[Paper Review] Quantum Brownian Motion in the Heisenberg picture: derivation of the adjoint master equation and applications

Matteo Carlesso, Angelo Bassi|arXiv (Cornell University)|Feb 16, 2016

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper presents a novel Heisenberg picture derivation of the master equation for quantum Brownian motion within the Hu-Paz-Zhang (HPZ) model, yielding an exact and analytic equation directly in terms of the spectral density. The approach enables efficient computation of time evolution for physically relevant observables without approximations, offering a more direct and systematic framework compared to traditional approaches.

ABSTRACT

Open quantum system theory is fundamental for a proper description of quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto-mechamics. Quantum brownian motion which is analytically described by the Hu-Paz-Zhang (HPZ) model represents one of the most important examples of an open quantum system. In this paper we propose a novel derivation of its master equation, based on the Heisenberg picture. We provide an exact and analytic equation both for the operators as well as for the states. The result, equivalent to the one derived originally by HPZ, is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the environment and allows to compute the time evolution of physically relevant quantities in a much easier way, since it is directly expressed in terms of the spectral density. An example is explicitly studied.

Motivation & Objective

• To develop a new derivation of the master equation for quantum Brownian motion using the Heisenberg picture framework.
• To express the time evolution of operators and states exactly and analytically in terms of the spectral density.
• To provide a computationally more efficient method for calculating physically relevant quantities in open quantum systems.
• To generalize the HPZ model's results by making the spectral density the central parameter, independent of coupling strength.

Proposed method

• Derive the master equation using the Heisenberg picture formalism, focusing on the time evolution of system operators.
• Employ the spectral density as the fundamental input, ensuring the derivation remains valid regardless of system-environment coupling strength.
• Utilize exact operator dynamics to obtain a closed-form equation for the time evolution of observables.
• Establish equivalence to the original HPZ master equation while expressing it in a form more amenable to analytical and numerical computation.
• Demonstrate the method through an explicit example, illustrating its practical utility.

Experimental results

Research questions

• RQ1 How can the master equation for quantum Brownian motion be derived directly in the Heisenberg picture?
• RQ2 Can the time evolution of system operators be expressed exactly and analytically using only the spectral density?
• RQ3 Does this approach yield a more efficient computational framework for predicting physical observables in open quantum systems?
• RQ4 To what extent is the derived master equation independent of the coupling strength between system and environment?

Key findings

• The derived master equation is mathematically equivalent to the original HPZ result but expressed directly in terms of the spectral density.
• The method provides an exact and analytic description of operator time evolution in the Heisenberg picture, valid for arbitrary coupling strength.
• The formalism allows for more straightforward computation of physically relevant quantities due to its direct dependence on the spectral density.
• An explicit example demonstrates the practical applicability and computational advantage of the proposed approach.