Exact Master Equation for Quantum Brownian Motion with Generalization to Momentum-Dependent System-Environment Couplings

TL;DR

This paper extends quantum Brownian motion models to include momentum-dependent system-environment couplings, deriving an exact master equation that improves understanding of initial correlations and renormalization effects in quantum systems.

Contribution

The paper introduces a generalized QBM model with momentum-dependent couplings and derives its exact master equation, addressing limitations of the conventional model and clarifying initial correlation issues.

Findings

Derivation of an exact master equation for the generalized QBM.

Reproduction of the Hu-Paz-Zhang master equation as a special case.

Identification of a momentum-dependent potential in the renormalized Hamiltonian.

Abstract

In this paper, we generalize the quantum Brownian motion to include momentum-dependent system-environment couplings. The conventional QBM model corresponds to the spacial case $W_k = V_k$. The generalized QBM is more complicated but the generalization is necessary. This is because the particle transition and the pair production between the system and the environment represent two very different physical processes, and usually cannot have the same coupling strengths. Thus, the conventional QBM model, which is well-defined at classical level, is hardly realized in real quantum physical world. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized Brownian particle Hamiltonian after traced out all the environmental states induced naturally a momentum-dependent potential, which also shows the necessity of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and and initial correlated states, the issues about the initial jolt which is a long-stand problem in the Hu-Paz-Zhang master equation is also re-examined. We find that the so-called "initial jolt", which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing do to with the initial decoupled state. The new exact master equation for the generalized QBM also has the potential applications to photonics quantum computing.