A photonic heat amplifier designed for cryogenic operations is introduced and analyzed. This device comprises two variable-range-hopping reservoirs connected by lossless lines, which allow them to exchange heat through photonic modes. This configuration enables negative differential thermal conductance, which can be harnessed to amplify thermal signals. To achieve this, one reservoir is maintained at a high temperature, serving as the source terminal of a thermal transistor. Concurrently, in the other reservoir, we establish tunnel contacts to metallic reservoirs, which function as the gate and drain terminals. With this arrangement, it is possible to control the heat flux exchange between the source and the drain by adjustment of the gate temperature. We present two different parameter choices that yield different performances: the first emphasizes modulation of the source-drain heat current, while the second focuses on the modulation of the lower-temperature variable-range-hopping reservoir. Lastly, we present a potential design variation in which all electronic reservoirs are thermally connected through only photonic modes, allowing interactions between distant elements. The proposed photonic heat amplifier addresses the lack of thermal transistors and amplifiers in the millikelvin range, while being compatible with the rich toolbox of circuit quantum electrodynamics. It can be adapted to various applications, including sensing and the development of thermal circuits and control devices at subkelvin temperatures, which are relevant to quantum technologies.

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• Heat radiation
• Heat transfer
• Photonics
• Quantum circuits
• Thermal conductivity
• Wireless power transfer
• Devices
• Semiconductors