Photon modes and field normalization

PDF p. 11: S3 Hamiltonian bullets, Eq. (S3.2)

Use units with $\mathrm{\hslash }=1$. In the article's box-normalized photon model, a mode is labeled by a nonzero wave vector $\mathbf{k}$ and one of two polarization labels $\lambda =1,2$. Its frequency is ${\omega }_{\mathbf{k}}=c\mathrm{\mid }\mathbf{k}\mathrm{\mid }$, where $c>0$ is the speed assigned to the field. For each $\mathbf{k}$, the two polarization vectors ${\mathit{ϵ}}_{\lambda }(\mathbf{k})$ are transverse, meaning $\mathbf{k}\cdot {\mathit{ϵ}}_{\lambda }(\mathbf{k})=0$, and together with $\widehat{\mathbf{k}}=\mathbf{k}\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$ form an orthonormal basis of the complexified three-dimensional vector space with its usual Hermitian inner product.

Let $a_{\mathbf{k}\lambda }$ and $a_{\mathbf{k}\lambda }^{†}$ be the annihilation and creation operators for the mode. Postulate the free bath Hamiltonian $H_B={\sum }_{\mathbf{k},\lambda }{\omega }_{\mathbf{k}}{a}_{\mathbf{k}\lambda }^{†}{a}_{\mathbf{k}\lambda }$ and the electric field at the fixed emitter position ${\mathbf{r}}_0$ as $\mathbf{E}({\mathbf{r}}_0)={\sum }_{\mathbf{k},\lambda }[{\mathcal{E}}_{{\omega }_{\mathbf{k}}}{\mathit{ϵ}}_{\lambda }(\mathbf{k})a_{\mathbf{k}\lambda }+{\mathcal{E}}_{{\omega }_{\mathbf{k}}}^{\ast }{\mathit{ϵ}}_{\lambda }(\mathbf{k}{)}^{\ast }{a}_{\mathbf{k}\lambda }^{†}]$, where ${\mathcal{E}}_{\omega }=i\sqrt{\omega \mathrm{/}(2{\epsilon }_{0}V)}$ up to a phase convention, ${\epsilon }_0>0$ is the vacuum permittivity, and $V>0$ is the quantization volume. The mode functions' phases at ${\mathbf{r}}_0$ are included in the chosen mode convention.

The electric-dipole approximation evaluates the field at ${\mathbf{r}}_0$ and neglects its variation across the emitter; with electric-dipole operator $\mathbf{d}$, the interaction is $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$. For a regulated realization, retain finitely many positive-frequency modes and a finite-dimensional emitter as in the cited bath model. The unrestricted mode sum is model notation and no removal of the finite-mode regulator is claimed.

Angular-momentum emitter, photon bath, and dipole approximation

Regularized transverse photon dipole bath model

Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases

For each nonzero wave vector, transversality makes the two polarization vectors orthogonal to the propagation direction; the stated orthonormal-basis convention fixes their normalization. Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases The dagger denotes the Hilbert-space adjoint, and complex conjugation of the coefficient and polarization vector makes each finite-mode Cartesian field component a Hermitian linear combination on the common finite-particle domain. Its positive-frequency mode energies and linear coupling have the form permitted by F2. F1 supplies the emitter interaction and its interpretation as a dipole approximation. The specific dispersion, field normalization and approximation are adopted model assumptions attributed to the article, not deductions of laboratory behavior. The finite-mode statement does not establish convergence of the unrestricted sum or the validity of a continuum point field.