2 ment and what role the photon-atom bound states play in this (note that photon-atom bound states have been discussed in a different context before [17]). To address this question, we have to go beyond station- ary calculations that determine the scattering matrices of photons in plane wave states for linearized dispersion relations where the photon-atom bound states are physi- cally inaccessible [12–14]. To do so, we employ our com- putational framework which we have described in detail elsewhere [9]. This framework allows us to analyze both the dynamics of multi-photon wave packets that interact with the TLS and the dynamics of the TLS itself. Fur- thermore, it takes into account all aspects introduced by the finite-bandwidth dispersion relation. First, we would like to note that on energetic grounds a single photon cannot excite the photon-atom bound states described above and, therefore, these states are of no relevance in single-photon scattering calculations from a TLS [8, 9]. In other words, the TLS (partially) absorbs an incoming single photon and a decomposition of the system’s initial state into the (polaritonic) single-particle eigenstates of the Hamiltonian (1) does not involve the bound photon- atom states. Thus, the excited TLS will eventually de- cay into its ground state. However, our reformulated Hamiltonian (2) suggests that, by virtue of the nonlin- ear interaction term, the bound states can, in principle, be energetically reached via multi-photon processes. In Fig. 1, we demonstrate that this is indeed possible: A two-photon wave packet interacts with the TLS and a siz- able fraction of the photon population becomes trapped at the TLS site. In other words, once the TLS is ap- preciably excited by one of the incoming photons, the remaining photon sees a modified (saturated) TLS and is thus (partially) scattered into the hitherto unreach- able bound photon-atom states via multi-particle scat- tering processes. After the scattering is complete, the bound photon-atom states are again decoupled from the continuum (such as is the case for the scattering of a single photon discussed above) and, thus, cannot decay. These bound states are of a polaritonic nature, i.e., they are multi-moded dressed eigenstates of (1) and (2) with complex wavenumbers solely induced by the existence of the waveguide’s finite bandwidth. This implies that a fraction of the radiation remains trapped at the TLS site in form of a partial occupation of the TLS. In order to verify the role of the multi-particle pro- cesses, we display in Fig. 2 the time evolution of the TLS’ excited-state occupation for the scattering of multi- photon wave packets with different particle numbers. The increase in the trapped photon population with the number of photons implies a corresponding increase in the rate at which radiation is scattered into the bound states. The strength of this interaction further depends on the detuning of the TLS relative to the photon fre- quency as well as on the strength of the coupling matrix element V between TLS and the waveguide modes. In With this reformulation several issues become appar- ent. Quantum interference processes associated with the coupling between TLS and the waveguide modes induce an effective interaction between photons as described by the nonlinear term Ub $^{†}$b ( b $^{†}$b − 1 ) . While this effective few-photon optical nonlinearity is spatially localized to the immediate vicinity of the TLS site, this system nev- ertheless represents a true quantum-mechanical many- particle problem. For instance, Hamiltonian (2) looks very similar to a bosonic version of the celebrated single- impurity Anderson model [16] that describes magnetic impurities in metals. Therefore, it is suggestive to apply methods that have been developed for correlated quan- tum systems to the Hamiltonians (1) and (2) [9, 12– 14]. From Hamiltonian (2), it becomes apparent that the TLS will induce correlations between two or more photons. This raises the fascinating question to what extent the TLS can be utilized to engineer this entangle- where we have replaced the TLS by an additional bosonic lattice site. More precisely, we have replaced the Pauli- operators of the TLS by appropriate combinations of bosonic creation and annihilation operators, b † and b . The ground and excited states of the TLS correspond, respectively, to none and a single boson on this addi- tional site (TLS site). Unphysical multiple occupancies of the TLS site have been addressed through the addition of the last term on the r.h.s. of (2). This term ensures that once the TLS site is occupied, i.e., the TLS is in its excited state, adding a further boson to the TLS site re- quires the energy U > 0. Thus, Hamiltonians (1) and (2) are equivalent in the limit U → ∞ and this is the only case we consider in this work. The U -term induces in- elastic scattering that allows us to discuss the physically relevant processes. For actual numerical calculations, we use Hamiltonian (1). While being physically intuitive, the above Hamilto- nian (1) does not allow for the most transparent discus- sion of the underlying physics. Instead, we find it most useful to reformulate the problem in terms of the Hamil- tonian denotes the corresponding hopping element. The quan- tum impurity is modeled as a two-level system (TLS) with transition frequency ω$_{0}$ = Ω / ℏ that is located at lattice site x$_{0}$ and couples with a coupling element V to the modes of the photonic band. When measuring energies from the center of the band, the correspond- ing dispersion relation is ℏ ω$_{k}$ = − 2 J cos( ka ), where a denotes the lattice constant and k stands for a wave number that lies within the first Brillouin zone. Finally, the TLS is described through the Pauli-operators σ$_{z}$ and σ$_{±}$ = σ$_{x}$ ± iσ$_{y}$ . ˆ H = − J N − 1 ∑ x =1 ( a † $_{x}$a$_{x}$$_{+1}$ + a † x $_{+1}$a$_{x}$ ) + Ω b $^{†}$b + V ( a$_{x}$$_{0}$b † + a † x $_{0}$b ) + Ub $^{†}$b ( b $^{†}$b − 1 ) , (2)