[Paper Review] Linear response theory for quantum open systems

Jianhua Wei, YiJing Yan|arXiv (Cornell University)|Aug 30, 2011

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper develops a linear response theory for quantum open systems using Feynman’s influence functional and hierarchical equations of motion (HEOM), enabling direct calculation of steady-state dynamical observables—such as spectra functions—without time-dependent driving. The key contribution is a formalism that generalizes Kubo’s linear response to open quantum systems, with the spectra function derived via a fluctuation-dissipation relation in HEOM space.

ABSTRACT

Basing on the theory of Feynman's influence functional and its hierarchical equations of motion, we develop a linear response theory for quantum open systems. Our theory provides an effective way to calculate dynamical observables of a quantum open system at its steady-state, which can be applied to various fields of non-equilibrium condensed matter physics.

Motivation & Objective

• To extend linear response theory beyond equilibrium closed systems to quantum open systems with time-translation symmetry.
• To address the challenge of calculating dynamical observables—such as correlation functions and spectra—of a static open system without applying time-dependent fields.
• To provide a practical framework for non-equilibrium condensed matter physics, molecular electronics, and nanophysics.
• To establish a connection between the influence functional formalism and linear response in open quantum systems via hierarchical equations of motion.

Proposed method

• Derives the hierarchical equations of motion (HEOM) for the reduced density operator (RDO) and auxiliary influence functionals based on Feynman’s influence functional formalism.
• Utilizes the factorization property of the influence functional: $ F = F_A \cdot F_B $, implying $ \Phi = \Phi_A + \Phi_B $, to handle multiple probe fields linearly.
• Introduces a Liouville-space propagator formalism to define the time-evolution superoperator $ \mathcal{G}_{\mathbf{j}}^{(n)}(t) $, enabling the solution of the dynamics in HEOM space.
• Constructs the response correlation function $ \widetilde{C}_{AB}(t) $ via the time-ordered evolution of the system under a perturbation, using the propagator $ \hat{\boldsymbol{\mathcal{G}}}_s(t) $.
• Derives the spectra function $ J_{AB}('\omega') $ using a fluctuation-dissipation relation in HEOM space: $ J_{AB}(\omega) = \frac{1}{2\pi}(1 + e^{-\beta\omega}) \langle\!\langle \boldsymbol{A}(0) | \hat{\boldsymbol{\mathcal{G}}}_s(\omega) \boldsymbol{B} | \boldsymbol{\rho}_{eq}(T) \rangle\!\rangle $.
• Solves the steady-state condition $ \hat{\boldsymbol{\varLambda}}_s \boldsymbol{\rho}_{eq}(T) = 0 $ to obtain the equilibrium density operator $ \boldsymbol{\rho}_{eq}(T) $, required for correlation functions.

Experimental results

Research questions

• RQ1 Can linear response theory be generalized to quantum open systems that are not in equilibrium and lack time-translation invariance in the conventional sense?
• RQ2 How can dynamical observables such as spectra functions be computed directly for a static open system without time-dependent perturbations?
• RQ3 What is the role of the influence functional’s factorization property in enabling linear response to multiple probe fields?
• RQ4 How can the fluctuation-dissipation theorem be reformulated in the context of HEOM for open quantum systems?
• RQ5 What is the precise connection between the Liouville-space propagator $ \hat{\boldsymbol{\mathcal{G}}}_s(\omega) $ and the spectral function $ J_{AB}(\omega) $?

Key findings

• The paper establishes a linear response theory for quantum open systems using the HEOM formalism, enabling direct computation of steady-state dynamical observables without time-dependent driving.
• The spectra function $ J_{AB}(\omega) $ is derived as $ J_{AB}(\omega) = \frac{1}{2\pi}(1 + e^{-\beta\omega}) \mathrm{Tr}[A^+ \sigma(\omega)] $, where $ \sigma(\omega) $ is the Fourier-transformed response operator.
• The fluctuation-dissipation relation in HEOM space is explicitly confirmed as $ J_{AB}(\omega) = \frac{1}{\pi}(1 + e^{-\beta\omega}) C_{AB}(\omega) $, linking the spectra function to the correlation function $ C_{AB}(\omega) $.
• The equilibrium density operator $ \boldsymbol{\rho}_{eq}(T) $ is obtained by solving $ \hat{\boldsymbol{\varLambda}}_s \boldsymbol{\rho}_{eq}(T) = 0 $, which is essential for computing all response functions.
• The formalism allows the calculation of the retarded Green’s function $ G_{AB}^r(t) $ from the correlation function $ \widetilde{C}_{AB}(t) $ via $ G_{AB}^r(t) = -i\theta(t)[\widetilde{C}_{AB}(t) + \widetilde{C}_{BA}(-t)] $, valid even for fermionic systems.
• The method is applicable to a broad range of non-equilibrium quantum systems, including those in molecular electronics, nanophysics, and many-body physics, due to its foundation in the HEOM framework.