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arXiv:2504.10631 · Physical Review Research

Heat operator approach to quantum stochastic thermodynamics in the strong-coupling regime

Sheikh Parvez Mandal, Mahasweta Pandit, Khalak Mahadeviya, Mark T. Mitchison, Javier Prior

Published Feb 2, 2026• 1 citation
arXivHTMLPDF

Abstract

Heat exchanged between an open quantum system and its environment exhibits fluctuations that carry crucial signatures of the underlying dynamics. Within the well-established two-point measurement scheme, we identify a “heat operator,” whose moments with respect to the vacuum state of a thermofield-doubled Hilbert space correspond to the stochastic moments of the heat exchanged with a bath. This recasts heat statistics as a unitary time evolution problem, which we solve by combining chain-mapped reservoirs with tensor-network propagation. In a multibath setup, all total and bath-resolved heat moments then follow from a single pure state evolution. We employ this approach to compute transient and steady-state heat fluctuations in Ohmic spin-boson models in and out of equilibrium, accessing the challenging low-temperature and long memory time regimes of the environment. In the nonequilibrium case, we show a crossover in the Fano factor from super-Poissonian to nearly Poissonian statistics under strong coupling asymmetry, corresponding to thermal rectification behavior. The method applies to noninteracting (bosonic or fermionic) nonequilibrium environments with arbitrary spectral densities, offering a powerful, nonperturbative framework for understanding heat transfer in open quantum systems.

DOI
10.1103/8plx-nfvq

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Heat exchanged between an open quantum system and its environment exhibits fluctuations that carry crucial signatures of the underlying dynamics. Within the well-established two-point measurement scheme, we identify a “heat operator,” whose moments with respect to the vacuum state of a thermofield-doubled Hilbert space correspond to the stochastic moments of the heat exchanged with a bath. This recasts heat statistics as a unitary time evolution problem, which we solve by combining chain-mapped reservoirs with tensor-network propagation. In a multibath setup, all total and bath-resolved heat moments then follow from a single pure state evolution. We employ this approach to compute transient and steady-state heat fluctuations in Ohmic spin-boson models in and out of equilibrium, accessing the challenging low-temperature and long memory time regimes of the environment. In the nonequilibrium case, we show a crossover in the Fano factor from super-Poissonian to nearly Poissonian statistics under strong coupling asymmetry, corresponding to thermal rectification behavior. The method applies to noninteracting (bosonic or fermionic) nonequilibrium environments with arbitrary spectral densities, offering a powerful, nonperturbative framework for understanding heat transfer in open quantum systems.
We present a numerically exact method to compute the full counting statistics of heat transfer in non-Markovian open quantum systems, which is based on the time-evolving matrix product operator (TEMPO) algorithm. This approach is applied to the paradigmatic spin-boson model in order to calculate the mean and fluctuations of the heat transferred to the environment during thermal equilibration. We show that system-reservoir correlations make a significant contribution to the heat statistics at low temperature and present a variational theory that quantitatively explains our numerical results. We also demonstrate a fluctuation-dissipation relation connecting the mean and variance of the heat distribution at high temperature. Our results reveal that system-bath interactions make a significant contribution to heat dissipation even when the dynamics of the open system is effectively Markovian. The method presented here provides a flexible and general tool to predict the fluctuations of heat transfer in open quantum systems in non-perturbative regimes.
We present a numerically exact method to compute the full counting statistics of heat transfer in non-Markovian open quantum systems, which is based on the time-evolving matrix product operator algorithm.This approach is applied to the paradigmatic spin-boson model in order to calculate the mean and fluctuations of the heat transferred to the environment during thermal equilibration.We show that system-reservoir correlations make a significant contribution to the heat statistics at low temperature and present a variational theory that quantitatively explains our numerical results.We also demonstrate a fluctuation-dissipation relation connecting the mean and variance of the heat distribution at high temperature.Our results reveal that system-bath interactions make a significant contribution to heat transfer even when the dynamics of the open system is effectively Markovian.The method presented here provides a flexible and general tool to predict the fluctuations of heat transfer in open quantum systems in nonperturbative regimes.
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.
We investigate the dynamics of a two-qubit open quantum system, in particular the two-qubit spin-boson model in the strong coupling regime, coupled to two thermal bosonic baths under non-Markovian and non-equilibrium conditions. Two complementary approaches, the Hierarchical Equations of Motion (HEOM) and Reaction Coordinate Mapping (RCM), are employed to examine various coupling regimes between the qubits and their respective baths. The dynamical features of the model and the impact of the tunneling amplitude on quantum coherence of the system are probed using the l1l_1l1​-norm of coherence. The model is further shown to have non-Markovian evolution. The nontrivial task of calculating entropy production in the strong-coupling regime is performed using auxiliary density operators in HEOM. Motivated by the realization of a quantum thermal device in the strong-coupling regime, the non-equilibrium steady-state behavior of the system is investigated. Furthermore, the relationship between the heat and spin currents and the tunneling amplitude is probed.
By solving the exact master equation of open quantum systems, we formulate the quantum thermodynamics from weak to strong couplings. The open quantum systems exchange matters, energies, and information with their reservoirs through quantum particle tunnelings that are described by the generalized Fano-Anderson Hamiltonians. We find that the exact solution of the reduced density matrix of these systems approaches a Gibbs-type state in the steady-state limit for the systems in arbitrary initial states as well as for both the weak and strong system-reservoir coupling strengths. When the couplings become strong, thermodynamic quantities of the system must be renormalized. The renormalization effects are obtained nonperturbatively after exactly tracing over all reservoir states through the coherent state path integrals. The renormalized system Hamiltonian is characterized by the renormalized system energy levels and interactions, corresponding to the quantum work done by the system. The renormalized temperature is introduced to characterize the entropy production counting the heat transfer between the system and the reservoir. We further find that only with the renormalized system Hamiltonian and other renormalized thermodynamic quantities can the exact steady state of the system be expressed as the standard Gibbs state. Consequently, the corresponding exact steady-state particle occupations in the renormalized system energy levels obey the Bose-Einstein and the Fermi-Dirac distributions for bosonic and fermionic systems, respectively. In the very weak system-reservoir coupling limit, the renormalized system Hamiltonian and the renormalized temperature are reduced to the original bare Hamiltonian of the system and the initial temperature of the reservoir. Thus, the conventional statistical mechanics and thermodynamics are thereby rigorously deduced from quantum dynamical evolution. In the last, this nonperturbative renormalization method is also extended to general interacting open quantum systems.
Quantum thermodynamics seeks to address the emergence of thermodynamic laws, which govern how energy is exchanged between physical systems, from a quantum mechanical description of light and matter. As such, the workhorse model for an underlying description of such energy exchange processes is the open quantum system: a quantum system which may interact with an external environment. Open quantum systems are well-understood in regimes where the interaction between the system and the environment is weak enough that perturbative methods may be employed to arrive at dynamical equations of motion for the evolution of the system. How to tackle the strong coupling regime is an open question and much research is still being devoted in this area. In this project we employ a technique developed for studying an open quantum system in regimes of strong reservoir coupling in problems in thermodynamics which have so far been restricted to the weak coupling regime. We spend some time developing our strong coupling framework and then use it to analyse a quantum heat engine to find that strong coupling leads to a particular set of operational costs which are not present in more well-understood weak coupling versions. We also identify effects purely quantum in nature, present in the engine cycle only at strong coupling, which lead to a degradation in the engine's performance. Finally we address the issue of quantum coherence more generally in archetypal open quantum system models beyond weak coupling, and identify a thermodynamic cost to its erasure.