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

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 multi-bath 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, non-perturbative framework for understanding heat transfer in open quantum systems.

Figures (5)

• Figure 1: In the two-point measurement (TPM) scheme for calculating heat moments, the bath Hamiltonian $\hat{H}_B$ is measured at times $t=0$ and $\tau$. The joint system-bath ($S$-$B$) state evolves under $\mathcal{U}_t$. In the proposed method (right), thermofield doubling (TFD) and Bogoliubov transformation (BT) take a Gaussian state on $B$ to a vacuum state on $B_O$-$B_A$. Measuring the 'heat operator' $\tilde{\mathcal{Q}}$ at $t=\tau$ on the unitarily evolved joint $B_A$-$S$-$B_O$ state then yields the heat moments of $B$.
• Figure 2: Diagram illustrating the evaluation of heat moments in Eq. \ref{['eq:main_result']}. The heat operator (shaded in orange) defined in \ref{['eq:heat_operator']} is applied $n$ times to the time-evolved state $\hat{\rho}(t)$ (denoted with circles) defined on $S$ and the extended bath $OA$.
• Figure 3: Evolution of (a) mean, (b) variance, and (c) Fano factor of heat fluctuations in the spin-boson model ($\epsilon_0 = 1,\ \Delta = 0,\ \omega_C = 5$). The initial state of the spin is the $|+\rangle$ eigenstate of $\hat{S}_x$. Curves corresponding to each $T$-$\alpha$ pair for the independent boson model (from benchmark results in Fig. \ref{['fig:exact_model']}) are shown as solid colored lines for reference.
• Figure 4: Current fluctuations in the nonequilibrium spin–boson model. Time evolution of the (a) mean, (b) variance, and (c) Fano factor of the heat current for two bosonic baths with cutoff frequency $\omega_C$ at temperatures $T_1 = 1$ and $T_2 = 0$. We set $\epsilon_0 = 1$, $\Delta = 0$, and initialize the spin in the excited eigenstate of $\hat{S}_z$. The double-headed vertical arrow in (a) highlights the difference between the rectified currents (green and brown dots). A gray dashed line in (c) marks $F=1$ corresponding to Poissonian statistics. All vertical axes are plotted on a logarithmic scale.
• Figure S1: Evolution of (a) mean, (b) variance, and (c) Fano factor of heat fluctuations in the independent-boson model ($\epsilon_0=0,\ \Delta=1, \ \omega_c = 5$). The initial state of the spin is the $|+\rangle$ eigenstate of $\hat{S}_x$. Exact solutions for this model are plotted as solid lines.