# Two-Excitation Routing via Linear Quantum Channels
## 3 The Model  
In this section, we apply the formalism of Section 2 to determine the two-excitation transition probability from a sender block to a receiver block, both composed of two sites, that are connected to a linear chain. The aim is to derive the conditions for the routing of the two excitations from the senders' to the receivers' location. We will analyze two networks: (a) the receiver blocks have switchable couplings to the wire (Figure 1); (b) the receiver blocks are permanently coupled to the wire and the hopping term \(J_{s}\) in the sender block is tunable (Figure 3).  
We consider Hamiltonians of the type given in Equation (1), which, decomposed into the different components of the network, i.e., sender \(S\), wire \(W\), and receivers \(R\), read  
\[\hat{H}=\hat{H}_{S}+\sum_{i}\hat{H}_{R_{i}}+\hat{H}_{W}+\hat{H}_{SW}+\sum_{i} \hat{H}_{R_{i}W}\;. \tag{6}\]  
The Hamiltonian of the sender block and the \(i\)-th receiver block are, respectively  
\[\hat{H}_{S}=J_{s}\left(\hat{c}_{1}^{\dagger}\hat{c}_{2}+h.c.\right)\,,\;\hat {H}_{R_{i}}=J_{i}\left(\hat{c}_{r_{i}}^{\dagger}\hat{c}_{r_{i}+1}+h.c.\right)\,, \tag{7}\]  
with \(r_{i}\) denoting the position on the graph which will be given in the following. The Hamiltonian for the quantum data bus reads  
\[\hat{H}_{w}=J\sum_{n=1}^{n_{w}-1}\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+h.c. \right)\,. \tag{8}\]  
where \(n_{w}\) denotes the length of the wire. Finally, the coupling between the sender block and the data bus site is assumed to be in the weak-coupling regime, \(J_{0}\ll J,J_{s},J_{i}\)  
\[\hat{H}_{SW}=J_{0}\left(\hat{c}_{2}^{\dagger}\hat{c}_{3}+h.c.\right)\,; \tag{9}\]  
as well as the coupling between the \(i\)-th receiver block at location \(r_{i}\) and the corresponding data bus site \(w_{i}\), where \(1\leq w_{i}\leq n_{w}\)  
\[\hat{H}_{R_{i}w}=J_{0}\left(\hat{c}_{r_{i}}^{\dagger}\hat{c}_{n_{\gamma_{i}}}+ h.c.\right)\,. \tag{10}\]  
For case (a) all couplings between the receiver blocks and the wire are switched off but one, embodying the recipient of the routing protocol and we set \(r_{i}=n_{w}+2\); see Figure 1 for an instance of the numbering choice of the sites following the sender-wire-receiver order; the same ordering is followed for case (b).  
The whole system sender+wire+receivers is made up of \(N=n_{w}+2(n_{r}+1)\) sites with \(r\) denoting the number of receiver blocks.  
From Equation (5), the two-body transition probability, with \(\mathbf{s}=\{s_{1},s_{2}\}\) and \(\mathbf{r_{i}}=\{r_{i,1},r_{i,2}\}\), is given by  
\[|f_{\mathbf{s}}^{\mathbf{n}}(t)|^{2}=\left|\langle 1,2|e^{-it\hat{H}}|N-1,N \rangle\right|^{2}=\begin{bmatrix}f_{1}^{N-1}(t)&f_{1}^{N}(t)\\ f_{2}^{N-1}(t)&f_{2}^{N}(t)\end{bmatrix}^{2}. \tag{11}\]  
For only one sender and one receiver block located at opposite edges of the quantum wire, the model is one-dimensional and, using the Jordan-Wigner mapping from spin-less fermions to spin-\(\frac{1}{2}\) particles, the Hamiltonian in Equation (6) with open boundary conditions is equivalent to the \(XX\) spin-\(\frac{1}{2}\) model with nearest-neighbor coupling.  
\[\hat{H}=\sum_{n=1}^{N}\frac{J_{n}}{2}\left(\partial_{n}^{x}\partial_{n+1}^{x} +\partial_{n}^{y}\partial_{n+1}^{y}\right)\,. \tag{12}\]  
In such a case, it has been shown that two-qubit quantum state transfer [13; 14; 15] as well as entanglement generation of two Bell states [27] is achieved with high fidelity. Modifications of the one-dimensional geometry have been investigated too. In Refs. [17; 18] each spin of the sender (receiver) block is coupled to the edges of the 1D quantum wire allowing for the transfer of a Bell state when operating in the single-excitation subspace. A similar geometry is adopted in Refs. [4; 6] with multiple sender (receiver) non-interacting spins coupled to the wire at the edges.