We depict the eigenvalues λ cl and λ QM of the Davies generator which corresponds to the Hamiltonian H and coupling operator S as given in (50) for different values of N = 4, …, 200. The eigenvalues and their inverses were plotted for different values of β = 0, 10−3, 10−2, 10−1. λ QM is independent of β and always scales as N −1, whereas the scaling of λ cl does indeed depend on the temperature. For illustration purposes, we have chosen a particular function G(ω) = (1 + e βω)−1, which is motivated from classical Glauber dynamics. The coupling γ was adjusted to obtain for β = 0 the value λ cl = 1. Note that the quantum mechanical function G(ω) depends on the particular bath operators in general and generally differs from the one given above. A good study of different functions can be found in Ref. 2
The figure depicts the construction of the tree T ν from the graph E ν. The tree in this example is obtained by removing the dashed link labeled by (2, 4). The two weights shown in the figure can be computed from the tree structure by ω2(7, 5) = ϕ5 + λ−(7, 5) and ω2(2, 1) = λ−(2, 1) + ϕ1 + ϕ5 + ϕ6 + ϕ7 + 2(λ−(7, 5) + λ−(1, 7) + λ−(1, 6)). This sum is uniquely determined by the tree T ν, and corresponds to summing up all ϕ n that live on the branches lower than the current node measured by the marked vertex (2 in this example). This sum is denoted by for a link (n, l), with the marked vertex m.
Interaction graph for the truncated harmonic oscillator. We consider a finite state space which is truncated at dimension D. The interaction graph depicted corresponds to E 0. All other graphs, i.e., E ν, can obtained from this one.
Interaction graph for a single particle hopping on a line. The transitions are induced by the local densities which become very non-local in the energy eigenbasis. Since the energies are very unevenly spaced, it is only the graph E 0 that is relevant.
Interaction graph for the D-level system with simple transition rules. The figure depicts the transition graph E ν for different blocks ν. We observe, that due to the particular form of S all vanish and we only consider the terms .
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