  We propose a novel cooling scheme for realising single photon sideband
cooling on particles trapped in a state-dependent optical potential. We develop
a master rate equation from an ab-initio model and find that in experimentally
feasible conditions it is possible to drastically reduce the average occupation
number of the vibrational levels by applying a frequency sweep on the cooling
laser that sequentially cools all the motional states. Notably, this cooling
scheme works also when a particle experiences a deeper trap in its internal
ground state than in its excited state, a condition for which conventional
single photon sideband cooling does not work. In our analysis, we consider two
cases: a two-level particle confined in an optical tweezer and Li atoms
confined in an optical lattice, and find conditions for efficient cooling in
both cases. The results from the model are confirmed by a full quantum Monte
Carlo simulation of the system Hamiltonian. Our findings provide an alternative
cooling scheme that can be applied in principle to any particle, e.g. atoms,
molecules or ions, confined in a state-dependent optical potential.
