Angular-momentum emitter, photon bath, and dipole approximation

PDF p. 3: Atomic and molecular systems, Hamiltonian paragraph; PDF p. 11: S3 introduction and Eq. (S3.1)

Use units with $\mathrm{\hslash }=1$. Let the emitter have angular-momentum basis states $\mathrm{\mid }J,m⟩$, where $J$ labels a spin block and $m=-J,-J+1,\dots ,J$ labels its magnetic states. Adopt the model Hamiltonian $H=H_S+H_B+H_{\mathrm{i}\mathrm{n}\mathrm{t}}$, where $H_S={\sum }_{J,m}{\omega }_0{\eta }_J\mathrm{\mid }J,m⟩⟨J,m\mathrm{\mid }$. Here ${\omega }_0>0$ sets the energy scale and the real dimensionless level factors ${\eta }_J$ increase monotonically with $J$. Thus all magnetic states in a given $J$ block have the same unperturbed energy. The rigid-rotor benchmark takes ${\eta }_J=J(J+1)$.

The bath Hamiltonian $H_B$ is the free photon energy. In the electric-dipole approximation, take $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$, where $\mathbf{d}$ is the emitter's electric-dipole operator and $\mathbf{E}({\mathbf{r}}_0)$ is the electric field evaluated at the emitter position ${\mathbf{r}}_0$. This approximation neglects the field's spatial variation across the emitter. The Hamiltonian is an effective platform-independent baseline for atomic and molecular models, rather than a species-specific microscopic description.

For a mathematically specified finite-emitter realization, choose a finite nonempty set $\mathcal{S}$ of spin labels and take the orthogonal direct sum ${\mathcal{H}}_S={⨁}_{J\in \mathcal{S}}{V}_J$, with componentwise vector operations and inner product $⟨x,y⟩={\sum }_{J\in \mathcal{S}}⟨x_J,y_J{⟩}_{V_J}$, linear in its first argument. Embed each block basis vector $e_{J,m}$ by zero in all other blocks and denote its image by $\mathrm{\mid }J,m⟩$. Order the combined basis by increasing $J$ and then increasing $m$; its coordinates identify ${\mathcal{H}}_S$ with ${\mathbb{C}}^d$, where $d={\sum }_{J\in \mathcal{S}}(2J+1)\ge 1$. Use the finite operator conventions for bras, outer products and adjoints: $⟨v\mathrm{\mid }x⟩=v^{†}x$, $(\mathrm{\mid }u⟩⟨v\mathrm{\mid })x=u(v^{†}x)$, and $A^{†}={\bar{A}}^T$ in these coordinates. In this realization the sum defining $H_S$ runs over $J\in \mathcal{S}$ and all magnetic labels in each retained block.

Use either of the regulated photon-bath constructions in the cited bath postulate, with its stated cutoff or finite-mode hypotheses, on this finite emitter. The displayed unrestricted expression is model notation; no regulator-removal result is asserted.

Angular-momentum basis and tensor labels

Regularized transverse photon dipole bath model

Finite operator and ket conventions

Hilbert direct sums of unitary representations

Given: The finite nonempty set $\mathcal{S}$, the spin-block bases in F1, the direct-sum construction F3, the operator conventions F4, real level factors, and the choice assumptions A1. The numbered argument establishes the finite emitter's mathematical realization.

1.1 By F3, the zero-extension embeddings preserve each block's inner product and have orthogonal images. Thus two embedded vectors from F1 have inner product one when both labels agree and zero otherwise. Every vector $x=(x_J{)}_{J\in \mathcal{S}}$ is a sum of its finitely many block components, each expanded in the corresponding basis from F1. Hence the combined vectors span and are orthonormal; taking their inner products with a vanishing linear combination also proves linear independence. Ordering them as stated identifies the direct sum isometrically with the standard coordinate space ${\mathbb{C}}^d$, and nonemptiness of $\mathcal{S}$ gives $d\ge 1$, as required by F4 and the finite-emitter bath construction. [F1, F3, F4, A1, algebra]

1.2 By F4, $P_{J,m}=\mathrm{\mid }J,m⟩⟨J,m\mathrm{\mid }$ acts on a vector by selecting its $(J,m)$ coordinate and multiplying that coordinate by $\mathrm{\mid }J,m⟩$. Its matrix has a single diagonal entry one and all other entries zero, so its conjugate transpose equals itself. Therefore the finite real linear combination $H_S={\sum }_{J\in \mathcal{S}}{\sum }_{m=-J}^J{\omega }_0{\eta }_J{P}_{J,m}$ is Hermitian and satisfies $H_S\mathrm{\mid }J,m⟩={\omega }_0{\eta }_J\mathrm{\mid }J,m⟩$. The eigenvalue is independent of $m$, giving the stated degeneracy within each retained block. [F4, step 1.1, given, algebra] ∎

F2 supplies the meaning and hypotheses of the regulated bath and its finite-emitter coupling, applied to this ${\mathbb{C}}^d$ realization. The identification of that coupling with the electric-dipole approximation is a physical modeling assumption, not a consequence of the basis construction. The rigid-rotor scaling is a specified example of the level factors; no species-specific accuracy or regulator-free limit is claimed.