Dipole-dipole interactions mediated by a photonic flat band

Flat bands (FBs) are energy bands with zero group velocity, which in electronic systems were shown to favor strongly correlated phenomena. Indeed, a FB can be spanned with a basis of strictly localized states, the so called <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">compact localized states</mml:mtext></mml:mrow></mml:math> (CLSs), which are yet generally non-orthogonal. Here, we study emergent dipole-dipole interactions between emitters dispersively coupled to the photonic analogue of a FB, a setup within reach in state-of the-art experimental platforms. We show that the strength of such photon-mediated interactions decays exponentially with distance with a characteristic localization length which, unlike typical behaviours with standard bands, saturates to a finite value as the emitter's energy approaches the FB. Remarkably, we find that the localization length grows with the overlap between CLSs according to an analytically-derived universal scaling law valid for a large class of FBs both in 1D and 2D. Using giant atoms (non-local atom-field coupling) allows to tailor interaction potentials having the same shape of a CLS or a superposition of a few of these.Flat Bands are dispersionless energy bands, feature causing a quenching of kinetic energy, making such systems extremely sensitive to small perturbations and non-linearities. Here, we examine the case in which the non-linearity is introduced through the coupling of two-level emitters (almost) resonant to the Flat Band. We find several features depending heavily on the orthogonality of the Flat Band basis, such as the emergence of compact bound states. We show that these compact states can be used to induce effective spin-Hamiltonians with exact finite-ranged interactions. Furthermore, in the non-orthogonal case we are able to observe the emergence of an exotic new class of bound states which, although resulting from the coupling in the bandgap of dispersive energy bands, can be completely studied through the Flat Band projector only. The localization length of these states is frequency-independent and is related to the overlap between states in the Flat Band basis. Finally, we also show that it is possible to engineer any shape of the bound state by controlling the coupling of a giant atom to the system.Flat bands (FBs) are band structures featuring zero-group velocity along the whole band, which in electronic systems have been shown to be source of strongly correlated phenomena. Here, we study the emergent photon-mediated interactions when emitter are tuned to the photonic analogue of FBs, a setup becoming within reach of state-of-art platforms In particular, we find that, unexpectedly, dispersive interactions between atoms are mediated by exponentially localized atom-photon bound states (BSs), whose range remains finite even for vanishing detuning from the FB. This represents a notable difference with respect to the cavity-QED case in which the presence of non-dispersive cavity modes does not lead to this kind of behaviour. We link this BS to a set of compact localized states (CLSs) spanning the FB eigenspace, which are generally non-orthogonal, and work out analytically the BS localization length as a function of the CLS non-orthogonality parameter for a large class of 1D and 2D lattices.We also show that photon-mediated interactions can be tailored in an exact way if giant atoms are coupled to the system in the vicinity of a FB.