We study a mobile impurity in a one-dimensional harmonically confined optical lattice interacting repulsively with a bosonic bath. The behavior of the impurity across baths with superfluid and Mott-insulator domains is examined, including its full back-action effect on the bath. We characterize the bath-impurity phase diagram and reveal the appearance of a correlated counterflow phase, which we support with an analytical model for a mobile impurity-hole pair. This phase shows an extended combined insulator domain of unity filling but no independent domain of constant density. The transition to this phase features a sudden orthogonality and the change of the shape of the impurity’s profile to that of a free particle in an infinite square well. The findings of this work suggest the appearance of unconventional counterflow in trapped imbalanced atomic mixtures.

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• Bose-Einstein condensates
• Cold gases in optical lattices
• Polarons
• Mott insulators
• Bose-Hubbard model