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We analyze the quantum motion of a cold, trapped two-level ion interacting with a quantized light field in a single-mode cavity. We show that in the nonclassical Lamb-Dicke limit the time evolution of the vibrational mode representing the quantized motion of the center of mass of the trapped ion is very sensitive to the quantum statistics of the light field. We also show that the system under consideration may evolve into the maximally entangled three-particle Greenberger-Horne-Zeilinger state. We briefly discuss the dynamics of a cluster of two-level ions trapped in a cavity and interacting with a quantized light field.

Quantum walks are powerful kernels in quantum computing protocols that possess strong capabilities in speeding up various simulation and optimisation tasks. One striking example is given by quantum walkers evolving on glued trees for their faster hitting performances than in the case of classical random walks. However, its experimental implementation is challenging as it involves highly complex arrangements of exponentially increasing number of nodes. Here we propose an alternative structure with a polynomially increasing number of nodes. We successfully map such graphs on quantum photonic chips using femtosecond laser direct writing techniques in a geometrically scalable fashion. We experimentally demonstrate quantum fast hitting by implementing two-dimensional quantum walks on these graphs with up to 160 nodes and a depth of 8 layers, achieving a linear relationship between the optimal hitting time and the network depth. Our results open up a scalable way towards quantum speed-up in complex problems classically intractable.Adapting well-known classical mathematical models in a way to include quantum mechanical laws has shown the emergence of new interesting behaviors. In some cases, the modified protocols have revealed an advantage with respect to the original ones in solving specific problems. This has clearly triggered the interest of the scientific community in the quest for a better understanding and exploitation of these new tools 1 . A striking example is given by quantum walks, the adaptation of the classical random walk to the world of quantum mechanics 2 . Quantum walks have already found applications in several scenarios, including spatial search problems 3,4 , the element distinctness problem 5 , testing matrix identities 6 , evaluating Boolean formulas 7 , judging graph isomorphism 8,9 , which all theoretically promise quantum speed-up and may inspire the breakthrough in real-life applications.One feature of quantum walks on complex graphs that is key in quantum algorithms is their ability to propagate from a node to a distant one in an efficient way. This is often denoted as fast hitting. In particular, fast hitting on a structure known as glued tree is extremely charming due to its exponential speed-up over its classical counterpart 10,11 . A glued tree is obtained by connecting the "final leaves" of two binary tree graphs 12 of the same depth, as shown in Fig.1(a). The process assumes a particle starting in the left-most vertex (called the Entry site), evolving through the graph, and finally hitting the right-most vertex (called the Exit site). It has been shown that, in a scenario where the central connections are randomly chosen, any algorithm exploiting a classical walker (i.e., a particle following the laws of classical mechanics) would require on average a time scaling exponentially with the graph depth to reach the Exit. On the other hand, a quantum walker will require a time that scales only linearly 11,13,14 . Due to the close relation between binary trees and decision trees in computer s...