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We experimentally investigate - polarization gradient cooling~(PGC) of a single Rb atom in a tightly focused dipole trap and show that the cooling limit strongly depends on the polarization of the trapping field. For optimized cooling light power, the temperature of the atom reaches~K in a linearly polarized trap, approximately five times lower than in a circularly polarized trap. The inhibition of PGC is qualitatively explained by the fictitious magnetic fields induced by the trapping field. We further demonstrate that switching the trap polarization from linear to circular after PGC induces only minor heating.

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