Controllable interatomic interaction mediated by diffractive coupling in a cavity

Photon-mediated interaction can be used for simulating complex many-body phenomena with ultracold atoms coupled to electromagnetic modes of an optical resonator. We theoretically study a method of producing controllable interatomic interaction mediated by forward-diffracted photons circulating inside a ring cavity. One example of such a system is the three-mode cavity, where an on-axis mode can coexist with two diffracted sidebands. We demonstrate how the self-organized stripe states of a Bose-Einstein condensate (BEC) occurring in this cavity geometry can exhibit supersolid properties, due to spontaneous breaking of the Hamiltonian's continuous translational symmetry. A numerical study of the collective excitation spectrum of these states demonstrates the existence of massles and finite-gap excitations, which are identified as phase (Goldstone) and amplitude (Higgs) atomic density modes. We further demonstrate how judicious Fourier filtering of intracavity light can be used to engineer the effective atom-atom interaction profile for many cavity modes. The numerical results in this configuration show the existence of droplet array and single droplet BEC states for commensurate and incommensurate cavity modes, respectively. Diffractive coupling in a cavity is thereby introduced as a novel route towards tailoring the photon-mediated interaction of ultracold atoms. Spatial features of the self-organized optical potentials can here be tuned to scales several times larger than the pump laser wavelength, such that the corresponding atomic density distributions could be imaged and manipulated using low numerical aperture optics. These calculations and insights pave the way towards quantum simulation of exotic nonequilibrium many-body physics with condensates in a cavity.