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we added a small amount of absorption across the net- work). Again, without prior knowledge of the receiver’s position, the network localises at the two marked vertices, thus making it possible to actually exchange information continuously between these two points. Changing the po- sition of the receiver leads to a sudden drop of the signal at the old receiver position and a build-up at the new po- sition, see Fig. 4. The system is thus capable of tracking a moving receiver position! The signal speed is limited by the transfer time T$_{s}$ ∼ √$_{N}$ ( T$_{s}$ ∼ √$_{N}$ ln N for d=2) and thus is the speed at which the receiver can move. 

The continuous sender/receiver protocol can also be used to search a marked item without knowing the search time T$_{0}$ ; this is a serious complication of Grover-type search algorithms when the precise number of marked items is not know (as T$_{0}$ depends on m ). Here, we find 

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the marked items as long as we wait for times T ≥ T$_{s}$ . Furthermore, the system can act as a switching device. Wave transport between two points on the grid can only be achieved, if the system is tuned to the avoided crossing. Slight detuning by for example changing the parameter λ in (2) will quickly cut-off the signal. 

The described effects open up completely new ways of transmitting signals across regular networks. While a construction of the map U is certainly feasible in a quantum setting and can be implemented efficiently on a quantum computer [7], an implementation using classical waves may be even more promising. For dispersionless wave dynamics and on regular lattices, the unitary matrix U$_{λ}$ is equivalent to a discretised version of the time dependent Green function (for times t = a/c with c being the wave velocity; corresponding time scales in a quantum setting would be given by the group velocity [20]). Indeed, localised states due to a local perturbation in a regular lattice are well known; (for example in optical crystals, see [19]). We predict that the effect will occur if “defect states” created by local phase perturbations (equivalent to the perturbed coin C ′ ) are pushed into the (discretised continuum) of the band close to the fully periodic state - the ⃗κ = 0 state, see Fig. 2 a. Furthermore, the interaction between the defect states and the lattice states must be small enough to lead to avoided crossings between these two states only. We expect that a signal (such as a laser or a microwave transmitter coupled into a periodic structure at a defect position) can be transmitted and focused onto another defect in the same lattice using the described effect. The totally symmetric state | Φ$_{0}$ 〉 acts then as a carrier state guiding the signal between the perturbations. 

We thank J Keating, U Kuhl, T Monteiro, U Peschel, H-J St¨ ockmann and H Susanto for helpful discussions and S Gnutzmann for carefully reading the manuscript. 

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