  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.
