  Macroscopic quantum phase coherence has one of its pivotal expressions in the
Josephson effect [1], which manifests itself both in charge [2] and energy
transport [3-5]. The ability to master the amount of heat transferred through
two tunnel-coupled superconductors by tuning their phase difference is the core
of coherent caloritronics [4-6], and is expected to be a key tool in a number
of nanoscience fields, including solid state cooling [7], thermal isolation [8,
9], radiation detection [7], quantum information [10, 11] and thermal logic
[12]. Here we show the realization of the first balanced Josephson heat
modulator [13] designed to offer full control at the nanoscale over the
phase-coherent component of thermal currents. Our device provides
magnetic-flux-dependent temperature modulations up to 40 mK in amplitude with a
maximum of the flux-to-temperature transfer coefficient reaching 200 mK per
flux quantum at a bath temperature of 25 mK. Foremost, it demonstrates the
exact correspondence in the phase-engineering of charge and heat currents,
breaking ground for advanced caloritronic nanodevices such as thermal splitters
[14], heat pumps [15] and time-dependent electronic engines [16-19].
