ULE system–bath coupling and short bath memory

PDF p. 9: Eqs. (S2.1)–(S2.2) and following paragraph; PDF p. 11: S3 introductory assumptions

Assume the Axiom of Choice and Countable Choice. In units with $\mathrm{\hslash }=1$, let the system and bath have nonzero complex Hilbert spaces ${\mathcal{H}}_S$ and ${\mathcal{H}}_B$, with $\dim {\mathcal{H}}_S<\mathrm{\infty }$; no finite-dimensionality of ${\mathcal{H}}_B$ is assumed. Inner products are linear in the first argument. The composite space is the Hilbert tensor product ${\mathcal{H}}_S\widehat{\otimes }{\mathcal{H}}_B$: the completion of the algebraic tensor product with $⟨u\otimes v,u^{\mathrm{\prime }}\otimes v^{\mathrm{\prime }}⟩=⟨u,u^{\mathrm{\prime }}⟩⟨v,v^{\mathrm{\prime }}⟩$. Tensor operators act by $(A\otimes D)(u\otimes v)=Au\otimes Dv$ wherever the factors are defined. A density operator here means a bounded self-adjoint trace-class operator $\rho$ with $⟨\rho v,v⟩\ge 0$ for all $v$ and $\mathrm{Tr}\rho =1$. The trace is the basis-independent trace on trace-class operators, including on an infinite-dimensional bath.

Take a self-adjoint system Hamiltonian $H_S:{\mathcal{H}}_S\to {\mathcal{H}}_S$ and a self-adjoint bath Hamiltonian $H_B:D(H_B)\to {\mathcal{H}}_B$. Let $n$ be a positive integer, $\gamma >0$, and let $X_{\alpha }$ be Hermitian operators on ${\mathcal{H}}_S$ with operator norm $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$. Each bath coupling operator $B_{\alpha }:D(B_{\alpha })\to {\mathcal{H}}_B$ is densely defined and Hermitian, meaning $⟨B_{\alpha }u,v⟩=⟨u,B_{\alpha }v⟩$ for $u,v\in D(B_{\alpha })$; bounded operators are included. Require a specified dense subspace $D_B\subseteq D(H_B)\cap {\bigcap }_{\alpha =1}^{n}D(B_{\alpha })$. With $\odot$ denoting the algebraic tensor product and $I_S,I_B$ the identity operators on the named factors, on ${\mathcal{H}}_S\odot D_B$ set $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=\sqrt{\gamma }{\sum }_{\alpha =1}^n{X}_{\alpha }\otimes B_{\alpha }$ and require a specified self-adjoint realization $H$ extending $H_S\otimes I_B+I_S\otimes H_B+H_{\mathrm{i}\mathrm{n}\mathrm{t}}$ there. Thus $H=H_S+H_B+H_{\mathrm{i}\mathrm{n}\mathrm{t}}$ uses the tensor-factor convention and this realization; existence of the realization is an assumption, not a consequence of Hermiticity.

Let ${\rho }_B$ be a bath density operator and write $V_B(t)=e^{-i{H}_{B}t}$ for the unitary group supplied by the self-adjoint bath Hamiltonian. Stationarity means $V_B(t){\rho }_B{V}_B(t{)}^{†}={\rho }_B$ for every real $t$; this is the meaning of the shorthand $[H_B,{\rho }_B]=0$ here, without asserting that the two unbounded products exist everywhere. Define ${\tilde{B}}_{\alpha }(t)=V_B(t{)}^{†}{B}_{\alpha }{V}_B(t)$ on $D({\tilde{B}}_{\alpha }(t))=V_B(t{)}^{†}D(B_{\alpha })$. Require that the products ${\rho }_B{B}_{\alpha }$, $B_{\alpha }{\rho }_B$, and ${\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B$ on their natural composition domains are densely defined and have bounded trace-class extensions for every channel and real $s,t$. A trace of such a product means the trace of that extension, which is unique by density. Require the same trace-existence condition for the higher products used in the Gaussian assumption. Assume the coupling observables are centered, ${\mathrm{Tr}}_B({\rho }_B{B}_{\alpha })={\mathrm{Tr}}_B(B_{\alpha }{\rho }_B)=0$, and Gaussian, meaning their higher correlations are determined by two-point correlations through Wick pairings. Assume the ULE supplier's remote-past factorized preparation ${\rho }_{SB}(t_0)={\rho }_S(t_0)\otimes {\rho }_B$, with ${\rho }_S(t_0)$ a system density operator and the remote-past formulation $t_0\to -\mathrm{\infty }$.

With these trace and domain hypotheses, define $C_{\alpha \beta }(t-s)={\mathrm{Tr}}_B({\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B)$. Adopt the stationary spectral and jump-kernel contract, including its integrability, positive spectral matrix, and finite nonzero zeroth and finite first kernel moments. Write $h$ for its kernel and put $\mathrm{\Gamma }=4\gamma (\int \mathrm{\parallel }h(t){\mathrm{\parallel }}_{2,1}dt{)}^2$ and $\tau =\int \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }h(t){\mathrm{\parallel }}_{2,1}dt\mathrm{/}\int \mathrm{\parallel }h(t){\mathrm{\parallel }}_{2,1}dt$. The short-memory regime is the additional approximation assumption $\mathrm{\Gamma }\tau \ll 1$: the kernel correlation time $\tau$ is much shorter than the inverse bath-induced rate scale ${\mathrm{\Gamma }}^{-1}$.

Under these assumptions the cited ULE contract supplies a Lindblad-form approximation with a correction; the postulate does not assert exact Markovian dynamics. The article characterizes Markovianity in this regime as independent of system details and dependent only on the bath. In the cited ULE formulation the criterion is independent of the system's level structure, but also depends on the coupling strength and on the stated normalization and preparation assumptions. The article's atomic and molecular application additionally invokes electric-dipole coupling and weak-coupling/Born–Markov assumptions; this postulate alone does not establish those assumptions for a particular photon bath.

The operator norm as the least bound and as the unit-sphere or unit-ball supremum

The Hilbert construction in [F3] supplies the composite space. The tensor action specifies that the system and bath operators act on separate factors. On the stipulated common domain the finite Hermitian sum is a formal interaction, using the operator meanings in [F4]. Its self-adjoint Hamiltonian realization is separately required; no self-adjointness theorem for arbitrary unbounded sums is inferred. The density-operator definition uses [F5] on either space and does not apply a finite-dimensional density definition to an infinite-dimensional bath. Each required product has at most one bounded extension because two bounded extensions agreeing on a dense domain agree everywhere. The trace-class-extension assumption and [F5] therefore make every displayed expectation and two-point trace a uniquely defined finite scalar.

By [F6], $V_B$ has the group law. Write $T_{\alpha \beta }(t,s)$ for the trace-class extension of ${\tilde{B}}_{\alpha }(t){\tilde{B}}_{\beta }(s){\rho }_B$. Stationarity and the transformed domains give $T_{\alpha \beta }(t+r,s+r)=V_B(r{)}^{†}{T}_{\alpha \beta }(t,s)V_B(r)$: the identity holds on the corresponding dense natural composition domains and then for their unique bounded extensions. By [F7] these extensions have the same trace. Taking $r=-s$ makes the trace depend only on $t-s$, as used in the definition of $C$. This argument moves only bounded unitaries around the trace-class extension, not the individual unbounded bath observables.

The kernel assumptions in [F1] make the two displayed moment expressions finite and the denominator nonzero. Thus $\mathrm{\Gamma }$ and $\tau$ are defined within that contract. The assumptions of [F2] are stated separately because a stationary bath or a short correlation time by itself does not supply Gaussianity, factorization, or the required kernel moments. The approximation is a modeling input with the stated hypotheses, not a measured property of every physical bath. The article's broad characterization is retained as attribution; the independently cited ULE statement specifies the coupling dependence and the level-structure scope.