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Fig. 3, we depict the corresponding dependence of the trapped photon population at the TLS for a fixed pho- ton wave number k = 3 π/ 4 a . Consistent with our above interpretation, trapping is most pronounced for zero de- tuning δ $^{= Ω}$− ℏ ω$_{k}$ (recall that ℏ ω$_{k}$$_{=3}$$_{π/}$$_{4}$$_{a}$ = √$_{2}$$_{J}$$_{). Fur-}$ thermore, proximity of the TLS resonance frequency to the band edge (or cut-off frequency of the waveguide) is clearly advantageous for realizing efficient trapping: For frequencies near a band edge the multi-particle scatter- ing mechanism has to provide less additional energy for exciting the energetically closest bound state. Less in- tuitive is the fact that there exists an optimal coupling strength V$_{opt}$ ∼ J between TLS and waveguide modes for which maximal trapping occurs. We have confirmed these findings for a number of different dispersion rela- tions. For instance, we have extended Hamiltonian (2) to include a next-nearest-neighbor hopping term J (2) ̸ = 0 that allows us to significantly modify the cos-type disper- sion relation of the tight-binding model (not shown). In addition, we have found analogous behavior for strictly linear dispersion relations with cut-off at finite energies (not shown). 

The above results suggest a certain robustness of the trapping effect which we have further analyzed by quali- tatively considering losses. This is accomplished by cou- pling the TLS to a second waveguide that can de-excite the TLS into modes other than those of the original 

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waveguide. The incorporation of this ”loss channel” into the Hamiltonian (1) thus proceeds by adding two addi- tional terms analogous to, respectively, the first (hopping term J $^{′}$) and third term (coupling term V $^{′}$) of the r.h.s. of (1). Clearly, the hopping term J ′ has to be chosen such that the energy of the bound photon-atom states that are associated with the first waveguide and the TLS alone lies in the band of the second waveguide. In Fig. 4, we display the time evolution of of the TLS’ excited-state occupation for different coupling strengths V ′ of the TLS to such a ”broad-band loss waveguide”. The trapping ef- fect persists even for rather strong coupling to the loss 

