Heat transfer in the spin-boson model: a comparative study in the incoherent tunneling regime

Bath-induced interactions and transient dynamics in open quantum systems at strong coupling: Effective Hamiltonian approach

Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.

Challenges in molecular dynamics simulations of heat exchange statistics

We study heat exchange in temperature-biased metal-molecule-metal molecular junctions by employing the molecular dynamics simulator LAMMPS. Generating the nonequilibrium steady state with Langevin thermostats at the boundaries of the junction, we show that the average heat current across a gold-alkanedithiol-gold nanojunction behaves physically, with the thermal conductance value matching the literature. In contrast, the full probability distribution function for heat exchange, as generated by the simulator, violates the fundamental fluctuation symmetry for entropy production. We trace this failure back to the implementation of the thermostats and the expression used to calculate the heat exchange. To rectify this issue and produce the correct statistics, we introduce single-atom thermostats as an alternative to conventional many-atom thermostats. Once averaging heat exchange over the hot and cold thermostats, this approach successfully generates the correct probability distribution function, which we use to study the behavior of both the average heat current and its noise. We further examine the thermodynamic uncertainty relation in the molecular junction and show that it holds, albeit demonstrating nontrivial trends. Our study points to the need to carefully implement nonequilibrium molecular dynamics solvers in atomistic simulation software tools for future investigations of noise phenomena in thermal transport.

Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines

We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.

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, 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.

Heat transport through a two-level system embedded between two harmonic resonators

We investigate heat transport through an assembly consisting of a two-level system coupled between two harmonic oscillators, which is described by the quantum Rabi model, as a prototype of nanoscale heat devices using controllable multi-level systems. Using the noninteracting-blip approximation, we find that the linear thermal conductance shows a characteristic temperature dependence with a two-peak structure. We also show that heat transport is sensitive to model parameters for weak system-bath coupling and strong hybridization between the two-level system and the harmonic oscillators. This property characteristic of the multi-level system is advantageous for applications such as a heat transistor, and can be examined in superconducting circuits.

Effect of Inter-System Coupling on Heat Transport in a Microscopic Collision Model

In this paper we consider a bipartite system composed of two subsystems each coupled to its own thermal environment. Based on a collision model, we mainly study whether the approximation (i.e., the inter-system coupling is ignored when modeling the system–environment interaction) is valid or not. We also address the problem of heat transport unitedly for both excitation-conserving system–environment interactions and non-excitation-conserving system–environment interactions. For the former interaction, as the inter-system interaction strength increases, at first this approximation gets worse as expected, but then counter-intuitively gets better even for a stronger inter-system coupling. For the latter interaction with asymmetry, this approximation gets progressively worse. In this case we realize a perfect thermal rectification, and we cannot find an apparent rectification effect for the former interaction. Finally and more importantly, our results show that whether this approximation is valid or not is closely related to the quantum correlations between the subsystems, i.e., the weaker the quantum correlations, the more justified the approximation and vice versa.

Size effects in energy transport between thermal contacts mediated by nanoparticles

We investigate size effects in phononic energy transport in a system of two nanoparticles interconnected by a molecule and attached to thermal contacts also by molecules. In the considered closed system, the nanoparticles and contacts are described by ensembles of finite numbers of harmonic oscillators within the Drude-Ullersma model. The macroscopic character of the contacts is simulated by a large value of the ratio Δ/Δ_{B}=n (n>100) of mode spacings Δ and Δ_{B} corresponding to the nanoparticles and contacts, respectively. Quasistatic energy transport on the timescale Δ^{-1} is investigated. Equations describing the dynamics of the averaged eigenmode energies that belong to the nanoparticles and contacts are derived and solved. The resulting expressions for the energy current exiting (entering) the contacts as well as the energy current between the nanoparticles are obtained and investigated. The latter current accounts for energy accumulation by (depletion from) the nanoparticles. The finite size effects result in reversibility features and peculiarities at time moments t=2πℓΔ^{-1} for non-negative integers ℓ. They are qualitatively the same as in a previously studied system of two equal nanoparticles mediated by a molecule, despite the presence of the macroscopic contacts. The thermal conductance of the whole nanojunction is derived and explored. The energy currents and thermal conductance of the nanojunction in a case when its parameters are known from the experiment are computed using the developed model.

Strong system-bath coupling induces negative differential thermal conductance and heat amplification in nonequilibrium two-qubit systems

Quantum heat transfer is analyzed in nonequilibrium two-qubits systems by applying the nonequilibrium polaron-transformed Redfield equation combined with full counting statistics. Steady-state heat currents with weak and strong qubit-bath couplings are clearly unified. Within the two-terminal setup, the negative differential thermal conductance is unraveled with strong qubit-bath coupling and finite qubit splitting energy. The partially strong spin-boson interaction is sufficient to show the negative differential thermal conductance. Based on the three-terminal setup, in which two qubits are asymmetrically coupled to three thermal baths, a giant heat amplification factor is observed with strong qubit-bath coupling. Moreover, the strong interaction of either the left or right spin-boson coupling is able to exhibit the apparent heat amplification effect.

Vibrational Heat Transport in Molecular Junctions

We review studies of vibrational energy transfer in a molecular junction geometry, consisting of a molecule bridging two heat reservoirs, solids or large chemical compounds. This setup is of interest for applications in molecular electronics, thermoelectrics, and nanophononics, and for addressing basic questions in the theory of classical and quantum transport. Calculations show that system size, disorder, structure, dimensionality, internal anharmonicities, contact interaction, and quantum coherent effects are factors that combine to determine the predominant mechanism (ballistic/diffusive), effectiveness (poor/good), and functionality (linear/nonlinear) of thermal conduction at the nanoscale. We review recent experiments and relevant calculations of quantum heat transfer in molecular junctions. We recount the Landauer approach, appropriate for the study of elastic (harmonic) phononic transport, and outline techniques that incorporate molecular anharmonicities. Theoretical methods are described along with examples illustrating the challenge of reaching control over vibrational heat conduction in molecules.

Modeling the quasistatic energy transport between nanoparticles

We consider phononic energy transport between nanoparticles mediated by a quantum particle. The nanoparticles are considered as thermal reservoirs described by ensembles of finite numbers of harmonic oscillators within the Drude-Ullersma model having, in general, unequal mode spacings Δ(1) and Δ(2), which amount to different numbers of atoms in the nanoparticles. The quasistatic energy transport between the nanoparticles on the time scale t∼1/Δ(1,2) is investigated using the generalized quantum Langevin equation. We find that double degeneracy of system's eigenfrequencies, which occurs in the case of identical nanoparticles, is removed when the mode spacings become unequal. The equations describing the dynamics of the averaged eigenmode energies are derived and solved, and the resulting expression for the energy current between the nanoparticles is obtained and explored. Unlike the case when the thermodynamic limit is assumed resulting in time-independent energy current, finite-size effects result in temporal behavior of the energy current that evinces reversibility features combined with decay and possesses peculiarities at time moments t=2πn/Δ(1)+2πm/Δ(2) for non-negative integers n and m. When Δ(1,2)→0, an expression for the heat current obtained previously under assumption of the thermodynamic limit is reproduced. The energy current between two platinum nanoparticles mediated by a carbon oxide molecule is considered as an application of the developed model.

Strongly Coupled Quantum Heat Machines

Energy conversion of heat into work at the quantum level is modeled by quantum heat machines (QHMs) generally assumed to operate at weak coupling to the baths. This supposition is grounded in the separability principle between systems and allows the derivation of the evolution equation. In the weak coupling regime, the machine's output is limited by the coupling strength, restricting their application. Seeking to overcome this limitation, we analyze QHMs in the virtually unexplored strong coupling regime here, where separability, as well as other standard thermodynamic assumptions, may no longer hold. We show that strongly coupled QHMs may be as efficient as their weakly coupled counterparts. In addition, we find a novel turnover behavior where their output saturates and disappears in the limit of ultrastrong coupling.

Improved Dyson series expansion for steady-state quantum transport beyond the weak coupling limit: divergences and resolution

We present a general theory to calculate the steady-state heat and electronic currents for nonlinear systems using a perturbative expansion in the system-bath coupling. We explicitly demonstrate that using the truncated Dyson-series leads to divergences in the steady-state limit, thus making it impossible to be used for actual applications. In order to resolve the divergences, we propose a unique choice of initial condition for the reduced density matrix, which removes the divergences at each order. Our approach not only allows us to use the truncated Dyson-series, with a reasonable choice of initial condition, but also gives the expected result that the steady-state solutions should be independent of initial preparations. Using our improved Dyson series we evaluate the heat and electronic currents up to fourth-order in system-bath coupling, a considerable improvement over the standard quantum master equation techniques. We then numerically corroborate our theory for archetypal settings of linear systems using the exact nonequilibrium Green's function approach. Finally, to demonstrate the advantage of our approach, we deal with the nonlinear spin-boson model to evaluate heat current up to fourth-order and find signatures of cotunnelling process.