Wave communication across regular lattices Birgit Hein and Gregor Tanner School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. (Dated: June 16, 2018) We propose a novel way to communicate signals in the form of waves across a d - dimensional lattice. The mechanism is based on quantum search algorithms and makes it possible to both search for marked positions in a regular grid and to communicate between two (or more) points on the lattice. Remarkably, neither the sender nor the receiver needs to know the position of each other despite the fact that the signal is only exchanged between the contributing parties. This is an example of using wave interference as a resource by controlling localisation phenomena effectively. Possible experimental realisations will be discussed. PACS numbers: 03.67.Hk, 03.65Sq, 03.67.Ac, 42.50.Ex Localization phenomena in linear wave systems are closely linked to wave interference effects. Anderson lo- calisation in disordered media is a prime example thereof still posing challenges to both theory and experiment [1] 60 years after its discovery [2]. Recently, a new re- search area has emerged focusing on interference as a resource and making use of localisation phenomena in a controlled way. Prominent examples are among others time reversal imaging [3] and reconstructing the Green function in terms of correlation functions [4], see also [5] . Here, information about the wave system is obtained by manipulating a seemingly ‘noisy’ signal using phase coherence. We will focus here on another class of wave localisation phenomena with counterintuitive properties, namely (quantum) search algorithms and (quantum) ran- dom walks. Wave search algorithms gained prominence with Grover’s work [6] demonstrating a √$_{N}$ speed-up compared to a classical search within an unsorted data base of N items. Even though search algorithms became an inherent part of quantum information theory [7], the speed-up is in effect caused by wave interference as has already been pointed out by Grover [8] and has been im- plemented for a classical wave system in [9]. Based on ideas from quantum random walks [10–12], Grover’s al- gorithm has been generalized to spatial search algorithms on networks such as on a hypercube [13, 14] and on reg- ular lattices [15]. Experimental realisations of quantum random walks have been achieved again both using clas- sical waves (optics) [16] and quantum devices [17]. Starting from wave search algorithms, we will demonstrate that localisation can be used to establish communication channels across a regular lattice with surprising properties: (i) signals can be exchanged exclusively between a source and a receiver point, where neither the sender nor the receiver know the position of each other; (ii) the signal can track a moving receiver in the network; (iii) the algorithm can be used as a searching device without the necessity to know the time of measurement, (a typical requirement for Grover’s search algorithm); (iv) the protocol can act as a sensitive switching device for wave transport through a lattice; (v) the algorithm can be effectively implemented both on a quantum computer and using classical waves only. We will first describe the set-up of the search algorithm. We then introduce a simplified model for the search and explain the wave communication protocol. We consider wave propagation across d -dimensional periodic lattices of identical scatterers or periodic po- tentials with fixed lattice parameter a , see Fig. 1. It is important that the lattice has a finite number of sites n along each axis with a total number n d of lattice sites. To simplify the calculations we will restrict ourselves to models with nearest-neighbour interaction only and con- sider periodic boundary conditions. The wave dynamics within each unit cell is given by a local scattering matrix σ mapping incoming channels onto outgoing channels, see Fig. 1. ( σ is also denoted a coin matrix in the con- text of quantum walks). The overall wave dynamics is then given in terms of an operator U$_{0}$ mapping incom- ing onto outgoing wave coefficients between unit cells. We have dim U$_{0}$ = m n d where m denotes the number of open scattering channels within a unit cell. Furthermore, U$_{0}$ is unitary when disregarding dissipation. Stationary solutions are obtained by the condition det ( 1 − e ı $^{ka}$U$_{0}$ ) ≡ 0 (1) where k is the wave length and exp(ı ka ) is a phase shift between incoming and outgoing waves. We ne- glect any (in general weak) k dependence of σ and thus U$_{0}$ . Note that the spectrum obtained from (1) is now periodic in k with period 2 π/a ; the eigenvalues are k$_{j,l}$ = (2 πl − θ$_{j}$ ) /a, l ∈ N , where θ$_{j}$ are the eigenphases of U$_{0}$ with j = 1 , . . . , N . To start with, we will consider a arXiv:1001.0335v1 [quant-ph] 2 Jan 2010 x 2 − 1 + 1 − 2 + a FIG. 1: Regular grid with d = 2 , n = 8; local scattering within a unit cell at vertex ⃗ x is described by the matrix σ .