imposed rf and dc magnetic fields may be written (in units of ℏ ) where R is the interatomic distance, µ is the reduced mass of the collision complex, ˆ ℓ is the orbital angular momentum for the collision, and ˆ V$_{sd}$ ( R ) is the spin-dependent interaction potential between the atoms. We neglect the weak magnetic dipole-dipole interaction. The asymptotic Hamiltonian is ˆ H$_{as}$ = ˆ H$_{A}$ + ˆ H$_{B}$ + ˆ H$_{rf}$ . The terms ˆ H$_{A}$ and ˆ H$_{B}$ account for the interaction of individual atoms with a homogeneous rf field B ( t ) = ˆ eB$_{rf}$ cos( ωt ) where γ$_{ν}$ is the hyperfine constant of atom ν ( ν = A , B) with total angular momentum ˆ F$_{ν}$ , g$_{F}$ ν is the magnetic g -factor, µ$_{0}$ is the Bohr magneton, and B is the dc magnetic field vector, whose direction defines the quantization axis z . We assume that the rf field is polarized in the x -direction (ˆ e = ˆ x ). The Hamiltonian of the rf field ˆ H$_{rf}$ = ℏ ω (ˆ a $^{†}$ˆ a − N ) is expressed in a second-quantized form using the photon creation and annihilation operators ˆ a † and ˆ a , the average number of photons in the field N , and the rf frequency ω . In Eq. (2), ˆ F$_{ν}$$_{±}$ are the atomic raising and lowering operators, and λ$_{ν}$ = − µ$_{0}$g$_{F}$ ν B$_{rf}$/ $_{2}$√$_{N}$ is the coupling constant, which quantifies the strength of the atom-field interaction. In the rotating-wave approximation, the Hermitian conjugate terms in Eq. (2) are neglected in calculating the matrix elements between the states within the F = 2 manifold (and vice versa for the F = 1 manifold). We consider ultracold collisions of two identical $^{87}$Rb atoms in superimposed dc and rf magnetic fields. Following Agosta et al. [16], we introduce a symmetrized basis where | τ$_{ν}$ 〉 are the hyperfine states | F$_{ν}$m$_{F}$ ν 〉 , | N + n 〉 are the photon number states, η accounts for exchange symmetry ( η = +1 for two identical $^{87}$Rb atoms), and | ℓm$_{ℓ}$ 〉 are the partial wave states. Diagonalization of ˆ H$_{as}$ in the basis (3) yields the energy levels of two non-interacting atoms dressed by the rf field. The field-dressed states occur in manifolds separated by multiples of the photon energy ℏ ω [21]. Throughout this work, we consider rf frequencies in the range ω/ 2 π = 18-40 MHz and moderate dc fields B < 10 G, which corresponds to the weak coupling regime, where the energy separation between different photon manifolds ˆ H = − 1 2 µR ∂ 2 ∂R $_{2}$R + ˆ ℓ 2 2 µR 2 + ˆ V$_{sd}$ ( R ) + ˆ H$_{A}$ + ˆ H$_{B}$ + ˆ H$_{rf}$, (1) ˆ H$_{ν}$ = 1 $_{2}$γ$_{ν}$ ˆ F 2 ν + g$_{F}$ ν µ$_{0}$ B · ˆ F$_{ν}$ z + 1 $_{2}$λ$_{ν}$ (ˆ a † ˆ F$_{ν}$$_{−}$ + ˆ a ˆ F$_{ν}$$_{+}$ + H.c.) , (2) 1 [2(1 + δ$_{τ}$$_{A}$$_{τ}$$_{B}$ )] 1 / 2 $^{[}$| τ$_{A}$τ$_{B}$ 〉 + η ( − ) $^{ℓ}$| τ$_{B}$τ$_{A}$ 〉 $^{]}$| N + n 〉| ℓm$_{ℓ}$ 〉 , (3) 3