We present an evaporative cooling technique for atoms trapped in an optical dipole trap that benefits from narrow optical transitions. For an appropriate choice of wavelength and polarization, a single laser beam leads to opposite light-shifts in two internal states of the lowest energy manifold. Radio-frequency coupling between these two states results in evaporative cooling at a constant trap stiffness. The evaporation protocol is well adapted to several atomic species, in particular to the case of Lanthanides such as Er, Dy, and fermionic Yb, but also to alkali-earth metals such as fermionic Sr. We derive the dimensionless expressions that allow us to estimate the evaporation efficiency. As a concrete example, we consider the case of $^{162}$Dy and present a numerical analysis of the evaporation in a dipole trap near the $J=J$ optical transition at 832 nm. We show that this technique can lead to runaway evaporation in a minimalist experimental setup.
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