case to the von Neumann equation, allowing for much larger system sizes to be studied compared to a direct spectral solution of the open quantum problem. We demonstrate this next.

Coherent hopping on a chain {#CoherentHop}
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The first example that we consider to illustrate our results is a simple model of coherent hopping on a closed periodic chain, described by the Hamiltonian $$\begin{aligned}
H=&\gamma\sum_{x=1}^{L-1} \left( \ket{x+1}\bra{x}+\ket{x}\bra{x+1} \right) 
\nonumber \\
&+ \gamma \left( \ket{1}\bra{L}+\ket{L}\bra{1} \right),\end{aligned}$$ and no jump operators, so that the reset-free system is closed. The dynamics generated by this Hamiltonian is similar to that considered in . The main difference is that we consider state-independent resets whereas consider resets induced by continuously measuring the system’s state on a particular site, leading to a single jump operator proportional to the projective measurement on that site.

![(a,b) Magnitude of the stationary density matrix on a log scale. (c,d) Occupation probability for each site of the chain compared with the probability of a classical hopping model with exponential distribution. Plots are for (a,c) $\Gamma=1$ and (b,d) $\Gamma=5$.[]{data-label="CoherentHoppingFig"}](CoherentHopping.pdf){width="1\linewidth"}

For closed quantum systems, the spectrum of the corresponding Lindblad equation is given by all possible outer products of the eigenvectors and corresponding differences 