Finite energy-resolved GKSL thermal model

Let $d\ge 1$ and let $\mathcal{H}={\mathbb{C}}^d$ with its standard complex inner product. Let $H=H^{\ast }$ on $\mathcal{H}$ be a Hamiltonian matrix with entries in joules, let $\mathrm{\hslash }>0$ have units joule-seconds, and let $S_1,\dots ,S_m$ be a finite family of dimensionless Hermitian matrices, where $m\ge 0$. Write the finite spectral decomposition $H={\sum }_{E\in \mathrm{spec}(H)}E{P}_E$, where $P_E$ are pairwise orthogonal spectral projections and ${\sum }_E{P}_E=I_d$. The Bohr-frequency set is

${\mathrm{\Omega }}_H:=\{(E^{\mathrm{\prime }}-E)\mathrm{/}\mathrm{\hslash }:E,E^{\mathrm{\prime }}\in \mathrm{spec}(H)\}\subset \mathbb{R},$

whose frequencies have units seconds${}^{-1}$. For each $\omega \in {\mathrm{\Omega }}_H$ and $1\le a\le m$, define the dimensionless energy-resolved coupling

$S_a(\omega ):={\sum }_{\begin{array}{c}E,E^{\mathrm{\prime }}\in \mathrm{spec}(H)\\ E^{\mathrm{\prime }}-E=\mathrm{\hslash }\omega \end{array}}{P}_E{S}_a{P}_{E^{\mathrm{\prime }}}\mathrm{.}$

Let $\beta \in \mathbb{R}$ have units joules${}^{-1}$. For each $\omega \in {\mathrm{\Omega }}_H$, choose a finite rate matrix $\mathrm{\Gamma }(\omega )=({\gamma }_{ab}(\omega ){)}_{a,b=1}^m$, with entries in seconds${}^{-1}$, that is positive semidefinite in the explicit sense that $\mathrm{\Gamma }(\omega )=\mathrm{\Gamma }(\omega {)}^{\ast }$ and $v^{\ast }\mathrm{\Gamma }(\omega )v\ge 0$ for every $v\in {\mathbb{C}}^m$, and require the thermal balance relation

${\gamma }_{ab}(-\omega )=e^{-\beta \mathrm{\hslash }\omega }{\gamma }_{ba}(\omega )\mathrm{.}$

Let $H_{\mathrm{L}\mathrm{S}}=H_{\mathrm{L}\mathrm{S}}^{\ast }$ be a Hermitian matrix with units joules satisfying $[H_{\mathrm{L}\mathrm{S}},H]=0$. For dimensionless $X\in M_d(\mathbb{C})$, the finite energy-resolved generator is the linear map

$\mathcal{L}(X)=-\frac{i}{\mathrm{\hslash }}[H+H_{\mathrm{L}\mathrm{S}},X]+{\sum }_{\omega \in {\mathrm{\Omega }}_H}{\sum }_{a,b=1}^m{\gamma }_{ab}(\omega )(S_b(\omega )X{S}_a(\omega {)}^{\ast }-\frac{1}{2}\{S_a(\omega {)}^{\ast }{S}_b(\omega ),X\}),$

where $[A,B]=AB-BA$, $\{A,B\}=AB+BA$, and $A^{\ast }$ is the conjugate transpose. The finite sums define a linear map with units seconds${}^{-1}$. This is an adopted finite effective model specification; the definition asserts no microscopic weak-coupling convergence or physical applicability.

Finite matrix spectral and state calculus

The sign convention makes $S_a(\omega )$ lower the system energy by $\mathrm{\hslash }\omega$ when $\omega >0$: it maps the $E^{\mathrm{\prime }}$ eigenspace to the $E$ eigenspace with $E^{\mathrm{\prime }}-E=\mathrm{\hslash }\omega$. The rate-matrix transpose in the thermal balance relation is tied to the displayed placement of $S_b(\omega )$ and $S_a(\omega {)}^{\ast }$ in the generator. No bath limit is part of this mathematical definition.