we rule out all the possible errors through post-selection. Indeed, the only way to have the detector atom in $a_{1}$ is that the steps two and three have occurred as desired. Any spontaneous emission in any of the atoms or photon absorption is incompatible with that event. Thus, to accumulate $m$ excitations we need to repeat an average number of times $R_{m}\propto p^{-m}$, but with $I_{m}=0$. As $p$ can be very close to $1$ for systems with $N,N_{d},P_{\mathrm{1d}}\gg 1$, we can still expect to achieve big number of excitations $m\gg 1$ in spite of the exponential scaling of $R_{m}$ as shown in Fig. 3 for systems with $N,N_{d}\gg m$. In case of unsuccessful detection, then, we pump all the target atoms back to $|g\rangle$ and re-initialize the process.

After this analysis one natural question arises, that is, whether the exponential number of operations is a fundamental problem of probabilistic protocols dakna99a . This leads to our third and final scheme in which we design a specific protocol which circumvents this exponential scaling with a judicious modification of the previous protocol using several additional atomic states $s_{n}$ [see Fig. 2(b)] and atomic number resolved detection. The idea is that with each successful detection, we transfer the heralded collective excitation in $s$ not to the same level $s_{1}$, but to one of several states $\{s_{n}\}$ to then merge them a posteriori with an adequate protocol that we explain below. For that note that, in contrast to our previous schemes, in case of unsuccessful detection, i.e., the detector atoms were not found in $a_{1}$, the $m$ excitations stored in $\{s_{n}\}_{n}$ are not destroyed, but only the one we wanted to add. We can pump back the target atoms in $s,a_{1}$ to $g$ and try again. The price one has to pay is that errors appear since one may not recover the original collective state in the ensemble, i.e., because a spontaneous emission event occurred within the target ensemble supmat . One can show that the main source of errors occurs when a collective excitation was indeed produced in the ensemble, but it was not detected (because, e.g., of spontaneous emission in the detector ensemble). In order to reduce this error, we have to ensure that undetected collective states return back to $g$ coherently. For that, we apply the following repumping procedure: first, we pump all levels but $s,\{s_{n}\}_{n}$ back to $g$. Then we coherently transfer the potential excitations from $s\to a_{1}$, and then apply a pumping through the waveguide, $a_{1}\to e_{1}\to g$. In order to further minimize the errors introduced by repumping and obtain the desired scaling of infidelity ($1/(NP_{\mathrm{1d}})$), we also need to modify the third step of the protocol for which then, interestingly, only a single detector atom is needed, $N_{d}=1$. It can be shown that once we are in $S_{a_{1}g}|\Phi_{m}^{\{s_{n}\}_{n}}\rangle|s\rangle_{d}$, the optimal strategy is to apply a fast $\pi$-pulse in the target atoms such $S_{e_{2}g}|\Phi_{m}^{\{s_{n}\}_{n}}\rangle_{t}$ is prepared. Then, the system is left free to interact for a time $T_{\mathrm{b}}=1/\Gamma_{\mathrm{1d}}$ without any external fields, and the dynamics are terminated by applying another $\pi$-pulse on the target and detector atoms. We find that the optimal probability for the system to be in $S_{sg}|\Phi_{m}^{\{s_{n}\}_{n}}\rangle_{t}\otimes|a_{1}\rangle_{d}$ is $p_{\mathrm{b}}\approx 0.1\left(1-1/P_{\mathrm{1d}}\right)$, which can be improved up to $p_{\mathrm{b}}\sim 0.33$ by repeating the fast $\pi$-pulses several times supmat . Finally, if we fail, we apply the repumping and repeat the process until success, of the order of $1/p$ times with $p=p_{\rm a}p_{\mathrm{b}}$. In Ref. supmat , we analyze the process in detail, identifying all errors and how they accumulate in the repetitions, leading to an (averaged) infidelity $I_{m\to m+1}\lesssim m/(pNP_{\mathrm{1d}})$ supmat . We confirm numerically (Fig. 4) the scaling of $p$ and $I_{m\to m+1}$ integrating the corresponding master equation, showing that our analytical estimations capture the right scalings.

Thus, the problem is reduced to merging the atomic excitations distributed in the levels $\{s_{n}\}_{n}$. For example, if instead of adding excitations one by one, as we did for the first scheme, we use $\log_{2}m$ metastable states $s_{n}$, and adopt a tree-like structure, we obtain a superpolynomial scaling fiurasek05a ; supmat in $R_{m}\propto m^{\log_{2}m}/p$ to merge $m$ collective excitations in a single atomic level. The key step is to double the number of excitations in each step using beam splitter transformations and post-selection conditioned no atomic detection in one of the levels. Moreover, combining the beam splitters with displacement transformations, arbitrary superpositions of collective atomic excitations can be obtained dakna99a ; fiurasek05a ; supmat .

Finally, we go one step further by using number-resolved atomic detection to obtain a polynomial scaling. The key point is to realize that if we have $n$ excitations stored in two atomic levels, after applying a beam splitter transformation, the probability to obtain exactly $2n$ excitations in one of them (by detecting no excitation in the other) decays with $n$; however, the probability of obtaining more than $3n/2$ in one level (by detecting $p<n/2$ in the other) is actually bounded below by 1/3, independent of $n$ supmat . Assuming the worst case scenario in which after detecting $q<m/2$ excitations in one of the two levels, we assume that the other state only goes up to $3n/2$ excitations, that gives us an upper bound for the number of operations supmat $R_{m}\lesssim m^{4.41}\log_{3/2}(m)/p$, that is already polynomial, and which in practical situations is better as the number of excitations in the other mode is always larger than $3n/2$. The number of atomic levels, $|s_{i}\rangle$, required to reach $m$ excitations scales logarithmically $\log_{3/2}m$, and the final infidelity scales as $I_{m}\propto\mathrm{poly}(m)/(NP_{\mathrm{1d}})$. We note that atomic detection may itself introduce errors, however, the aim of this scheme is just to show that polynomial scaling can be achieved, despite not being currently the most efficient method. However, we expect that by suitable modification and adaptation to specific setups, polynomial and efficient scaling can be constructed.

State-of-the art technologies already provide systems with $N\sim 1-2$ emitters coupled to engineered dielectrics thompson13a ; tiecke14a ; yu14a ; goban15a ; laucht12a ; lodahl15a with $P_{\mathrm{1d}}\sim 1-100$. Advances in both fabrication and trapping techniques foresee implementations with $N\sim 100$ atoms and $P_{\mathrm{1d}}\gtrsim 100$ in the near future. The use of atomic internal levels may be replaced by motional levels of each of the atoms, if they are trapped in pseudo-harmonic potentials parkins95a , since there can be many of them at our disposal. There are several sources of errors that have not been considered here and that will give limitations to our proposal, such as decoherence of hyperfine levels, laser fluctuations, non perfect atom-resolving detection, although we expect that they can be controlled to a large extent. Other effects such as retardation, imperfect atomic cooling, or fluctuations in atomic positions may have to be considered in the analysis to achieve large fidelities (see, however gonzaleztudela15a where part of these errors were already considered).

In conclusion, we have presented several probabilistic protocols to generate heralded highly entangled atomic states that afterwards can be mapped to photonic states at will with very high fidelities. In particular, we show how to accumulate $m$ collective atomic excitations with infidelity $I_{m}=0$ and an exponential number of operations $R_{m}=p^{-m}$, being $p$ the heralding probability for adding a single excitation. We design a protocol where $p$ can be close to unity for systems with large $N,P_{\mathrm{1d}}$, which would allow to accumulate $m\sim 20-50$ excitations with systems $P_{\mathrm{1d}}\sim 10^{2}-10^{3}$ using $R_{m}\sim 10^{4}$ operations with unit fidelity [see Fig. 3]. Moreover, we also present a protocol with polynomial scaling in the number of operations $R_{m}$ by using atomic excitation number resolved and overall low infidelity $I_{m}\propto\mathrm{poly}(m)/(NP_{\mathrm{1d}})$. Though, we discus our protocols in the context of waveguide QED, the ideas can also be exported to other systems such as low mode cavities specht11a ; thompson13a ; tiecke14a ; haas14a , superconducting circuits mlynek14a or optical fibers vetsch10a ; goban12a ; petersen14a ; beguin14a with suitable modification of the protocols.

Acknowledgements. The work of AGT, VP and JIC was funded by the European Union integrated project Simulators and Interfaces with Quantum Systems (SIQS). AGT also acknowledges support from Intra-European Marie-Curie Fellowship NanoQuIS (625955). VP acknowledges the Cluster of Excellence NIM. HJK acknowledges funding by the Air Force Office of Scientific Research, Quantum Memories in Photon-Atomic-Solid State Systems (QuMPASS) Multidisciplinary University Research Initiative (MURI), by the Institute of Quantum Information and Matter, a National Science Foundation (NSF) Physics Frontier Center with support of the Moore Foundation by the Department of Defense National Security Science and Engineering Faculty Fellows (DoD NSSEFF) program, by NSF PHY1205729 and support as a Max Planck Institute for Quantum Optics Distinguished Scholar.

Supplemental Material: Reliable multiphoton generation in waveguide QED

In this Supplementary Material, we provide the full details on how to characterize the different protocols described within the main manuscript. For the sake of understanding, we devote different Sections to the each protocol including in all of them the atom-waveguide tools required and the calculation of probabilities and fidelities such that each Section is self-contained. The summary of the Supplementary Material reads as follows:

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  In Section SM1, we discuss the first protocol of the main manuscript which adapts a protocol envisioned for entangling atomic ensembles duan01a to herald the generation of $m$ collective excitations using single photon detections within the waveguide. We discuss both the situation where we accumulate directly the excitations into the same atomic level and where we store it in another hyperfine level and combine them a posteriori.

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  In Section SM2, we discuss the second protocol of the main manuscript which accumulate excitations directly within the same level using a two-step protocol. The protocol overcomes the trade-off between probability of success and fidelities of the first one but still requires an exponential number of operations.

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  In Section SM3, we discuss the third protocol which show to overcome the exponential scaling of the number of operations of the second protocol by using extra auxiliary levels to store the excitations after each heralded transfer while keeping the overall infidelity still low

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  For completeness, in Section SM4 we discuss a variation of the previous two protocol that relaxes part of the requirements of the protocols two and three, e.g., closed transition, at expense of slightly worse scaling of probabilities and infidelities.

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  Finally, in Section SM5 we discuss on how to use beam splitters transformations and atomic detection to merge atomic excitations and reach high photon numbers.