GLYPH<K62> GLYPH<K7C> GLYPH<K62>GLYPH<K2C> GLYPH<K20>GLYPH<K4E> GLYPH<K3E> GLYPH<K7C> GLYPH<K69>GLYPH<K2C> GLYPH<K20>GLYPH<K4E> GLYPH<K3E> GLYPH<K7C> GLYPH<K62>GLYPH<K2C> GLYPH<K20>GLYPH<K4E> GLYPH<K20> GLYPH<K2B> GLYPH<K31> GLYPH<K3E> GLYPH<K7C> GLYPH<K61>GLYPH<K2C> GLYPH<K20>GLYPH<K4E> GLYPH<K20> GLYPH<K2B> GLYPH<K31> GLYPH<K3E> GLYPH<K56> GLYPH<K73>GLYPH<K64> Fig. 4: A schematic illustration of the rf-induced Feshbach resonance. The rf field induces coupling between the initial two-atom dressed state | i, N 〉 and a bound molecular state | b, N + 1 〉 , which is coupled with a low-lying open channel | a, N + 1 〉 via the spin-dependent interaction potential. Δ = ℏ ω − ϵ$_{b}$ is the detuning from resonance. while keeping inelastic losses to a minimum, it is necessary to operate at large rf detunings Δ ≫ γ$_{ba}$ , where α − α$_{bg}$ ∼ γ$_{ib}$/ Δ and β ∼ γ$_{ib}$γ$_{ba}$/ Δ $^{2}$. In this regime, the real part of the scattering length is independent of γ$_{ba}$ . For practical applications, however, it may be advantageous to minimize inelastic losses quantified by the parameter γ$_{ba}$ , so that the condition Δ ≫ γ$_{ba}$ is satisfied at smaller Δ, where α − α$_{bg}$ is most sensitive to Δ. Indeed, as shown in Fig. 3, β is never too large except at very small detunings, which indicates that the regime Δ ≫ γ$_{ba}$ is within reach. In summary, we have developed a rigorous quantum theory of atomic collisions in the presence of rf fields, and applied it to explore the effect of rf fields on s -wave collisions of Rb atoms. Our results show that rf fields generated in experiments with atom chips can be used as an efficient tool to control collisions of ultracold atoms, which has a number of advantages over the methods that use dc magnetic fields. In particular, shifting magnetic Feshbach resonances with rf fields may be used to selectively tune scattering lengths in ultracold atomic mixtures, thereby enabling the preparation of exotic many-body states and novel quantum phases [29]. As illustrated in Fig. 2(a), applying an rf field with B$_{rf}$ = 4 G at B = 9 . 206 G results in a three-fold enhancement of the real part of the scattering length. The calculated inelastic decay rates do not exceed ∼ 10 − 10 cm $^{3}$/s, so elastic collisions of Rb atoms can be efficiently manipulated on timescales of hundreds of microseconds for typical atom densities of 10 13 cm − $^{3}$. Further advantages of rf control include the possibility of 8