of eigenvalues of $H$. Given $$H\ket{e_k}=\lambda_k\ket{e_k},$$ the matrices $R_{kk'}=\ket{e_k}\bra{e_{k'}}$ then provide the right eigenmodes of the Lindblad equation $${\mathcal{L}}(R_{kk'})=-i(\lambda_{k}-\lambda_{k'})R_{kk'},$$ with the corresponding left eigenmodes also given by $L_{kk'}=\ket{e_k}\bra{e_{k'}}$. In this context, our results of Sec. \[secresqtheory\] can be modified to handle degenerate modes with ${\mathcal{L}}(R_i)=0$, as is the case for closed quantum systems, analogously to the procedure described at the end of Sec. \[StatStateResets\]. The resulting modification to Eq.  for coherent dynamics contains a sum over the non-zero eigenvalue modes as before, with the unique stationary density matrix replaced by a diagonal matrix in the energy eigenbasis of probabilities for the reset state to be measured in each eigenstate of the Hamiltonian.

Using the outer product structure of the eigenstates, we can rewrite this sum more compactly as the matrix product $$\rho_{ss}^{\Gamma}=E\Lambda^\Gamma E^\dagger,$$ where $E$ is the matrix of eigenvectors defined by $HE=E\Lambda$ with $\Lambda_{ii}=\lambda_i$, and $\Lambda^\Gamma$ has elements defined by $$\Lambda^\Gamma_{ij}=\frac{\Gamma\left\langle\psi|e_j\right\rangle\left\langle e_i|\psi\right\rangle}{\Gamma+i(\lambda_i-\lambda_j)}.$$ These matrices can be efficiently constructed numerically and used to calculate the stationary state of a closed system after the addition of resets.

For the coherent hopping model we have $$\ket{e_n}=\frac{1}{\sqrt{L}}\sum_{x=1}^{L}e^{i\frac{2\pi n}{L}(x-1)}\ket{x},$$ and $\lambda_k=2\Omega \text{cos}(2\pi n/L)$. Choosing resets to the state $\ket{0}$ for a chain of $L=2001$ 