Dissipative Quantum Dynamics in Static Network with Different Topologies

TL;DR

This work addresses how static network topology influences dissipative quantum dynamics in a quantum Ising spin network coupled to a thermal bath. It combines an exact Lindblad master equation treatment for small networks across different topologies with a mean-field approximation to extend the analysis to larger, more homogeneous networks, linking topology to quantum coherence and decoherence. The authors show that decoherence times and the persistence of coherence depend on topology, notably the mean degree $\bar{k}$ and degree disparity $\overline{\Delta k^2}$, with higher $\bar{k}$ and lower disparity generally prolonging coherence; they also introduce a scalable MF framework where observables depend on an effective dimensionality $z$. These results offer a principled route for topology-driven control of dissipative quantum dynamics and suggest applications to complex systems, including social, biological, and condensed-m matter contexts, while highlighting limitations related to finite-size effects and heterogeneity.

Abstract

We investigate the dissipative dynamics of quantum population and coherence among different network topologies of a quantum network using a quantum spin model coupled to a thermal bosonic reservoir. Our study proceeds in two parts. First, we analyze a small network of Ising spins embedded in a large dissipative bath, modeled via the Lindblad master equation, where temperature arises naturally from system-bath coupling. This approach reveals how network topology shapes quantum dissipative dynamics, providing a basis for controlling quantum coherence through tailored network structures. Second, we propose a mean-field approach that extends the network to larger scales and captures dissipative dynamics in large-scale networks, connecting network topology to quantum coherence in complex systems and revealing the sensitivity of quantum coherence to network structure. Our results highlight how dissipative quantum dynamics depend on network topology, providing insight into the coherent dynamics of entangled states in networks. These results may be extended to dynamics in complex systems such as opinion propagation in social models, epidemiology, and various condensed-phase and biological systems.

Figures (9)

• Figure 1: A Ising spin network immersed in the dissipative bath.
• Figure 2: Ohmic spectral density with strength $\eta=0.4$ and cut-off $\omega_c=1.2$ in exponential form (blue) and in Drude-Lorentz form (red)
• Figure 3: Superpostion of the spin states
• Figure 4: Time evolution of the diagonal part of the density matrix (population) of 6 different spin states on four different topologies: (a) a network with 4 nodes, 5 edges, and degree disparity $\overline{\Delta k^2}=0.33$; (b) a network with 4 nodes, 4 edges, and degree disparity $\overline{\Delta k^2}=0.67$; (c) a network with 7 nodes, 13 edges, and degree disparity $\overline{\Delta k^2}=1.24$; (d) a network with 7 nodes, 7 edges, and degree disparity $\overline{\Delta k^2}=1.33$. The parameters are fixed by $J=0.4$, $h=0.1$, $\beta=1.2$ with the spectral density fixed by $\eta=0.4$ and $\omega_c=1.2$. The spin states are labeled by colors matching their evolution trajectory and spin values on each node are labeled up (blue) and down (red).
• Figure 5: The spin-spin correlation $\langle\sigma_i\sigma_j\rangle(t)$ between node $i$ and $j$ on the two different 7-node network (a) and (b). Red node: spin down, blue node: spin up. The system parameters is given by $J=0.4$, $h=0.1$, $\beta=1.2$ with spectral density fixed by $\eta=0.4$ and $\omega_c=1.2$.