# Two-Excitation Routing via Linear Quantum Channels
## 3 The Model
### Routing with Switchable Weak Couplings  
Here we consider only one receiver block coupled to the wire for each execution of the routing protocol and, as we shall see, this allows us to assume uniform coupling within each component of the setup, i.e, the sender, the wire and the receiver blocks. We choose \(J_{s}=J=J_{r}=1\) as our energy and time unit. On the other hand, the couplings between the sender (receiver) block and the wire will be in the weak-coupling regime, which we set throughout the paper to \(J_{0}=0.01\). The 1D-case where only one block of senders and one of receivers is each coupled at the edge of the quantum wire has been addressed in Ref. [25]. There it has been shown that, although each length of the quantum wire \(n_{w}\) allows for high-fidelity excitation transfer, for \(n_{w}=3l+2\) (\(l=0,1,2,\dots\)), resonances between the sender (receiver) and the wire single-particle energy levels give rise to a faster transfer with respect to the instances \(n_{w}=3l,3l+1\) where off-resonant transfer takes place. In the former case, the single-particle transfer occurs on a time scale of the order of \(J_{0}^{-1}\), yielding to a two-excitation transfer time scale of the order of \(10\int_{0}^{-1}\) with the reason for the multiplicative factor being  
Figure 1: Quantum routing of excitations by means of a linear chain quantum data bus with switchable interactions. The sender and receiver sites are depicted in green and red, respectively, while the quantum data bus sites are in blue. Continuous lines represent permanent couplings \(J=1\), while dotted lines encode switchable weak couplings \(J_{0}\ll 1\); also shown is the numbering choice of the sites position adopted in Section 3.1.  
that the transfer dynamics involves a difference between eigenenergies that are perturbed to first-order in \(J_{0}\). On the other hand, for the off-resonant dynamics, the two-particle transfer time is of order \(J_{0}^{-2}\). Considering that the excitation transfer mechanism holds in the perturbative regime \(J_{0}\ll 1\), this may translate in severals of magnitude. Here we address the case where the receiver block is coupled to the quantum wire at a different position \(w_{i}\) than the edge of the chain; see Figure 1. We also omit the suffix \(i\) since only one receiver block is present in this protocol. We aim at finding the conditions on the position \(w\) for which resonant transfer of the excitations from the sender to the receiver block at takes place. Following the argument for faster (resonant) transfer in Ref. [25], we set the length of the quantum wire \(n_{w}=3l+2\) so that perturbative transfer is achieved for \(w=n_{w}\), i.e., a receiver block can be coupled to the edge of the wire. For this wire length, we find that it is possible to couple a receiver block at each site \(w\
eq 3p\) of \(n_{w}\), with \(p\) integer. The fact that these latter sites of the wire cannot act as connection points for the receiver block can be explained by looking at the eigenstates of the wire's Hamiltonian \(\hat{H}_{W}\) (Equation (8)) that are resonant with the eigenstates of the sender (receiver) block. For \(J=1\), the unperturbed energy level of the sender (receiver) is \(E_{res}=\pm 1\), and, because of the mirror-symmetry of the Hamiltonians in Equation (7), they have the identical (absolute value) overlap on each site [28] so that it suffices to consider only one of them. The unperturbed energy levels of the wire that are resonant with the sender (receiver) are given by \(E_{k}=2\cos\frac{kR\pi}{n_{w}+1}=1\). Therefore, we obtain that, ordering \(E_{k}^{\prime}\)s in decreasing order, the \(k=\frac{n_{w}+1}{3}\)-energy level of the wire is the resonant one, see left panel of Figure 2 for a schematic representation of the resonance condition.