[Paper Review] On the structure of the exact master equation

Inés de Vega|arXiv (Cornell University)|Mar 31, 2014

Advanced Thermodynamics and Statistical Mechanics15 references4 citations

TL;DR

This paper derives an exact, non-perturbative master equation for open quantum systems with arbitrary system Hamiltonians—including anharmonicities—by analytically averaging over noise trajectories in the stochastic Liouville-von Neumann (SLN) equation. The resulting master equation exhibits a hierarchical structure, providing a criterion to assess the feasibility of master equation approaches based on the relative weight of successive terms, thus offering a non-approximate framework to diagnose convergence issues in SLN simulations.

ABSTRACT

We derive a master equation from the exact stochastic Liouville von-Neumann (SLN) equation \cite{stockburger2002,wallsbook}. The latter depends on two correlated noises and describes exactly the dynamics of an oscillator (which can be either harmonic or present an anharmonicity) coupled to an environment at thermal equilibrium. The newly derived master equation is obtained by performing analytically the average over different noise trajectories. It is found to have a complex hierarchical structure that might be helpful to explain the convergence problems occurring when performing numerically the stochastic average of trajectories given by the SLN equation \cite{koch2008,koch2010}.

Motivation & Objective

• To derive an exact master equation for open quantum systems with arbitrary system Hamiltonians, including anharmonicities, without approximations.
• To resolve persistent convergence problems in stochastic Liouville-von Neumann (SLN) simulations, especially for long times and super-ohmic reservoirs.
• To provide a criterion for determining whether a master equation approach is feasible for a given system based on the relative weights of hierarchical terms.
• To establish a non-approximate framework for reduced dynamics that avoids reliance on projection operator techniques or weak-coupling assumptions.

Proposed method

• Derives the master equation by analytically performing the average over noise trajectories in the exact SLN equation, which describes an oscillator coupled to a thermal bath.
• Uses a hierarchical expansion of the master equation, where terms are organized by increasing complexity involving multiple time integrals.
• Applies a consistency condition between time derivatives and functional derivatives with respect to noise variables to ensure mathematical validity of the derivation.
• Introduces a criterion based on the relative weight of successive terms in the hierarchy to assess the feasibility of truncating the equation for numerical use.
• Extends the formalism to include non-Hermitian coupling operators and general initial states via a stochastic Schrödinger equation (SSE) framework.
• Connects the derived hierarchy to existing stochastic hierarchy methods, suggesting potential for broader applicability to exponential correlation functions and fermionic environments.

Experimental results

Research questions

• RQ1 Can an exact master equation be derived for open quantum systems with anharmonic Hamiltonians without approximations?
• RQ2 Why do stochastic Liouville-von Neumann (SLN) simulations often fail to converge, especially for long times and super-ohmic reservoirs?
• RQ3 Is there a systematic way to determine whether a master equation approach is feasible for a given open quantum system?
• RQ4 How can the hierarchical structure of the derived master equation inform the truncation of the equation while preserving non-approximate accuracy?
• RQ5 Can the formalism be generalized to include non-Hermitian coupling operators or arbitrary initial states of the environment?

Key findings

• The derived master equation is exact and non-perturbative, valid for arbitrary system Hamiltonians, including those with anharmonicities.
• The master equation takes the form of an infinite hierarchy of terms, with increasing complexity involving multiple time integrals.
• The relative weight of successive terms in the hierarchy determines the feasibility of truncation: if higher-order terms grow, the master equation approach is not suitable.
• If the weights of higher-order terms diminish, the equation can be truncated with a well-defined error, yielding a non-approximate description of the reduced dynamics.
• The method provides a diagnostic tool for SLN simulations, explaining convergence failures in terms of the hierarchical structure of the exact dynamics.
• The formalism can be extended to stochastic Schrödinger equations with general initial states and non-Hermitian coupling operators, broadening its applicability.