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Thermal Markovian dynamics is typically obtained by coupling a system to asufficiently hot bath with a large heat capacity. Here we present a scheme forinducing Markovian dynamics using an arbitrarily small and cold heat bath. Thescheme is based on injecting phase noise to the small bath. Several uniquesignatures of small bath are studied. We discuss realizations in ion traps andsuperconducting qubits and show that it is possible to create an ideal settingwhere the system dynamics is indifferent to the internal bath dynamics.

Markovian heat sources with the smallest heat capacity