We investigate the heat transport and the control of heat current among two spatially separated trapped Bose-Einstein Condensates (BEC), each of them at a different temperature. To allow for heat transport among the two independent BECs we consider a link made of two harmonically trapped impurities, each of them interacting with one of the BECs. Since the impurities are spatially separated, we consider long-range interactions between them, namely a dipole-dipole coupling. We study this system under theoretically suitable and experimentally feasible assumptions/parameters. The dynamics of these impurities is treated within the framework of the quantum Brownian motion model, where the excitation modes of the BECs play the role of the heat bath. We address the dependence of heat current and current-current correlations on the physical parameters of the system. Interestingly, we show that heat rectification, i.e., the unidirectional flow of heat, can occur in our system, when a periodic driving on the trapping frequencies of the impurities is considered. Therefore, our system is a possible setup for the implementation of a phononic circuit. Motivated by recent developments on the usage of BECs as platforms for quantum information processing, our work offers an alternative possibility to use this versatile setting for information transfer and processing, within the context of phononics, and more generally in quantum thermodynamics.
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