Interaction-picture ULE jumps and Lamb shift

PDF p. 10: Eqs. (S2.11)–(S2.12)

Assume the Axiom of Choice and Countable Choice. Work in units $\mathrm{\hslash }=1$ with a finite-dimensional system and Hamiltonian $H=H_S(t)+H_B+\sqrt{\gamma }{\sum }_{\alpha =1}^n{X}_{\alpha }{B}_{\alpha }$, where $n$ is finite, $\gamma >0$, the coupling operators are Hermitian, and $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$. Let ${\mathcal{H}}_B$ be the bath Hilbert space and let ${\rho }_B$ be a density operator on it, meaning a self-adjoint positive trace-class operator with ${\mathrm{Tr}}_B{\rho }_B=1$. Assume that every bath operator product appearing inside a trace in the centering, correlation and Wick-pairing hypotheses is well-defined and trace class, including when bath operators are unbounded. Assume a centered stationary Gaussian bath: its state ${\rho }_B$ commutes with $H_B$, $\mathrm{Tr}({\rho }_B{B}_{\alpha })=0$, and its correlation functions are determined by two-point functions through Wick pairings. Use factorized preparation in the remote-past formulation of the ULE contract and assume its weak Markov regime $\mathrm{\Gamma }{\tau }_B\ll 1$.

With ${\tilde{B}}_{\alpha }(t)=e^{i{H}_{B}t}{B}_{\alpha }{e}^{-i{H}_{B}t}$, put $C_{\alpha \beta }(t-s)={\mathrm{Tr}}_B({\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B)$ and $J(\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}C(t)e^{i\omega t}dt$ entrywise. Assume the kernel contract: the entries of $C$ are integrable, $J(\omega )$ is Hermitian positive semidefinite, $g(\omega )=[J(\omega )\mathrm{/}(2\pi ){]}^{1\mathrm{/}2}$ has integrable entries, and $g(t)={\int }_{\mathbb{R}}g(\omega )e^{-i\omega t}d\omega$ has finite nonzero zeroth and finite first mixed-norm moments. Here $\mathrm{\parallel }M{\mathrm{\parallel }}_{2,1}={\sum }_b({\sum }_a\mathrm{\mid }{M}_{ab}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, $\mathrm{\Gamma }=4\gamma (\int \mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt{)}^2$, and ${\tau }_B=\int \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt\mathrm{/}\int \mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$.

Let $U_S(t)$ be the unitary system propagator and define the interaction-picture coupling ${\tilde{X}}_{\alpha }(s)=U_S(s{)}^{†}{X}_{\alpha }{U}_S(s)$. The jump operators and Lamb-shift Hamiltonian correction in the ULE approximation are ${\tilde{L}}_{\lambda }(t)=\sqrt{\gamma }{\sum }_{\alpha }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}_{\lambda \alpha }(t-s){\tilde{X}}_{\alpha }(s)ds$ and $\tilde{\mathrm{\Lambda }}(t)=\frac{\gamma }{2i}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}ds{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}d{s}^{\mathrm{\prime }}{\sum }_{\alpha \beta }{\tilde{X}}_{\alpha }(s){\tilde{X}}_{\beta }(s^{\mathrm{\prime }}){\phi }_{\alpha \beta }(s-t,s^{\mathrm{\prime }}-t)$, where $\phi (u,v)=g(u)g(-v)\mathrm{sgn}(u-v)$ is a matrix product. These formulas do not assert that the exact reduced dynamics have no residual correction.

The article reference attributes Eqs. (S2.11)–(S2.12). The independent ULE contract supplies the physical input; its microscopic derivation is imported there and is not reconstructed here.