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We propose two experimental setups for fermionic atoms in a high-finesseoptical resonator in which either a superconducting state with s-wave symmetryof the pairs or a 4k F charge density wave can self-organize. In order tostabilize the s-wave pairing, a two component attractively in- teractingfermionic gas is confined to a one dimensional chain structure by an opticallattice. The tunneling of the atoms along the chains is suppressed initially byan energy offset between neighbor- ing sites. A Raman transition using thecavity mode and a transversal pump laser then reintroduces a cavity-assistedtunneling. The feedback mechanism between the cavity field and the atoms leadsto a spontaneous occupation of the cavity field and of a state of the fermionicatoms which is dominated by s-wave pairing correlations. Extending the setup toa quasi-one-dimensional ladder structure where the tunneling of atoms along therungs of the ladder is cavity-assisted, the repul- sively interacting fermionicatoms self-organize into a 4k F charge density wave. We use adiabaticelimination of the cavity field combined with state-of-the-art density matrixrenormalization group methods in finite systems in order to identify the steadystate phases of the system.

Cavity-induced superconducting and $4k_F$ charge-density-wave states