1. Introduction

The challenge of developing a reliable optical quantum network [1,2] essentially relies on two key ingredients, one being photons carrying quantum information as flying qubits and the other being stationary QEs residing at the nodes of the network to store and process the received information [3]. The burgeoning field of waveguide quantum electrodynamics (wQED) [4–6] encompasses both of these ingredients. Additionally, wQED platforms allow us to establish strong light-matter interaction at the level of single photons and single QEs [6]. Furthermore, wQED also enables long-distance quantum communication protocols (thanks to the advancement in optical fiber technology ) [7,8]. Due to these remarkable features, the field of wQED has witnessed a tremendous amount of research activity in the last decade, both at theoretical and experimental frontiers (see the reviews [4–6] and the papers indicated therein).

The case of a single photon and a waveguide coupled to an arbitrary number of QEs is a well-studied problem in quantum optics [9–12]. As is the case of adding more photons to the system [13–22]. However, a real quantum network would require more than a single waveguide, instead a series of waveguides coupled via QEs which will allow for the routing/switching/redirection of photons among different spatially distant nodes, a key feature in such networks. In this context, the case of single-photon routing either in the presence of two-level quantum emitters coupled to bidirectional [23–26] or chiral/non-bidirectional waveguides [27] have been reported in the past [28–31]. There have also been studies where a single-photon routing is reported using three-level QEs [32–37]. However, the routing of more than one photon in many-emitter waveguide geometries has, to the best of our knowledge, not been studied in depth (see, for instance, Refs. [38–40] where two-photon routing has been studied in wQED architectures but only for the single QE case). As we emphasized above, a practical quantum network should be able to route more than one photon in the presence of several QEs; therefore, in this work, we present a detailed examination of the novel and important problem of two-photon routing using few-emitter wQED ladders.

In our theoretical approach, we use the chiral light-matter couplings where the process of photon absorption and emission would be suppressed in one of the two directions of the waveguide due to the phenomenon of spin-momentum locking of light [27]. We then calculate the two-photon routing probability, i.e., the likelihood of the two photons emerging at two ports of the wQED ladder. We divide the present study into two regimes, namely, under a purely plane wave approximation (well known as the scattering case) and for the case in which bound states are incorporated in the analysis. For the scattering case, we discuss our routing scheme up to five QEs, while for the bound state problem (due to the mathematically involved nature of the results), single and two QEs cases will be presented. In both cases, we discuss the ideal and non-ideal regimes where the spontaneous emission loss from the quantum emitters has been or has not been ignored, respectively.

As some of the key findings, we noticed a general pattern in the scattering case and in the absence of losses (ideal case), where the routing probability is maximized when one photon is off-resonance and the other is on resonance for an odd number of QEs. While for an even number of QEs, both photons being off-resonance results in better routing probabilities. We also constructed an algorithm to extend the results to an arbitrary number of quantum emitters. Our bound state analysis, for both one and two QEs, shows that in the critical coupling regime (when spontaneous emission rate is set equal to the emitter-waveguide coupling rate), the routing probability is maximized (and takes a higher value as compared to the corresponding plane wave scenario) where both photons are off-resonant.

The rest of the paper is organized as follows. In Sec. 2, we present the theoretical description (Hamiltonian, two-photon state, amplitude equations, and boundary conditions) obeyed by the system. In Sec. 3, we briefly recap the scattering case for the single-photon routing problem in the presence of a single QE. In Sec. 4 and Sec. 5, we presented the results for two-photon routing for a single and many (up to 5 QEs) under a purely scattering regime. Next, in Sec. 6, we examine the two-photon routing in the case of bound state inclusion. Finally, in Sec. 7, we close with a summary of the main results of this work and indicate some of the possible future directions of this work.

2. Two-photon transport equations in chiral wQED ladders

The theoretical description of a number of two-level QEs side-coupled to two waveguides can be encapsulated with a Hamiltonian of the form

Note that waveguide fields have been quantized in the real-space formalism of quantum optics [9]. Here $\widetilde {\omega }_{eg} = \omega _{eg}-i\frac {\gamma }{2}$ is the modified emitter transition frequency (assumed same for all QEs) with $\gamma$ being the spontaneous emission rate. $\hat {\sigma }_{j}$ is the atomic lowering operator of the $j$-th QE. $v_{g}$ is the group velocity of the photon in the waveguides which for the sake have been assumed to be the same for both waveguides. $\hat {b}(y)$ and $\hat {c}(x)$ are position-dependent photon annihilation operators in the upper and lower waveguides and $W_{j}$ and $V_{j}$ are the QE-coupling strengths in the upper and lower waveguides respectively. The location of the $j$th emitter given by $L_{j} = (j-1)L/2, \forall j=1,2,3,\ldots, N$ with respect to the origin which we have set at the location of the first emitter. Non-vanishing commutation relations are given by: $[\hat {c}(x), \hat {c}(x^{'})]=\delta (x-x^{'})$, $[\hat {b}(y), \hat {b}(y^{'})]=\delta (y-y^{'})$, and $\lbrace \hat {\sigma }_j,\hat {\sigma }^\dagger _k\rbrace =\delta _{jk}$. Next, we specify the state of the system restricted to two-excitation sector of the Hilbert space as

From the general structure of these equations, after eliminating $e_{u/d}$ amplitude, one would expect to generate $4N+3$ coupled equations for the general problem of $N$ number of QEs. In the equations above we have defined $\Gamma _{j}^{u} = \frac {W_{j}^{2}}{2v_{g}}$ and $\Gamma _{j}^{d} = \frac {V_{j}^{2}}{2v_{g}}$ as the emitter-waveguide coupling rates respectively for the upper and lower waveguide. Furthermore, we have eliminated the mixed amplitudes using the expressions

Lastly, to solve the aforementioned set of equations, we express the boundary conditions obeyed by $\psi$, $\phi$, $\varphi$, and $e_{u/ld}$ amplitudes as

With the amplitude equations and boundary conditions specified for the two-photon problem, in the next section, we briefly discuss the single-photon routing for the single QE case [28,38], which will aid in expressing our two-photons routing results in the later sections.

3. One photon routing with one QE: scattering case

To set the stage for the two-photon routing calculations in wQED ladders, we recap the simple case of single-photon routing in a single-QE wQED ladder setup. The Hamiltonian for the one-photon one-QE case can be recovered from Eq. (1) by setting $j=1$. However, the quantum state of the system restricted to a single excitation is different than the two-photon state and can be written in the following as

In the last equation, $\xi$, $\zeta$, and $\mathcal {A}$ are the probability amplitudes of the photon in the lower waveguide, upper waveguide, and QE being excited, respectively. Like before, $| {\varnothing }\rangle$ represents the system’s ground state. Assuming $\xi (x)$ and $\zeta (y)$ obey the plane-wave ansatzes, one can express

 

Fig. 1. Two-photon waveguide QED ladder setup: Two bidirectional waveguides are side coupled to many two-level QEs, with the lower and upper waveguides having left ($V_{L}$, $W_{L}$) and right ($V_{R}$, $W_{R}$) QE-coupling strengths. This work considers a perfectly chiral case in which all left/back-reflected directions are entirely suppressed. As the initial conditions, we assume the two photons enter Port 1 all QEs and are in their ground state, and there are no photons in the waveguides. For further details about the system description, see the text below.

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Here $\omega$ is the frequency of the launched single photon. Due to the symmetry of the waveguide setup, we observe $\widetilde {t}^{~u} = \widetilde {t}^{~d}$, which allows us to drop the superscripts in the discussion to follow. The environmental decay rates should be addressed in an actual physical setup. Therefore, it is helpful to cast the previous probability amplitudes in a form that accounts for the possible ratios between the QE coupling with the waveguide and environmental decay rates. Such a ratio can be expressed as a Purcell factor [38]

Proceeding further, assuming $2\Gamma _{1}^{d} = 2\Gamma _{1}^{u} + \gamma$ and $\omega = \omega _{eg}$, the probability amplitudes mentioned in Eq. (10), can be expressed in terms of Purcell factor in the following form

Focusing on the case in which the photon was launched from Port-(1), the routing probability of redirecting the photon from Port-1 to Port-3 is expressed as $\widetilde {T} = \left\| {\widetilde {t}} \right\|^{2}$. In Fig. 2, we plot the routing probability as a function of the Purcell factor. The routing probability begins to plateau and reaches a unit value (deterministic routing) around $P_{F} = 100$. Note that the realistic Purcell factors in the 1D wQED platforms can be found within a range of $10$ to $20$ [41], which from our plot correspond to routing probabilities ranging from $\sim 0.812$ to $\sim 0.905$. This is an appealing finding consistent with what Gonzalez-Ballestero et al. have reported [38].

 

Fig. 2. The dependence of single-photon routing probability $\widetilde {T}$ from Port 1$\rightarrow$ Port 3 on the Purcell factor $P_{F}$ in a single-emitter wQED ladder problem.

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4. Two photon routing with one QE: scattering case

We now focus on the two-photon routing case and the simplest possible problem of a single QE chirally coupled to both waveguides. By two-photon routing here, we mean that out of the two photons launched at Port 1, one photon is detected at Port 2 and the other at Port 3. For the sake of simplicity, we assume both QEs to be identical and the inter-emitter separation to be much smaller than the length $L$ of the waveguide. The first assumption ensures that the Purcell factor remains the same for each QE, while the second condition ensures that only the plane-wave solutions contribute to the overall probability. We define the required two-photon routing probability $P_{23}$ in the following way

Here, $\langle {\Psi }| {\Psi }\rangle _{\gamma = 0}$ has the same bounds of integration as the numerator. The form of $\phi$ can be obtained from equation set Eq. (4) along with the given initial incident wave. Thus, in the region $\lbrace x_{1},x_{2}\rbrace < 0$ we obtain

Here, the subscripts $1$ and $2$ indicate photons 1 and 2, respectively. Next, the solution to $\psi$ when $\lbrace y_{1},y_{2}\rbrace < 0$ and $y_{1/2} < 0 < y_{2/1}$ is zero due to the choice of initial condition that the photons being incident from Port 1. The solution for $\psi$ when $\lbrace y_{1},y_{2}\rbrace >0$ can be expressed in terms of the ansatz mentioned below

Similarly, the solution for $\varphi$ when $\lbrace x,y\rbrace > 0$ takes the following form

We note that in the plane wave limit $\langle {\Psi }\left\| {\Psi }\rangle\right\| _{\gamma = 0} \propto L^{2}$, thereby $P_{23}$ can be reduced in the final form as

Note that using Eq. (4) and Eq. (5), one can also find the corresponding emitter excitation probabilities. Since we focus only on the output detection probability, we leave the emitter excitation probability calculations as an exercise for the interested reader. Next, in the plane wave case we notice that the routing probability is proportional to the sum of all paths that photons can take. Since $P_{23}$ can be expressed in terms of single-photon probabilities, it can be arranged to depend on the Purcell factor. While discussing the single-photon routing case in the previous section (see below Eq. (11)), we pointed out a condition on emitter-waveguide coupling rates, a similar condition can be introduced here such that $2\Gamma _{1}^{u} = 2\Gamma _{1}^{d} + \gamma$ when $\omega = \omega _{eg}$. Under this condition, we find that the routing amplitude remains unchanged while the upper and lower transmission amplitudes become

For the sake of clarity, we’ll refer to this situation (when $2\Gamma _{1}^{u} = 2\Gamma _{1}^{d} + \gamma$) as case 1. Alternatively, one can also assume the scenario when $2\Gamma _{1}^{d} = 2\Gamma _{1}^{u} + \gamma$ which was the condition considered for the single-photon case. We call this condition case 2.

Interestingly, it turns out that case 2 yields a vanishing routing probability (i.e., $P_{23} = 0$) for the two-photon problem for all values of Purcell factor, as shown in Fig. 3 (a). However, for case 1, the routing probability $P_{23}$ starts from a null value at $P_F=0$, but then as $P_F$ is increased, $P_{23}$ begins to increase and reaches a maximum of $\sim 14.8{\%}$ at $P_{F} = 2$. Past this point, $P_{23}$ decays and approaches zero around $P_{F}=100$. Going beyond the resonant problem reported in Fig. 3 (a), in Fig. 3 (b) and Fig. 3 (c) we plot the behavior of $P_{23}$ for a range of frequencies, $\omega _1$ for one photon and $\omega _2$ for the other. From Fig. 3 (b), we notice that unlike the single-photon case, the two-photon case becomes optimal in a symmetric system ($\Gamma ^{d} = \Gamma ^{u}$) where under the ideal conditions $P_{F} \xrightarrow []{}\infty$ when $\gamma \xrightarrow []{} 0$. A more realistic system, under the critical coupling regime $\Gamma ^{d} = \Gamma ^{u} = \gamma$, as seen in Fig. 3 (c) still displays this behavior where the maximum value of $P_{23}$ when the frequency of one photon is fixed to $\omega _{eg}$ and the frequency of the other photon is taken to be twice of the resonance, i.e., 2$\omega _{eg}$. In particular, Fig. 3 (b) reveals that in the ideal/no-loss case within the symmetric scenario, the choice of $\omega _{1/2} = \omega _{eg}$ and $\omega _{2/1} = 2\omega _{eg}$ achieves a routing probability of $\sim 92.6{\%}$. Whereas, in the realistic case ($\gamma \neq 0$) under the critical coupling regime (Fig. 3 (c)), the same frequency choice degrades the maximum routing probability considerably and produces a routing probability of $\sim 58.3 {\%}$.

 

Fig. 3. Two-photon routing with one QE: (a) $P_{23}$ as a function of Purcell factor $P_F$. Blue and red curves represent case 2 ($2\Gamma _{1}^{d} = 2\Gamma _{1}^{u} + \gamma$) and case 1 ($2\Gamma _{1}^{u} = 2\Gamma _{1}^{d} + \gamma$) of emitter-waveguide couplings relation with the spontaneous emission rate. For this plot for both photons, a resonant condition ($\omega =\omega _{eg}$) has been assumed. (b) The density plot of $P_{23}$ as a function of the frequencies $\omega _1$ and $\omega _2$ of the two photons. Here we have assumed a symmetric system with $\Gamma ^{d} = \Gamma ^{u} = 0.1\omega _{eg}$, and $\gamma = 0$. (c) $P_{23}$ density plot again for a non-ideal case in which $\gamma = 0.1\omega _{eg}$. Emitter-waveguide coupling rates $\Gamma ^{d}$ and $\Gamma ^{u}$ take the same values as in part (b).

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5. Two photon routing with many QEs: scattering case

5.1 Routing in the presence of two QEs

For the case of many QEs, the mathematics of tracking the number of paths increases in complexity. Notwithstanding, in the case of two QEs, the probability of $\phi$, $\psi$, and $\varphi$ can be expressed with an easily reported number of terms normalized against $L^{2}/4$, as shown below

In the above set of equations, whenever a photon path was identical under the permutation of photon swapping, we counted such a path twice due to the Bosonic symmetry. For the many-emitter problem, we assume that all QEs are identical. From Eq. (20a) to Eq. (20c), we can then infer certain properties that many QE scenarios will have in common with the two-emitter problem. For instance, we expect that only $\left | {\phi } \right |^{2}$ and $\left | {\psi } \right |^{2}$ will have terms that have a factor of two in front of them and that $\left | {\varphi } \right |^{2}$ will always have twice as many terms as $\left | {\phi } \right |^{2}$ and $\left | {\psi } \right |^{2}$. Additionally for a general problem of $N$ QEs, we expect the number of paths for $\left | {\phi } \right |^{2}$ and $\left | {\psi } \right |^{2}$ to grow like $4^{N-1}$ while for $\left | {\varphi } \right |^{2}$ to follow $2(4^{N-1})$ pattern. Proceeding further, similar to the single-QE problem, we introduce cases 1 and 2 relating the emitter-waveguide coupling rates to the spontaneous emission rate for the two-emitter problem. The routing probability $P_{23}$ in this case can then be expressed in terms of the Purcell factor as

In Fig. 4, we plot the two-photon routing probability for the two-QE case. From Fig. 4(a), we notice that when $P_{23}$ is plotted against the Purcell factor, a drastic decrease in the routing probability is obtained as compared to the single QE problem. For example, we observe that in case 1, $P_{23}$ achieves a maximum value of $\sim 5.6{\%}$ at $P_{F} = 4/3$, while for case 2, even a smaller maximum value of $\sim 3.5{\%}$ is attained at $P_{F} = 4$. Motivated by these observations, we concluded that due to the stringent manner in which coupling and spontaneous emission rates are connected in the Purcell factor definition, for problems with two or more emitters, the Purcell factor can no longer serve as an optimal parameter to plot $P_{23}$ against of. Therefore, starting the next section, we vary the emitter-waveguide coupling and spontaneous emission rates without relating these to $P_F$.

 

Fig. 4. Two-photon routing in a two QE wQED ladder. (a) $P_{23}$ in terms of the Purcell factor, with case 1 represented by a blue-colored curve and case 2 with a red-colored curve. (b,c) $P_{23}$ as a density plot as a function of the two-photon frequencies. (b) Ideal case ($\gamma = 0$) with $\Gamma ^{d} = \Gamma ^{u} = 0.1\omega _{eg}$. (c) Non-ideal situation ($\gamma = 0.1\omega _{eg}$) with $\Gamma ^{d} = \Gamma ^{u} = \gamma$. Note that both emitters’ parameters are chosen to be identical in all plots.

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Moving forward, we plot the two-photon redirection probability as a density plot. As shown in Fig. 4(b), in the no-loss or ideal case, there still exist regions where only a single photon being on resonance achieves a maximal probability. However, in comparison to the single QE problem, the presence of a second QE creates a larger region where both photons being off-resonance can achieve a maximum probability of $\sim 41.4{\%}$ for $\omega _{1/2} \cong 1.17\omega _{eg}$ and $\omega _{2/1} \cong 1.72\omega _{eg}$. However, unlike the single QE case, in the critical coupling regime, we observe a behavior change in $P_{23}$ as shown in Fig. 4(c). For instance, we point out that not only the regions of maximum routing probability are relatively shrunk but also the maximum routing probability value also reduces by a factor of $\sim 3/8$ (see $P_{23}$ reaching the value of $\sim 24.9{\%}$ when $\omega _{1/2} \cong 0.815\omega _{eg}$ and $\omega _{2/1} \cong 2\omega _{eg}$). These numerical findings indicate the general trend of deterioration of the two-photon routing probability in the presence of two QEs where the routing is comparatively worsened more when QEs are coupled to environmental degrees of freedom (non-ideal case).

5.2 Routing in the presence of many QEs

We now turn our attention to the case of two-photon routing in the plane wave (scattering) setting in the presence of many QEs. In this situation, the number of terms appearing in different position-dependent amplitudes grows significantly. However, under the assumption of chiral light-matter interaction, all possible terms can be tracked using the following compact matrix notation: $\bf {S}^{N}\bf {\Psi _{i}} = \bf {Q}_{N}$, where $\bf {S}$ is the scattering matrix, $\bf {\Psi _{i}}$ is the input state, and $\bf {Q}_{N}$ is the output describing all possible scattered states post $N$-th QE as defined below

Here $Q^{dd}_{N}$, $Q^{du}_{N}$, $Q^{ud}_{N}$, $Q^{uu}_{N}$ represent the possible paths the photons can take after scattering at the $N$-th QE. At this stage, we notice that the matrix mentioned above construct can keep track of all possible paths, but the weighting factors for each path have been predetermined to be one. At the single-photon level, this may not cause any issues (for instance, note that for the case of one photon in each waveguide after scattering at the $N$-th QE $\left | {\varphi } \right |^{2} = Q^{du}_{N} + Q^{ud}_{N}$ which carries a prefactor of 1). However, an algorithm must be created for a two-photon problem (in particular for the paths in which both photons propagate in the same waveguide) to account for the correct prefactors. To this end, we start by defining a new input state $\bf {\Phi }_{j}$, $\forall j=1,2,3,\ldots, N$ which will calculate for $\left | {\phi } \right |^{2}$ and $\left | {\psi } \right |^{2}$. The new output state will be $\bf {O}_{j}$. The algorithm is defined through the transformation $\bf {S}\bf {\Phi }_{j} = \bf {O}_{j}$ with

With this algorithm at hand, the density plots of $P_{23}$ can be plotted for an arbitrary number of QEs. In Fig. 5, we plot the resulting density plots for three, four, and five QE chains both for the ideal ($\gamma =0$) and realistic ($\gamma \neq 0$) scenarios. We notice the emergence of an interesting pattern starting at $N=3$. As seen in Fig. 5(a), the optimal probability of $\sim 89.3{\%}$ is achieved when similar to the single-QE problem a single photon is on resonance ( $\omega _{1/2}=\omega _{eg}$) and the other is off resonance ($\omega _{2/1}=2\omega _{eg}$). As seen in Fig. 5(a), the overall profile of the density plot also matches the behavior for the single QE problem (see Fig. 3(b)) where the maximum value of routing probability was obtained at the same frequency conditions.

 

Fig. 5. A panel of $P_{23}$ density plots drawn against frequencies of the photons for (a/d) three, (b/e) four, and (c/f) five QEs wQED ladders. For (a) through (c), the system is ideal such that $\Gamma ^{d} = \Gamma ^{u} = 0.1\omega _{eg}$, and $\gamma = 0$. For (d) through (f), the spontaneous emission losses have been incorporated under a critical coupling regime, i.e., $\Gamma ^{d} = \Gamma ^{u} = \gamma = 0.1\omega _{eg}$. In all plots, all QEs and their couplings with the waveguides have been assumed to be identical (symmetric situation).

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This motivates us to plot the odd number of QE cases when $N=5$. We plot this case in Fig. 5(c). Interestingly, the regions with the highest routing probabilities are shrunk compared to the $N=3$ problem. Still, five QEs case produces an optimal probability of $\sim 83.5{\%}$ under the same frequency conditions found for a single and triple QE situation. A similar trend is observed for an even number of QEs. For instance, extending to four QEs (as seen in Fig. 5(b)), we observe an optimal probability of $\sim 48.9{\%}$ can be gained at $\omega _{1/2} = 0.852\omega _{eg}$ and $\omega _{2/1} = 2\omega _{eg}$ which is similar to what we have found for two QEs (see and compare with Fig. 4(b)).

This leads us to conjecture that, for the ideal chiral wQED ladders considered in this work, the odd number of QEs follow a pattern where maximum routing probability is achieved when one photon is on- while the other photon is off-resonant. On the other hand, the even number of QEs case follows a pattern in which both photons being off-resonance generates optimal routing probability. This trend can be explained as follows. The initial condition of two photons entering the same waveguide port introduces an asymmetry in the $P_{23}$ probability. Notably, adding more QEs to the odd case sharpens the two-photon frequency regions where elevated values of routing probabilities are achievable. On the other hand, the additional QEs in the even case spread out such frequency regions. From this behavior, it can also be concluded that out of these two cases (even versus the odd number of QEs), a higher maximum routing probability can be obtained for an odd number of QEs given the initial condition of both photons being incident from Port 1.

In Fig. 5(d) to Fig. 5(f), we plot $P_{23}$ for $N=3,4$ and $5$ QEs subjected to a non-ideal situation in the critical coupling regime. We observe with more QEs in the chain; the routing probability structure begins to spread out in addition to lowering in magnitude due to spontaneous emission loss. For example, our numerical results indicate that for the case of three QEs, a maximal probability of $\sim 22.9{\%}$ is achieved when $\omega _{1/2} = \omega _{eg}$ and $\omega _{2/1} = 2\omega _{eg}$. A more spread pattern emerges for the five QEs problem, as shown in Fig. 5(f). This pattern indicates that a $\gamma \neq 0$ doesn’t seem to promote the similarity in the $P_{23}$ plots for an odd number of QEs (unlike pointed out in the ideal case above).

Finally, in Fig. 6, we plot $P_{23}$ for $N=8,9$ and $10$ QEs subjected to ideal symmetric situation. This plot shows that our numerical algorithm can be applied to a higher number of QEs. When we compare Fig. 6 with Fig. 5, we observe that the alternating behavior of $P_{23}$ with even and odd number of emitters observed in Fig. 5 extends down to eight, nine, and ten emitters as well. However, the maximum value of $P_{23}$ shows a general decreasing trend compared to the ideal three QEs problem.

 

Fig. 6. A panel of $P_{23}$ density plots drawn against frequencies of the photons for (a) eight, (b) nine, and (c) ten QEs wQED ladders. For all plots, the system is ideal such that $\Gamma ^{d} = \Gamma ^{u} = 0.1\omega _{eg}$, and $\gamma = 0$ in an identical (symmetric) situation.

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In the qualitative explanation of Fig. 5 and Fig. 6 there are two noticeable points. One is that our initial launching configuration has an asymmetry (both photons launched from the left side of the bottom waveguide). Additionally, in our model (see the two-photon state), only one emitter can be excited. Now, when a photon is on-resonance, it has a higher probability of absorption, while the off-resonant photon is likely to propagate in the waveguide freely. Putting all of these considerations, our numerical results seem to indicate that for an odd number of emitters, the interference between probability amplitudes works out in such a way that the freely propagating photon in the bottom waveguide reaches the Port-2 without being impacted much by the destructive interference of the amplitudes. At the same time, the photon absorbed by the emitters tends to be more likely to show up on Port-(3) due to constructive interference. Finally, we note that the presence of spontaneous emission (see lower panel of plots in Fig. 5) destroys the constructive interference such that for both even and odd numbers of emitters, we obtain more or less the same routing probability.

6. Two photon routing with two QEs: inclusion of bound states

The assumption that only the plane waves contribute to the overall probability breaks down when the length of the waveguide begins to approach the inter-emitter separation. To calculate the accurate probability of this physical setup, the possibility of photon-photon bound state formation must be considered [20,42,43]. To report the solutions to the Bosonic wave functions in a concise manner, we introduce a new notation in which $g_{j}^{p/b}$ represents a wave function at the $j$-th QE with $p$ and $b$ being either the plane wave or bound state solutions. The overall solution is merely the sum such that $g_{j} = g_{j}^{p} + g_{j}^{b}$. Additionally, to report the bound state solutions at the $N$-th QE in a compact format, new functions are introduced in the following form

Next, we report the bound state wavefunctions under the assumption of short-length waveguides ($L$ being the waveguide length) as compared to the characteristic wavelength $\lambda _0=2\pi /k_0$ of the photon emitter by the QEs, i.e., $e^{ik_0L}\approx 1$. The solution to the bound states can then be concisely stated in the following fashion

The coefficients in front of the bound state functions turn out to be somewhat mathematically involved after the second QE and are given by

The routing probability can be worked out now that the entire wave functions, including the bound states, have been expressed explicitly. For the case of one QE, Eq. (13) is still sufficient to calculate $P_{23}$. In the case of two QEs, the bounds of integration need to be re-expressed such that,

We plot the routing probability in the presence of bound states in Fig. 7 for shorter $\frac {10^{-4}}{2} \omega _{eg}v^{-1}_g$ (plot (a) and (b)) and longer $\frac {10^{-3}}{2} \omega _{eg}v^{-1}_g$ (plot (c) and (d)) inter-emitter separations. It is apparent from Fig. 7(c) and Fig. 7(d) that, for longer inter-emitter separations, the inclusion of bound states for both one and two QEs cases under the critical coupling regime increases the routing probability when compared to the corresponding pure plane wave scattering case (see and compare with Fig. 3(c) and Fig. 4(c)). For instance, our numerical results indicated that, for longer inter-emitter scenario, in the case of one QE, an optimal probability of $\sim 67.9{\%}$ is achieved when $\omega _{1/2} = 0.65\omega _{eg}$ and $\omega _{2/1} = 1.35\omega _{eg}$. While in the case of two QEs, an optimal probability of $\sim 61.6{\%}$ is achievable when $\omega _{1/2} << \omega _{eg}$ with $\omega _{2/1} = 2\omega _{eg}$. Additionally, we note that including the bound states in the probability shifts the routing probability from being symmetric around $\omega _{1/2} = \omega _{eg}$ to being symmetric around $\omega _{1} = \omega _{2}$. For smaller inter-emitter separation (plots (a) and (b)), our numerical results indicate that not only the pattern of the density plot changes (as compared to larger separations as manifested in plots (c) and (d)) but also the highest value of routing probability decreases by approximately $2{\%}$ and $5{\%}$ for one and two QEs cases, respectively. We close this section by making this remark that the intricate form of the bound state coefficients for two QEs is enough to indicate the complexity of the bound state problem for more than two QE cases.

 

Fig. 7. The routing probability $P_{23}$ including the bound state solutions is expressed as a function of the frequencies of the photons past (a) one and (b) two QEs with a length parameter of $L = 10^{-4}(\omega _{eg}/v_{g})$; (c) one and (d) two QEs with a longer length parameter of $L = 10^{-3}(\omega _{eg}/v_{g})$. The system is symmetric and in the critical coupling regime such that $\Gamma ^{d} = \Gamma ^{u} = \gamma = 0.1\omega _{eg}$.

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7. Summary, conclusions and outlook

We have studied the problem of two-photon redirection through two waveguides coupled with a few two-level QEs in the purely plane wave case and when the bound states were included. In the plane wave case, the behavior of up to five QEs was analyzed, and an algorithm was reported to generalize to an arbitrary number of QEs. We found that under the initial condition of two photons being incident from Port 1 of the wQED ladder and for an ideal (no loss) symmetric situation, the routing probability manifested an alternate behavior for even and odd numbers of QEs. We concluded that an odd number of QEs offered maximum routing for the set of optimized parameters when one of the photons was on resonance and the other was off-resonant. However, we noted that the optimal routing probability dropped off in the critical coupling when a finite spontaneous emission was incorporated. Finally, we demonstrated the general feature that including the bound states in the critical coupling regime increased the routing probability for one and two QEs compared to the corresponding purely plane wave problem.

The four-port wQED device considered in this work can be thought of as one of the essential building blocks of more complex quantum networks [1] where multiple photon redirection, strong light-matter interactions, and preferential photonic emissions (chirality) can be accomplished in a single system. Considering this and the main results obtained, we envision quantum circuitry and quantum networking protocols as two areas of application of this work. Despite being a theoretical work, many experimental platforms are now available where our two-photon routing study can be implemented. On the waveguide QED end, examples include quantum dots coupled with nanowires [45], Josephson junctions in microwave transmission lines [46], naturally occurring atoms (for example, Ce atoms) coupled with photonic crystal waveguides [47], and Si-vacancy centers coupled with diamond waveguides [48]. On the chiral waveguide QED end, silica nanofiber interacting with gold nanoparticle [49] and quantum dot interacting with cross waveguide architectures [50] provide appealing experimental setups.

Finally, we would like to mention that the rich architecture of the wQED ladder platform offers several areas of exploration. For instance, one can investigate how collective atomic effects induced by dipole-dipole interaction among QEs can aid the two-photon routing. Also, the problem of extending the two-photon bound state routing scheme for more than two QEs is an open area requiring both a formal mathematical understanding and the development of efficient numerical routines. We leave these interesting questions as possible future directions of this work.