Dimensionless dipole and ULE coupling identification

PDF p. 11: Eq. (S3.3) and final bullets

Work in units with $\mathrm{\hslash }=1$ and use the finite-mode photon field at the fixed emitter position. Take real transverse polarization vectors ${\mathit{ϵ}}_{\lambda }(\mathbf{k})$ and a nonzero real dipole scale $d_0$. Postulate Hermitian dimensionless system operators $X_x,X_y,X_z$ with $\mathbf{d}=d_0\mathbf{X}$. The system space is a finite direct sum of spin blocks, and the spherical components $T_0=X_z$ and $T_{\pm 1}=\mp (X_x\pm i{X}_y)\mathrm{/}\sqrt{2}$ form a rank-one spherical tensor: $[J_z,T_q]=q{T}_q$ and $[J_{\pm },T_q]=\sqrt{(1\mp q)(2\pm q)}{T}_{q\pm 1}$, with $T_{\pm 2}=0$. These are model assumptions; they do not specify nonzero matrix elements.

For the regulated realization let $M$ be the finite nonempty set of retained mode labels $j=(\mathbf{k},\lambda )$ and use the bosonic Fock space ${\mathcal{H}}_B={\mathcal{F}}_s({\mathbb{C}}^M)$. Use the second-variable-linear inner product convention and Fock sector maps. For the standard orthonormal mode vectors $e_j$, let $a_j=a(e_j)$ and $a_j^{†}=a^{†}(e_j)$ be the maximal mutually adjoint operators. Let ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ consist of Fock vectors with only finitely many nonzero particle-sector components. It is dense and invariant under these operators. Write ${\mathcal{H}}_S$ for the emitter space and take the total Hilbert space ${\mathcal{H}}_S\widehat{\otimes }{\mathcal{H}}_B$, with common dense invariant domain $\mathcal{D}={\mathcal{H}}_S{\otimes }_{\mathrm{a}\mathrm{l}\mathrm{g}}{\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. Each field component $E_{\alpha }({\mathbf{r}}_0)$ is initially the finite field sum on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$; all interaction sums and tensor products below have domain exactly $\mathcal{D}$.

Let ${\widehat{e}}_{\alpha }$ be the Cartesian unit vector in direction $\alpha \in \{x,y,z\}$ and put $\kappa (\omega )=d_0{\mathcal{E}}_{\omega }$, allowing the chosen field-normalization convention to be included in $\kappa$. The interaction is $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$ and, in this real-polarization convention, $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=-{\sum }_{(\mathbf{k},\lambda )\in M,\alpha }[X_{\alpha }({\widehat{e}}_{\alpha }\cdot {\mathit{ϵ}}_{\lambda }(\mathbf{k}))\kappa ({\omega }_{\mathbf{k}}{)}^{\ast }\otimes a_{\mathbf{k}\lambda }^{†}+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}]$. Here $-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$ means $-d_0{\sum }_{\alpha }{X}_{\alpha }\otimes E_{\alpha }({\mathbf{r}}_0)$. Here $\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$ means the formal conjugate term $X_{\alpha }({\widehat{e}}_{\alpha }\cdot {\mathit{ϵ}}_{\lambda }(\mathbf{k}))\kappa ({\omega }_{\mathbf{k}})\otimes a_{\mathbf{k}\lambda }$ on $\mathcal{D}$. It is the restriction to $\mathcal{D}$ of the preceding term's Hilbert-space adjoint, rather than an assertion that the unrestricted adjoint has domain $\mathcal{D}$. For complex polarization vectors the creation coefficient instead contains ${\mathit{ϵ}}_{\lambda }(\mathbf{k}{)}^{\ast }$ and the annihilation coefficient its conjugate.

For the ULE interface choose $\gamma >0$, use the three system operators $X_{\alpha }$, and set $B_{\alpha }=-d_0{E}_{\alpha }({\mathbf{r}}_0)\mathrm{/}\sqrt{\gamma }$. Initially $B_{\alpha }$ has domain ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. Then the interaction has the interface form $\sqrt{\gamma }{\sum }_{\alpha }{X}_{\alpha }\otimes B_{\alpha }$ as an equality on $\mathcal{D}$. The field, bath and interaction expressions are symmetric on their respective stated domains. No equality of closed-operator realizations is asserted by this domain-level identification; a concrete Hamiltonian application must also specify appropriate self-adjoint realizations. Applying the normalized ULE contract additionally requires $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$, a centered stationary Gaussian bath with the specified preparation, and the contract's spectral and kernel regularity and short-memory assumptions. These conditions are additional assumptions, not consequences of the operator identification.

Photon modes and field normalization

ULE system–bath coupling and short bath memory

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Schwartz/distribution spaces and Hilbert Fock conventions

Bosonic and fermionic Fock operators with exact domains

Hilbert tensor construction: inner product, completion and product bases

Unbounded linear operators: domain, graph and extension

Adjoint of a densely defined operator

Symmetric, self-adjoint and essentially self-adjoint operators

Apply [F4] to the finite-dimensional one-particle space ${\mathbb{C}}^M$. Creation raises the particle sector by one and annihilation lowers it by one, so each maps ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ to itself. Density of $\mathcal{D}$ follows from [F6] and density of ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$: in any finite sum of elementary tensors approximate each bath factor by finite-particle truncations, using $\mathrm{\parallel }s\otimes \psi \mathrm{\parallel }=\mathrm{\parallel }s\mathrm{\parallel }\mathrm{\parallel }\psi \mathrm{\parallel }$. For a system map $X$ and a bath map $A$ on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$, $X\otimes A$ acts by $s\otimes \psi \mapsto Xs\otimes A\psi$ and linear extension; this is well defined by bilinearity of the algebraic tensor product. Thus all finite sums and products used here preserve the stated common domains.

The finite-mode expansion in [F1], substitution of $d_{\alpha }=d_0{X}_{\alpha }$, and real polarization vectors give the displayed creation coefficient. For a term $T=c{X}_{\alpha }\otimes a_j^{†}$ and its formal conjugate $T^{\mathrm{♯}}=\bar{c}{X}_{\alpha }\otimes a_j$, both restricted to $\mathcal{D}$, the product pairing in [F6] and mutual adjointness in [F4] give $⟨Tu,v⟩=⟨u,T^{\mathrm{♯}}v⟩$ for $u,v\in \mathcal{D}$. By [F5], $\mathcal{D}\subseteq D(T^{\ast })$ and $T^{\ast }{\mathrm{\mid }}_{\mathcal{D}}=T^{\mathrm{♯}}$. Exchanging the two terms gives symmetry of their sum on $\mathcal{D}$. The same pairing argument on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ makes each finite field component and its real multiple $B_{\alpha }$ symmetric. With complex polarization vectors the conjugate in the creation term follows directly from the field expansion in [F1]. The spherical-component relations instantiate the rank-one definition in [F3].

The bath and system operators act on separate tensor factors on $\mathcal{D}$: both orders send $s\otimes \psi$ to $X_{\alpha }s\otimes E_{\alpha }\psi$. Substituting the stated $B_{\alpha }$ into the ULE coupling reproduces $-d_0{\sum }_{\alpha }{X}_{\alpha }\otimes E_{\alpha }$ with exactly that domain. This algebraic identification does not establish the norm bound, bath preparation, kernel conditions or short-memory regime in [F2]. Nor does symmetry alone supply a self-adjoint realization of a total Hamiltonian. In particular, a finite collection of oscillator modes is not asserted to satisfy the ULE kernel assumptions; a bath model satisfying them must be supplied separately.