ULE jump construction under independent bath hypotheses

Assume Countable Choice for the cited kernel-operator and static-jump constructions. For a finite-dimensional system, take $H=H_S(t)+H_B+\sqrt{\gamma }{\sum }_{\alpha =1}^n{X}_{\alpha }{B}_{\alpha }$, with Hermitian $X_{\alpha },B_{\alpha }$, $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$, $\gamma >0$, and units $\mathrm{\hslash }=1$. Assume a centered stationary Gaussian bath: $[H_B,{\rho }_B]=0$, $\mathrm{Tr}({\rho }_B{B}_{\alpha })=0$, and bath correlation functions are determined by two-point correlations by Wick pairings. The preparation is ${\rho }_{SB}(t_0)={\rho }_S(t_0)\otimes {\rho }_B$. For the ULE approximation use the source's remote-past formulation $t_0\to -\mathrm{\infty }$, not arbitrary newly factorized preparation. Set $C_{\alpha \beta }(t-s)={\mathrm{Tr}}_B({\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B)$, with ${\tilde{B}}_{\alpha }(t)=e^{i{H}_{B}t}{B}_{\alpha }{e}^{-i{H}_{B}t}$. Require the spectral and time-kernel regularity of the named kernel contract. Write its kernel as $h=g$, so $h(\omega )=\sqrt{J(\omega )\mathrm{/}(2\pi )}$, $\mathrm{\Gamma }=4\gamma (\int \mathrm{\parallel }h(t){\mathrm{\parallel }}_{2,1}dt{)}^2$ and $\tau =\int \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }h(t){\mathrm{\parallel }}_{2,1}dt\mathrm{/}\int \mathrm{\parallel }h(t){\mathrm{\parallel }}_{2,1}dt$, and assume $\mathrm{\Gamma }\tau \ll 1$ as the weak Markov approximation regime.

With $U(t)$ the system propagator and ${\tilde{X}}_{\alpha }(s)=U(s{)}^{†}{X}_{\alpha }U(s)$, the imported ULE coefficients are ${\tilde{L}}_{\lambda }(t)=\sqrt{\gamma }{\sum }_{\alpha }\int h_{\lambda \alpha }(t-s){\tilde{X}}_{\alpha }(s)ds$ and $\tilde{\mathrm{\Lambda }}(t)=\frac{\gamma }{2i}{\sum }_{\alpha \beta }\int \text{\hspace{0.17em}⁣}\int {\tilde{X}}_{\alpha }(s){\tilde{X}}_{\beta }(s^{\mathrm{\prime }}){\varphi }_{\alpha \beta }(s-t,s^{\mathrm{\prime }}-t)dsd{s}^{\mathrm{\prime }}$, where $\varphi (u,v)=h(u)h(-v)\mathrm{sgn}(u-v)$. These are the coefficients of a Lindblad-form approximation; this statement does not assert exact reduced dynamics satisfy a residual-free Lindblad equation.

For static $H_S\mathrm{\mid }m⟩=E_m\mathrm{\mid }m⟩$, assume the Fourier identities at the required Bohr frequencies and the principal values below exist. Then $L_{\lambda }=2\pi \sqrt{\gamma }{\sum }_{m,n,\alpha }{h}_{\lambda \alpha }(E_n-E_m)(X_{\alpha }{)}_{mn}\mathrm{\mid }m⟩⟨n\mathrm{\mid }$. With the erratum's corrections, $\mathrm{\Lambda }={\sum }_{\alpha \beta ,lmn}{f}_{\alpha \beta }(E_l-E_m,E_n-E_l)(X_{\alpha }{)}_{ml}(X_{\beta }{)}_{ln}\mathrm{\mid }m⟩⟨n\mathrm{\mid }$, where $f(E_1,E_2)=-2\pi \gamma \mathrm{PV}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }^{-1}h(\omega -E_1)h(\omega +E_2)d\omega$.

For finite $t_0$ and $t>t_0$, the source separately bounds the BR preparation-tail correction acting on the reduced density matrix by $\mathrm{\parallel }{D}_T(t)(\rho (t))\mathrm{\parallel }\le \mathrm{\Gamma }\tau \mathrm{/}(t-t_0)$, where the norm is the spectral operator norm and $D_T$ is the tail map defined in External input. For any $\epsilon >0$, on an observation window $t\ge t_{\ast }>t_0$, the explicit sufficient condition $t_{\ast }-t_0\ge 1\mathrm{/}(\epsilon \mathrm{\Gamma })$ ensures $\mathrm{\parallel }{D}_T(t)(\rho (t))\mathrm{\parallel }\le \epsilon {\mathrm{\Gamma }}^2\tau$. This identifies a preparation condition; it does not transfer the source's exact remote-past ULE residual constant to finite preparation or provide an all-time trajectory error bound.

Imported input 1: Nathan–Rudner's stationary Gaussian, normalized finite-channel model in its remote-past preparation formulation yields the coefficients (28)–(29) and static formula (33). Define ${\mathrm{\Delta }}_B(t,s)$ by importing exactly the Born superoperator's defining expression in Appendix A4, Eq. (A25). The conventions included in this definition are left/right multiplication ${\widehat{A}}^l(M)=AM$, ${\widehat{A}}^r(M)=MA$ from (A2), ${\nu }_l=1$, ${\nu }_r=-1$ following (A6), and the bath superoperator correlations $J_{\alpha \beta }^{mn}(t-s)$ defined in (A9). The source's time-domain $J(t-s)$ is our $C(t-s)$, its system and bath operators in Appendix A are our interaction-picture ${\tilde{X}}_{\alpha }$ and ${\tilde{B}}_{\alpha }$, and its $g$ is our $h$. The definition retains the same system–bath model, normalized couplings, Gaussian correlations and factorized preparation hypotheses stated above; it does not infer this map from a norm bound. All superoperator norms here mean (A3), namely $\mathrm{\parallel }T\mathrm{\parallel }={\sup }_{\mathrm{\parallel }M\mathrm{\parallel }\le 1}\mathrm{\parallel }T(M)\mathrm{\parallel }$ with spectral operator norms on matrices, and $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}={\sum }_{\alpha \beta }\mathrm{\mid }{A}_{\alpha \beta }\mathrm{\mid }$ is the source's entrywise norm. For finite $t_0$ and $t>t_0$, (A38) defines $D_T(t)=-{\int }_{-\mathrm{\infty }}^{t_0}{\mathrm{\Delta }}_B(t,s)ds$, and Appendix A6 gives $\mathrm{\parallel }{\mathrm{\Delta }}_B(t,s)\mathrm{\parallel }\le 4\gamma \mathrm{\parallel }C(t-s){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}$ and $\mathrm{\parallel }{D}_T(t)(\rho (t))\mathrm{\parallel }\le \mathrm{\Gamma }\tau \mathrm{/}(t-t_0)$, the latter after (A41) using (B2). This imports the source-defined map and density-matrix estimate, including its existence in that model. The convolution identity immediately before (B4), the resulting norm inequality (B4), and moment relation (B2) apply in this model. Source: Universal Lindblad equation for open quantum systems, Sec. IIIC, (24)–(29), Sec. IVA, (33), Appendix A opening and A1–A2, (A2)–(A3), signs following (A6), (A9), Appendix A4, (A25), Appendix A6, (A38)–(A41), and Appendix B, (B2)–(B4). The microscopic derivation and finite-preparation tail bound are imported, not reconstructed here.

Imported input 2: The static Lamb formula and coefficient above use the corrected energy ordering, sign and coupling factor. Source: Erratum, p. 1, corrected expressions (1)–(2), and the time-kernel correction (5). Principal-value existence remains an explicit hypothesis.

The four cited local lemmas establish kernel symmetry and rate identities, coefficient well-definedness, static evaluation and the correlation-moment estimate. The finite-preparation tail bound is imported directly from Appendix A6, and [F6] gives its scalar waiting-time consequence. They do not reconstruct the microscopic ULE derivation. The supplier and source use the same convention, $h(\omega )=g(\omega )=\sqrt{J(\omega )\mathrm{/}(2\pi )}$; the new letter only distinguishes frequency and time-kernel notation in these intermediate lemmas. The static Lamb formula follows the independently read erratum. No Gaussianity, factorization, memory bound or weak-coupling regime for a particular photon model follows merely from its vocabulary or spectral cutoff.