# Two-Excitation Routing via Linear Quantum Channels
## 3 The Model
### Routing with Permanent Weak Couplings  
A much more desirable routing configuration would be one without the need of switching on and off the couplings as described in the previous section. For the routing of a single particle, this has been achieved in Ref. [3] where in both the linear and the circular geometry, the sender chooses the receiver site tuning the only single-energy level on resonance with the receiver (and the quantum wire) energy level by means of a local magnetic field, i.e., the value of the local magnetic field acting on the receiver qubit is the routing address. In the case of a sender block embodied by two particles, the same strategy does not work as a local magnetic field produces an uniform shift of both of the two energy levels and the simple sinusoidal excitation dynamics is lost. However, it is still possible to perform resonant routing with a sender block of two sites by using the intraspin coupling, which results in a symmetric energy shrinking or dilatation of the two single-energy levels. In such a case, the routing address of each receiver block is given by their intraspin coupling \(J_{r}\); see Figure 3 for the geometry of the network and the left panel of Figure 4 for the single-excitation energy levels. In Section 3.1, we have shown that, for \(J_{s}=1\), the wire's \(k=\frac{n_{w}+1}{3}\)-energy level is resonant with the sender block, and allows for the transfer of the two excitations to a receiver block with intraspin coupling \(J_{r}=J_{s}\) provided that the connection point along the wire is \(w\
eq 3p\). The very same argument can be applied by tuning \(J_{s}\) to a different value so that the single-energy levels of the sender block \(E_{s}=\pm 2J_{s}\) are resonant with two (symmetric) energy levels of the wire. In order to match the resonance condition, an integer solution for \(k\) has to satisfy the following equation  
\[J_{s}=\cos\frac{k\pi}{n_{w}+1}\;. \tag{14}\]  
That is, the \(k=\frac{n_{w}+1}{\pi}\) accross \(J_{s}\)-th energy eigenvalue of the wire is resonant with the sender. For example, for \(J_{s}=\frac{\sqrt{3}}{2}\), \(k=\frac{n_{w}+1}{6}\). Hence, the allowed contact points \(w_{i}\) along the wire have to fulfill the condition that the resonant eigenstate spatial component of the contact point of the sender has to be equal to that of the receiver's contact point, i.e.,  
\[\sqrt{\frac{2}{n_{w}+1}}\sin\frac{k_{res}s\pi}{n_{w}+1}=\sqrt{\frac{2}{n_{w}+ 1}}\sin\frac{k_{res}w_{i}\pi}{n_{w}+1}\;. \tag{15}\]  
To conclude this section, we recap the main results for routing a pair of excitations across a wire with uniform couplings to a desired location, specifying the resonance conditions on the sender and receiver couplings, respectively \(J_{s}\) and \(J_{r_{i}}\) and the respective allowed contact points. Figure 3: Quantum routing of excitations by means of a linear chain quantum data bus with permanent interactions and assuming \(J_{s}\) tuneable. The sender and receiver sites are depicted in green and red, respectively, while the quantum data bus sites are in blue; also shown is the numbering choice of the sites position adopted in Section 3.2. For a wire of length \(n_{w}\), the possible communication parties are \(\frac{n_{w}}{2}\) for even length chain and \(\frac{n_{w}-1}{2}\) for odd length ones. Setting the intrawire coupling \(J=1\), the single energy levels for which first-order excitation transfer occurs are \(E_{k}=2\cos\frac{k\pi}{n_{w}+1}\), \(k=1,2,\ldots,\frac{n_{w}}{2}\) (\(k=1,2,\ldots,\frac{n_{w}-1}{2}\) for odd length chains).