The scattering of a flying photon by a two-level system ultrastrongly coupled to a one-dimensional photonic waveguide is studied numerically. The photonic medium is modeled as an array of coupled cavities and the whole system is analyzed beyond the rotating wave approximation using matrix product states. It is found that the scattering is strongly influenced by the single-and multiphoton dressed bound states present in the system. In the ultrastrong coupling regime a new channel for inelastic scattering appears, where an incident photon deposits energy into the qubit, exciting a photon-bound state, and escaping with a lower frequency. This single-photon nonlinear frequency conversion process can reach up to 50% efficiency. Other remarkable features in the scattering induced by counterrotating terms are a blueshift of the reflection resonance and a Fano resonance due to long-lived excited states. Introduction.-As light-matter interaction controls an immense variety of physical processes, its modification usually leads to new phenomena. One strategy to increase this interaction is to confine the electromagnetic field in waveguides and make it interact with few level systems. It is possible nowadays to reach in this way the situation where the coherent light-matter coupling predominates over decoherence processes (the so-called strong-coupling regime), and to generate, manipulate, and store a single (or a few) photon. The ability of performing tasks with just one photon has already been demonstrated [1,2], opening the path for proposals such as optical transistors [3][4][5], singlephoton routers [6], one-photon lasers [7], qubit-mediated entanglement [8], or efficient photodetectors [9].All these results have been analyzed within the rotatingwave approximation (RWA) for the photon-dipole interaction [10]. The RWA only considers the processes where light and matter exchange excitations, which is valid when the couplings are much smaller than the typical photon and qubit energies. For sufficiently strong couplings, processes involving spontaneous creation and annihilation of pairs of excitations are relevant and the RWA picture breaks down [11]. This regime of ultrastrong coupling opens the door to new physics [12,13], which is within reach for many different experimental implementations [14].From the theoretical viewpoint, within the RWA the scattering of multiphoton wave packets by qubits is a complex problem [15][16][17][18][19][20], but the one-photon scattering is trivial. Beyond the RWA, computing the scattering of even one flying photon is difficult as subspaces with different photon numbers mix in the dynamics. This converts the problem into a many-body one for which only partial solutions exist for models that consider linear (unbounded)

Complex networks are formal frameworks capturing the interdependencies between the elements of large systems and databases. This formalism allows to use network navigation methods to rank the importance that each constituent has on the global organization of the system. A key example is Pagerank navigation which is at the core of the most used search engine of the World Wide Web. Inspired in this classical algorithm, we define a quantum navigation method providing a unique ranking of the elements of a network. We analyze the convergence of quantum navigation to the stationary rank of networks and show that quantumness decreases the number of navigation steps before convergence. In addition, we show that quantum navigation allows to solve degeneracies found in classical ranks. By implementing the quantum algorithm in real networks, we confirm these improvements and show that quantum coherence unveils new hierarchical features about the global organization of complex systems.

We study the spontaneous decay of an impurity coupled to a linear array of bosonic cavities forming a single-band photonic waveguide. The average frequency of the emitted photon is different from the frequency for single-photon resonant scattering, which perfectly matches the bare frequency of the excited state of the impurity. We study how the energy of the excited state of the impurity influences the spatial profile of the emitted photon. The farther the energy is from the middle of the photonic band, the farther the wave packet is from the causal limit. In particular, if the energy lies in the middle of the band, the wave packet is localized around the causal limit. Besides, the occupation of the excited state of the impurity presents a rich dynamics: it shows an exponential decay up to intermediate times, this is followed by a power-law tail in the long-time regime, and it finally reaches an oscillatory stationary regime. Finally, we show that this phenomenology is robust under the presence of losses, both in the impurity and the cavities