Static ULE formulas and principal-value kernel

PDF p. 10: Eqs. (S2.13)–(S2.14)

Assume the Axiom of Choice and Countable Choice. Work in units $\mathrm{\hslash }=1$ with a nonzero finite-dimensional complex system Hilbert space ${\mathcal{H}}_S$ and a complex bath Hilbert space ${\mathcal{H}}_B$. Let $H_S$ be a static Hermitian system Hamiltonian, $H_B$ a densely defined self-adjoint bath Hamiltonian, and $B_{\alpha }$ densely defined symmetric bath coupling operators. Take the total Hamiltonian $H_S+H_B+\sqrt{\gamma }{\sum }_{\alpha =1}^n{X}_{\alpha }{B}_{\alpha }$, where $n$ is finite, $\gamma >0$, the $X_{\alpha }$ are Hermitian system operators and $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$. Identity tensor factors are understood, and a self-adjoint realization of this total Hamiltonian on the system–bath Hilbert space is part of the model data.

Let ${\rho }_B$ be a bounded trace-class bath operator, with ${\rho }_B^{\ast }={\rho }_B$, $⟨{\rho }_{B}v,v⟩\ge 0$ for every $v\in {\mathcal{H}}_B$, and ${\mathrm{Tr}}_B{\rho }_B=1$. Put $U_B(t)=e^{-it{H}_B}$ using self-adjoint functional calculus, and assume strong stationarity $U_B(t){\rho }_B{U}_B(t{)}^{\ast }={\rho }_B$ for all real $t$; this is the meaning of the formal notation $[H_B,{\rho }_B]=0$ here. Define ${\tilde{B}}_{\alpha }(t)=U_B(t{)}^{\ast }{B}_{\alpha }{U}_B(t)$ on its natural domain $U_B(t{)}^{\ast }D(B_{\alpha })$.

For every finite ordered product $W={\tilde{B}}_{{\alpha }_1}(t_1)\cdots {\tilde{B}}_{{\alpha }_k}(t_k)$ used in a bath moment, require that the composition $W{\rho }_B$ on its natural domain is densely defined and closable and that its closure is a bounded trace-class operator on ${\mathcal{H}}_B$. Impose the same requirement on ${\rho }_B{B}_{\alpha }$ for each $\alpha$. All traces of these products mean the basis-independent trace of their trace-class closures. Assume a centered stationary Gaussian bath: ${\mathrm{Tr}}_B({\rho }_B{B}_{\alpha })=0$, its two-point moments depend only on time differences, and its higher moments obey Wick pairings of the two-point functions. Use the factorized preparation ${\rho }_{SB}(t_0)={\rho }_S(t_0)\otimes {\rho }_B$ with a system density operator ${\rho }_S(t_0)$, in the remote-past formulation $t_0\to -\mathrm{\infty }$, and the weak Markov regime $\mathrm{\Gamma }{\tau }_B\ll 1$ of the ULE contract.

Let $C_{\alpha \beta }(t-s)={\mathrm{Tr}}_B({\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B)$ in this closure convention, and $J(\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}C(t)e^{i\omega t}dt$ entrywise. Require integrable entries of $C$, positive semidefinite Hermitian $J(\omega )$, integrable entries of $g(\omega )=[J(\omega )\mathrm{/}(2\pi ){]}^{1\mathrm{/}2}$, and finite nonzero zeroth and finite first moments of $\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}$, where $g(t)=\int g(\omega )e^{-i\omega t}d\omega$, $\mathrm{\parallel }A{\mathrm{\parallel }}_{2,1}={\sum }_b({\sum }_a\mathrm{\mid }{A}_{ab}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, $\mathrm{\Gamma }=4\gamma (\int \mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt{)}^2$, and ${\tau }_B=\int \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt\mathrm{/}\int \mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$. Assume the pointwise Fourier identities at the required Bohr frequencies and the principal values below exist.

Choose an orthonormal eigenbasis $H_S\mathrm{\mid }m⟩=E_m\mathrm{\mid }m⟩$ and write $X_{mn}^{(\alpha )}=⟨m\mathrm{\mid }{X}_{\alpha }\mathrm{\mid }n⟩$. The static ULE jump and Lamb-shift coefficients are $L_{\lambda }=2\pi \sqrt{\gamma }{\sum }_{m,n,\alpha }{g}_{\lambda \alpha }(E_n-E_m)X_{mn}^{(\alpha )}\mathrm{\mid }m⟩⟨n\mathrm{\mid }$ and $\mathrm{\Lambda }={\sum }_{l,m,n,\alpha ,\beta }{f}_{\alpha \beta }(E_l-E_m,E_n-E_l)X_{ml}^{(\alpha )}{X}_{ln}^{(\beta )}\mathrm{\mid }m⟩⟨n\mathrm{\mid }$, with $f(E_1,E_2)=-2\pi \gamma \mathrm{PV}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\omega }^{-1}g(\omega -E_1)g(\omega +E_2)d\omega$. The product in the integrand is matrix multiplication. Here $\mathrm{PV}$ is the symmetric Cauchy principal value at zero, with convergence at infinity also required.

The article’s Eq. (S2.13) gives the jump formula. Its Eq. (S2.14) instead states the first Lamb-kernel argument as $E_m-E_l$ and defines the kernel with prefactor $2\pi$; this original formulation is preserved here for attribution. The displayed conditional deduction follows the independent ULE supplier’s erratum-corrected formula, not the article as proof.