Conditional specified thermal bath relaxation

Assume full AC, the adopted finite effective-bath model Specified effective GKSL bath and instrument assumption on its physical interval $0\le t\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}$, and the connected finite Gibbs-jump hypotheses of Connected Gibbs jump generators satisfy detailed balance. For an initial density $\rho$, let ${\rho }_t=e^{t\mathcal{L}}\rho$ denote the mathematical all-time continuation of the fixed finite Schrödinger generator, and let $\sigma =Z_{\beta }^{-1}{e}^{-\beta H_{\mathrm{s}}}$. If $d>1$, the finite detailed-balance gap $\gamma >0$ gives the conditional model prediction for every $t\ge 0$ in that continuation

$\mathrm{\parallel }{\rho }_t-\sigma {\mathrm{\parallel }}_1\le \mathrm{\parallel }{\sigma }^{-1\mathrm{/}2}(\rho -\sigma ){\mathrm{\parallel }}_2{e}^{-\gamma t}\mathrm{.}$

Under the adopted physical model this bound is applicable only for $0\le t\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}$; its all-time form is a mathematical consequence of continuing the same fixed generator, not a claim about physical evolution for $t>t_{\mathrm{r}\mathrm{e}\mathrm{s}}$.

For the two-level specialization, let $H_{\mathrm{s}}=\mathrm{\Delta }Z\mathrm{/}2$, $\mathrm{\Delta }>0$ joules, $Z=\mathrm{\mid }0⟩⟨0\mathrm{\mid }-\mathrm{\mid }1⟩⟨1\mathrm{\mid }$, and $X=\mathrm{\mid }0⟩⟨1\mathrm{\mid }+\mathrm{\mid }1⟩⟨0\mathrm{\mid }$, $Y=-i\mathrm{\mid }0⟩⟨1\mathrm{\mid }+i\mathrm{\mid }1⟩⟨0\mathrm{\mid }$. Adopt downward jump $\sqrt{a}\mathrm{\mid }1⟩⟨0\mathrm{\mid }$ with $a>0$ seconds${}^{-1}$, upward jump $\sqrt{b}\mathrm{\mid }0⟩⟨1\mathrm{\mid }$ with $b\mathrm{/}a=e^{-\beta \mathrm{\Delta }}$, and optionally a dephasing jump $\sqrt{{\gamma }_{\varphi }\mathrm{/}2}Z$ with ${\gamma }_{\varphi }\ge 0$ seconds${}^{-1}$. Put

$\omega :=\mathrm{\Delta }\mathrm{/}\mathrm{\hslash },{\gamma }_1:=a+b,{\gamma }_2:=\frac{a+b}{2}+{\gamma }_{\varphi },z_{\mathrm{e}\mathrm{q}}:=\frac{b-a}{a+b}=-\tanh (\beta \mathrm{\Delta }\mathrm{/}2)\mathrm{.}$

For every $t\ge 0$ in the mathematical continuation, the exact model expectations and coherence satisfy

$⟨Z{⟩}_t=z_{\mathrm{e}\mathrm{q}}+e^{-{\gamma }_{1}t}(⟨Z{⟩}_0-z_{\mathrm{e}\mathrm{q}}),{\rho }_{01}(t)=e^{-({\gamma }_2+i\omega )t}{\rho }_{01}(0)\mathrm{.}$

Equivalently, $e^{tG}X=e^{-{\gamma }_{2}t}(X\cos \omega t-Y\sin \omega t)$ and $e^{tG}Z=z_{\mathrm{e}\mathrm{q}}I+e^{-{\gamma }_{1}t}(Z-z_{\mathrm{e}\mathrm{q}}I)$. For $d=1$, $\rho =\sigma$ and relaxation is trivial. These are conditional model deductions, not experimental findings or claims that a physical bath realizes the postulate.

The qubit formulas above are stated for the mathematical continuation; their physical model applicability is limited to $0\le t\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}$.

The positive-temperature relation fixes the rate ratio, while the numerical values of $a$, $b$, and ${\gamma }_{\varphi }$ remain model inputs. No observation, calibration uncertainty, or empirical agreement is claimed here.