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Quantum Thermal Transport Beyond Second Order with the Reaction Coordinate Mapping

Standard quantum master equation techniques such as the Redfield or Lindblad equations are perturbative to second order in the microscopic system-reservoir coupling parameter $λ$. As a result, characteristics of dissipative systems, which are beyond second order in $λ$, are not captured by such tools. Moreover, if the leading order in the studied effect is higher-than-quadratic in $λ$, a second-order description fundamentally fails, even at weak coupling. Here, using the reaction coordinate (RC) quantum master equation framework, we are able to investigate and classify higher-than-second order transport mechanisms. This technique, which relies on the redefinition of the system-environment boundary, allows for the effects of system-bath coupling to be included to high orders. We study steady-state heat current beyond second-order in two models: The generalized spin-boson model with non-commuting system-bath operators and a three-level ladder system. In the latter model heat enters in one transition and it is extracted from a different one. Crucially, we identify two transport pathways: (i) System's current, where heat conduction is mediated by transitions in the system, with the heat current scaling as $j_q \propto λ^2$ to lowest order in $λ$. (ii) Inter-bath current, with the thermal baths directly exchanging energy between them, facilitated by the bridging quantum system. To the lowest order in $λ$, this current scales as $j_q \propto λ^4$. These mechanisms are uncovered and examined using numerical and analytical tools. We contend that the RC mapping brings, already at the level of the mapped Hamiltonian, much insights on transport characteristics.