[Paper Review] Nonequilibrium system-bath entanglement theorem versus heat transport

Peng-Li Du, Chen, Zi-Hao+4 more|arXiv (Cornell University)|Dec 2, 2021

Advanced Thermodynamics and Statistical Mechanics5 citations

TL;DR

This paper extends the system-bath entanglement theorem (SBET) to nonequilibrium quantum heat transport by deriving a generalized Langevin equation framework that connects entangled system-bath correlation functions to heat current in molecular junctions. The key result is a closed-form expression for the heat current (Eq. 26) that exactly reproduces the Meir-Wingreen NEGF formalism, validated numerically with excellent agreement between direct and indirect evaluations.

ABSTRACT

In this work, we extend the recently established system-bath entanglement theorem (SBET) [J. Chem. Phys. 152, 034102 (2020)] to the nonequilibrium scenario, in which an arbitrary system couples to multiple Gaussian baths environments at different temperatures. While the existing SBET connects the entangled system-bath response functions to those of local systems, the extended theory is concerned with the nonequilibrium steady-state quantum transport current through molecular junctions. The new theory is established on the basis of the generalized Langevin equation, with a close relation to nonequilibrium thermodynamics in the quantum regime.

Motivation & Objective

• To extend the system-bath entanglement theorem (SBET) from equilibrium to nonequilibrium quantum transport scenarios involving multiple thermal baths at different temperatures.
• To establish a theoretical framework linking entangled system-bath correlation functions to nonequilibrium steady-state heat current in molecular junctions.
• To provide an alternative to the nonequilibrium Green's function (NEGF) formalism for quantum heat transport using a Langevin equation approach grounded in nonequilibrium quantum thermodynamics.
• To validate the new formalism numerically by comparing indirect evaluation via the extended SBET with direct evaluation using the hierarchical equations of motion (DEOM) method.

Proposed method

• Derives a generalized Langevin equation for hybrid bath dynamics by decomposing the system-bath interaction into forward and backward components using the Gauss–Wick’s environment ansatz.
• Introduces a decomposition of the hybrid bath operator $\hat{F}_{\alpha u}$ into $\hat{F}_{\alpha u}^{\sigma}$ ($\sigma = +, -$) corresponding to absorptive and emissive processes, enabling treatment of nonequilibrium correlations.
• Expresses the heat current $J_\alpha$ via a spectral representation involving the bath spectral density $J^{\alpha}_{uv}(\omega)$ and the system correlation function $C_{vu}(\omega)$, as shown in Eq. (26).
• Relies on the generalized Langevin equation to relate the system's dynamics to the entangled bath response, allowing evaluation of nonequilibrium correlation functions beyond the scope of the fluctuation-dissipation theorem.
• Uses the identity $\langle \hat{Q}_u \dot{\hat{F}}_{\alpha u} \rangle = 2\,\text{Re} \sum_v \int_0^\infty d\tau\, \dot{\phi}_{uv}^{\alpha;+}(\tau) \langle \hat{Q}_v(0) \hat{Q}_u(\tau) \rangle$ to derive the final current expression.
• Validates the extended SBET numerically by comparing the heat current computed via Eq. (26) (indirect method) with direct DEOM simulations for a two-level system coupled to left and right baths at different temperatures.

Experimental results

Research questions

• RQ1 Can the system-bath entanglement theorem (SBET) be extended to describe nonequilibrium steady-state quantum heat transport through molecular junctions?
• RQ2 How can entangled system-bath correlation functions be systematically evaluated in the quantum regime far from equilibrium?
• RQ3 Does the generalized Langevin equation framework yield a heat current formula equivalent to the established Meir-Wingreen NEGF formalism?
• RQ4 Can the extended SBET provide an alternative, numerically validated route to computing heat currents in open quantum systems with multiple thermal reservoirs?
• RQ5 What is the role of the hybrid bath spectral density $J^{\alpha}_{uv}(\omega)$ and system correlation function $C_{vu}(\omega)$ in determining the heat current in nonequilibrium conditions?

Key findings

• The extended SBET successfully derives a closed-form expression for the heat current, Eq. (26), which exactly matches the Meir-Wingreen NEGF formalism for quantum heat transport.
• Numerical validation shows excellent agreement between the indirect method (via Eq. 26) and the direct DEOM evaluation: for $T_R/T_L = 0.5$, direct gives $J_L = 0.01484$, indirect gives $0.01487$, with relative error below 0.2%.
• The method enables the indirect evaluation of heat current through local system correlation functions $C_{vu}(\omega)$, bypassing the need for full system-bath entanglement calculations.
• The framework is general and applicable to arbitrary systems coupled to multiple thermal baths at different temperatures, as demonstrated in the two-level model with $H_S = V(|1\rangle\!\langle 2| + |2\rangle\!\langle 1|)$.
• The derivation establishes a direct link between the generalized Langevin equation and nonequilibrium quantum thermodynamics, enabling access to correlation functions not governed by the conventional fluctuation-dissipation theorem.
• The spectral function $J^{\alpha}_{uv}(\omega)$ and the system correlation $C_{vu}(\omega)$ fully determine the heat current, providing a physically transparent and computationally accessible route to quantum transport.