A Comprehensive Approach to Finite-Bath Open Quantum Systems: Exact Dynamics

TL;DR

This work develops a rigorous framework to obtain exact master equations for finite-bath open quantum systems via a minimal-dissipation construction, then applies it to the central spin model. By extracting the generator from the dynamical map and using Choi-matrix spectral decomposition, it yields a canonical Hamiltonian and a minimal dissipator that together reproduce exact dynamics, including a novel phase-covariant master equation for a dissipative central-spin coupling. It also provides a microscopic derivation of random telegraph noise as pure dephasing arising from stochastic system-bath coupling, with explicit Kraus operators and a clear non-Markovian regime. Finally, the exact dynamics are connected to quantum thermodynamics through heat currents and ergotropy, demonstrating practical utility and offering a foundation for finite-bath quantum technologies and metrology.

Abstract

Here, we develop the exact dynamics of the central spin model, modeling a finite-bath open quantum system. Particularly, two different types of interactions are investigated between the system and the bath: Heisenberg interaction with constant interaction strength, and a stochastic time-dependent interaction. In the former case, a new quantum channel is characterized, while the latter elucidates the microscopic understanding of a very well-known non-Markovian quantum channel. Exact master equations are provided in both scenarios. This is achieved by developing a new technique for obtaining a master equation from the map, making use of the concept of a minimal dissipator. This paves the way for a foundational understanding of finite-bath open quantum systems and a number of novel applications in the vast domain of quantum physics, one of which, implemented here, is in quantum thermodynamics.

Figures (2)

• Figure 1: Variation of the heat current, $\mathcal{J}^\textrm{CS}$, passive heat current $\mathcal{J}^\textrm{CS}_{\rm passive}$, and charging power $\mathcal{P}[\rho_S(t)]$ for the dissipative central spin model, Eq. \ref{['eq_CS_Ham']} in (a), (b), (c), and (d) for different values of mixing probability $p$, and variation of the charging power $\mathcal{P}^\textrm{RTN}[\rho_S(t)]$ with time for the RTN model in (e). In (a)-(d), the parameters are: $\omega = 1.0, \omega_0 = 1.5, N = 50, \epsilon = 0.5$, and $\beta = 0.5$. In (e), the parameters are: $\omega_0 = 1.5, N = 50, \epsilon = 0.5$, and the initial state is taken to be $\ket{\psi_S(0)} = \frac{\sqrt{3}}{2}\ket{0} + \frac{1}{2}\ket{1}$.
• Figure 2: Variation of the matrix elements $h_{jk}(t)$ of the canonical Hamiltonian $H^{\textrm{can}}_\textrm{CS}(t)$ obtained using exact dynamics, and the elements $\tilde{h}_{jk}$ of the canonical Hamiltonian $\tilde{H}^{\textrm{can}}_\textrm{CS}(t)$ obtained using perturbative method. The interaction strength $\lambda = \epsilon/\sqrt{N}$ is varied across the subplots. In (a), $\epsilon = 0.5$, in (b) $\epsilon = 0.1$, and in (c) $\epsilon = 0.05$. The parameters are taken to be: $\omega = 1.0, \omega_0 = 1.5, N = 50$, and $\beta = 0.5$.