[Paper Review] Quantum and statistical mechanics in open systems: theory and examples

David Zueco|ArXiv.org|Aug 25, 2009

Advanced Thermodynamics and Statistical Mechanics84 references3 citations

TL;DR

This thesis develops a unified theoretical framework for open quantum systems using Hamiltonian-based dissipation, master equations, and phase-space methods, demonstrating quantum-to-classical crossovers in Brownian motion and superparamagnetic relaxation. It derives thermodynamically consistent master equations, applies continued-fraction techniques to compute linear and nonlinear responses, and provides analytical expressions for decoherence times, magnetization, and current rectification in periodic and spin systems.

ABSTRACT

Using the system-bath model Hamiltonian this thesis covers the equilibrium and out of equilibrium properties of quantum open systems. Topics included are the calculation of thermodynamical quantities of open systems, derivation of quantum master equations, phase space and numerical methods and Linear and non Linear Response Theory. Applications are the transport in periodic potentials and the dynamics of spins.

Motivation & Objective

• To establish a Hamiltonian-based theory of dissipation in open quantum systems, resolving irreversibility paradoxes through bath-of-oscillators formalism.
• To derive and validate quantum master equations that are thermodynamically consistent, with stationary solutions matching quantum statistical mechanics.
• To develop and apply continued-fraction methods for solving master equations, enabling computation of linear and nonlinear responses in complex systems.
• To analyze quantum transport and relaxation in periodic potentials and spin systems, including rectified currents and decoherence effects.
• To provide analytical expressions for key observables—magnetization, relaxation times, and susceptibility—applicable to real materials like Mn12.

Proposed method

• Formalism based on the Caldeira–Leggett bath-of-oscillators model to describe system-bath coupling and derive quantum master equations.
• Use of Wigner phase-space representation to map density matrix dynamics into phase-space Fokker–Planck-like equations.
• Application of Bopp operators and Stratonovich postulates to extend phase-space methods to spin systems with large spin quantum number S.
• Employment of continued-fraction techniques to solve recurrence relations arising from master equations, enabling analytical solutions for stationary states and linear response.
• Derivation of the Caldeira–Leggett master equation in Wigner phase space and its solution via continued fractions for periodic potentials.
• Use of linear response theory (LRT) and Bloch equations to compute relaxation times T1 and T2, and nonlinear susceptibility for tunneling detection.

Experimental results

Research questions

• RQ1 How can quantum dissipation be consistently derived from a Hamiltonian framework while preserving irreversibility and thermodynamic consistency?
• RQ2 What are the quantum corrections to classical rectified current in ratchet potentials, and how do they arise from tunneling and wave reflection?
• RQ3 How can continued-fraction methods be generalized to solve master equations for spin systems with S ≫ 1?
• RQ4 What is the role of system-bath entanglement in modifying equilibrium magnetization in quantum superparamagnets?
• RQ5 How do nonlinear susceptibility and transverse response reveal tunneling dynamics in single-molecule magnets like Mn12?

Key findings

• The master equation derived from the bath-of-oscillators model is thermodynamically consistent, with a stationary solution matching quantum statistical mechanics, and explicitly depends on damping in the quantum regime.
• Wigner phase-space solutions for a damped particle in a cosine potential show quantum interference effects, including negative quasi-probability regions, persisting at long times.
• The rectified current in a ratchet potential receives first-order quantum corrections due to tunneling and wave reflection, deviating from the classical limit as damping decreases.
• A continued-fraction solution of the Caldeira–Leggett equation in phase space enables analytical computation of the linear susceptibility and decoherence times in nonlinear potentials.
• For superparamagnets, the longitudinal linear susceptibility is derived in closed form for arbitrary spin S, with relaxation times T1 and T2 expressible analytically.
• The transverse susceptibility is expressed in closed form, providing a direct link to experimental absorption spectra in Mn12-type molecules.