Phase-controlled thermal amplification in a quantum thermal transistor

We investigate phase-controlled heat transport in a quantum thermal transistor composed of three coupled two-level systems, each connected to an independent thermal reservoir. Using a nonsecular Redfield master equation, we show that a gauge-invariant phase in the triangular coupling loop provides an efficient control knob beyond reservoir temperatures: at fixed temperature bias, tuning the phase strongly modulates the steady-state heat currents and can generate pronounced thermal amplification, whereby a small variation of the control current induces large changes in the currents at the other two terminals. We further analyze how the amplification depends on the reservoir temperature as well as on the intersystem couplings, identifying regimes of enhanced performance. To clarify the underlying mechanism, we develop a complementary weak-coupling theory based on a local Lindblad equation and second-order cumulant expansion. This analysis shows that the phase first enters the exchange coherences and is then transferred to the populations through loop interference, while the heat currents are dominated by the resulting population corrections. Our results establish phase engineering as a flexible route to regulating heat flow and thermal amplification in multi-terminal quantum thermal devices.