  Superconducting circuits provide a favorable platform for quantum
thermodynamic experiments. An important component for such experiments is a
heat valve, i.e. a device which allows one to control the heat power flowing
through the system. Here we theoretically study the heat valve based on a
superconducting quantum interference device (SQUID) coupled to two heat baths
via two resonators. The heat current in such system can be tuned by magnetic
flux. We investigate how does the heat current modulation depend on the
coupling strength g between the SQUID and the resonators. In the weak coupling
regime the heat current modulation grows as g2, but, surprisingly, at the
intermediate coupling it can be strongly suppressed. This effect is linked to
the resonant nature of the heat transport at weak coupling, where the heat
current dependence on the magnetic flux is a periodic set of narrow peaks. At
the intermediate coupling, the peaks become broader and overlap, thus reducing
the heat modulation. At very strong coupling the heat modulation grows again
and finally saturates at a constant value.
