Quantum thermal transport beyond second order with the reaction coordinate mapping

Abstract

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, the 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 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 ∝ λ2 to the 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 ∝ λ4. 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 insight into transport characteristics.

Nonequilibrium quantum heat transport between structured environments

“…Future work could potentially examine the effect of asymmetry on the transport properties of the model, including rectification of the currents that arises with noncommutative system-bath coupling operators [31,37] and-or asymmetric coupling strengths [1,3], and in particular, whether the diode effect is enhanced or diminished for environments possessing gapped spectra. Alternatively, generalisations of the model to include couplings between components of the different baths in the spirit of [52,53] could be analyzed.…”

“…Hence, in this case a second-order treatment fails to describe the correct physics even in the asymptotically weak coupling limit. However, it should be emphasized that the quartic scaling here stems from a different transport mechanism to that explored in [31,37], which as mentioned above, is due to the noncommutativity of the system-bath coupling operators.…”

“…To date, numerous analytical and numerical techniques have been employed to study the NESB, each providing varying levels of insight into its heat transport characteristics. These include perturbative master equations [1,21], nonequilibrium Green's function methods [24,25], path integral techniques [26][27][28][29], the reaction-coordinate method [30,31], the hierarchical equations of motion (HEOM) [32][33][34][35][36][37], and the time-evolving matrix product operator technique [38], among others. Perturbative master equations constitute a standard tool of open quantum systems theory [39] and have been shown to capture resonant heat transfer in the NESB up to second-order in the system-bath coupling.…”

“…Recently, an extension to the NESB was proposed and studied in [31,37,48,49], where a unitary rotation is applied to one of the interaction terms so as to make the system-bath coupling operators noncommutative. This feature introduces a source of asymmetry into the model that facilitates two competing transport mechanisms: (i) one in which heat transfer occurs between baths through direct excitation of the system, termed system currents, and (ii) another pathway independent of the system in which a heat flow occurs directly between the baths.…”

“…This feature introduces a source of asymmetry into the model that facilitates two competing transport mechanisms: (i) one in which heat transfer occurs between baths through direct excitation of the system, termed system currents, and (ii) another pathway independent of the system in which a heat flow occurs directly between the baths. Notably, since the inter-bath current scales quartically with respect to the system-bath coupling parameter λ as λ → 0, it was established that a second-order approach is insufficient to capture the correct steady-state behavior in cases where the system current is zero [31,37]. At the same time, other extensions to the NESB were examined in [50][51][52] to study heat transport through superconducting circuits, consisting of a transmon qubit and pair of resonators in a resonator-qubit-resonator configuration (quantum Rabi model) embedded between two thermal reservoirs [18,53].…”