Abstract
We introduce an approach to characterize the dynamics of disordered quantum networks. Each quantum element (i.e., each node) of the network experiences the other nodes as an effective environment that can be self-consistently represented by a Feynman-Vernon influence functional. For networks having the topology of locally treelike graphs, these Feynman-Vernon (FV) functionals can be determined by a new version of the cavity or belief propagation (BP) method. Here, we find the fixed point solution of this version of BP for a network of uniform quantum harmonic oscillators. Then, we estimate the effects of the disorder in these networks within the replica symmetry ansatz. We show that over a large time interval, at small disorder, the real part of the FV functional induces decoherence and classicality while at sufficiently large disorder the Feynman-Vernon functional tends to zero and the coherence survives, signaling in a time setting, the onset of an Anderson's transition.
Highlights
This work proposes an approach to describe real-time dynamics of quantum networks
In this work we have introduced a real-time version of the quantum cavity method
We have shown that when all systems in a network are harmonic oscillators interacting linearly, this real-time quantum cavity can be represented as transforms of Feynman-Vernon kernels
Summary
The cavity method in its most widely used incarnation is a way to simultaneously compute all marginals of a GibbsBoltzmann distribution when the interaction graph has no short loops [1,2,3]. In our generalization pi(Xi, Yi ), which has to depend on two variables of the same type, is the probability amplitude of system i, and ma→i(Xi, Yi ) are the Feynman-Vernon influence functionals from integrating out all variable in the network starting from a. In the real-time quantum cavity method which we introduce m j→i and n j→i are very high-dimensional objects, and the update step (n to m iteration) is computationally expensive. For harmonic networks this aspect is mitigated as one can use the Feynman-Vernon theory with closedform expressions for the actions in the influence functionals. More advanced approximations have been explored, in both discrete and continuous time [33,34,39]
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