arXiv 2602.035342026-02-03quant-ph

开放量子布朗运动的微观推导：用于描述处于势场中的粒子的完全正主方程

Microscopic derivation of a completely positive master equation for the description of Open Quantum Brownian Motion of a particle in a potential

University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）；North-West University（西北大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；Stellenbosch University（斯坦伦布什大学）

Ayanda Zungu, Ilya Sinayskiy, Francesco Petruccione

备注：: 19 pages, 6 figures

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AI总结：

本文推导了开放量子布朗运动的完全正主方程，揭示了非高斯动力学特性。

摘要：

开放量子布朗运动（OQBM）被引入为离散时间开放量子行走的极限情况。这种极限定义了一种新的量子布朗运动，它结合了布朗粒子的外部和内部自由度。我们考虑一个被弱驱动并被谐振势限制的布朗粒子，并与热浴发生耗散耦合。通过对系统-浴相互作用哈密顿量应用旋转波近似（RWA），我们推导出一个完全正的Born-Markov主方程，用于描述减少的动力学。我们将得到的主方程表示为坐标表示，并利用对快变量的绝热消除，推导出一个完全正的混合量子-经典主方程，该方程定义了OQBM。我们通过初始高斯和非高斯分布的例子来说明所得到的动力学。这两个例子揭示了OQBM动力学在极限行为中出现高斯分布，这与标准OQBM非常接近。借助所获得的OQBM主方程，我们推导出开放布朗漫步者位置分布的第n阶矩和累积量的方程。随后，我们针对各种参数区域对这些方程进行数值求解。值得注意的是，我们发现第三阶累积量非零，表明动力学的内在生成器是非高斯的。

英文摘要：

Open Quantum Brownian Motion (OQBM) was introduced as a scaling limit of discrete-time open quantum walks. This limit defines a new class of quantum Brownian motion, which incorporates both the external and internal degrees of freedom of the Brownian particle. We consider a weakly driven Brownian particle confined in a harmonic potential and dissipatively coupled to a thermal bath. Applying the rotating wave approximation (RWA) to the system-bath interaction Hamiltonian, we derive a completely positive Born-Markov master equation for the reduced dynamics. We express the resulting master equation in the coordinate representation and, utilizing the adiabatic elimination of fast variables, derive a completely positive hybrid quantum-classical master equation that defines OQBM. We illustrate the resulting dynamics using examples of initial Gaussian and non-Gaussian distributions of the OQBM walker. Both examples reveal the emergence of Gaussian distributions in the limiting behavior of the OQBM dynamics, which closely matches that of the standard OQBM. With the help of the obtained OQBM master equation, we derive the equations for the $n$-th moments and the cumulants of the position distribution of the open Brownian walker. We subsequently solve these equations numerically for Gaussian initial distributions across various parameter regimes. Notably, we find that the third-order cumulant is nonzero, indicating that the dynamics' intrinsic generator is non-Gaussian.

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