$\nu_{lmn}$ |  resonance frequency of the cavity mode with longitudinal index $l$ and transverse mode indices $(mn)$
$\omega_{c}$ |  cavity resonance frequency: $\omega_{c}=2\pi\nu_{lmn}$
$\Delta_{c}$ |  cavity detuning, relative frequency between the pump field and the cavity resonance: $\Delta_{c}=\omega_{p}-\omega_{c}$
$\tilde{\Delta}_{c}$ |  complex cavity detuning including cavity dissipation: $\tilde{\Delta}_{c}\equiv\Delta_{c}+i\kappa$
$\delta_{c}$ |  dispersively-shift cavity detuning: $\delta_{c}\equiv\Delta_{c}-\int U(\mathbf{r})\hat{n}(\mathbf{r})d\mathbf{r}$
$\tilde{\delta}_{c}$ |  dispersively-shift complex cavity detuning: $\tilde{\delta}_{c}\equiv\Delta_{c}-\int U(\mathbf{r})\hat{n}(\mathbf{r})d\mathbf{r}+i\kappa=\delta_{c}+i\kappa=\tilde{\Delta}_{c}-\int U(\mathbf{r})\hat{n}(\mathbf{r})d\mathbf{r}$
$\lambda_{c}$ |  wave length of the cavity field
$k_{c}$ |  wave number of the cavity field: $k_{c}=2\pi/\lambda_{c}$
$\boldsymbol{\epsilon}_{c}$ |  polarization vector of the cavity field
$\mathcal{E}_{mn}(\mathbf{r})$ |  Hermite-Gaussian wavefunction of $\text{TEM}_{mn}$ mode
$\mathcal{E}_{0}$ |  maximum electric field strength of a single photon in the TEM00 mode with volume $\mathcal{V}$: $\mathcal{E}_{0}=\sqrt{\hbar\omega_{c}/2\epsilon_{0}\mathcal{V}}$
$\varphi^{\rm Gouy}_{mn}(x)$ |  Gouy phase shift associated with $\text{TEM}_{mn}$ mode: $\varphi^{\rm Gouy}_{mn}(x)=(1+m+n)\arctan(\lambda_{c}x/\pi w_{0}^{2})$
$\varphi^{\rm offset}_{mn}$ |  Phase offset due to boundary conditions of the cavity: $\varphi^{\rm offset}_{mn}=(m+n)\arctan(\lambda_{c}l_{\rm res}/2\pi w_{0}^{2})$
$\hat{a}_{\nu}$ |  photonic annihilation operator for mode $\nu$ with mode indices $\nu=(lmn)$
$\alpha_{\nu}$ |  coherent field amplitude for mode $\nu$ with mode indices $\nu=(lmn)$: $\alpha_{\nu}=\langle\hat{a}_{\nu}\rangle$
$\mathcal{G}_{0}$ |  maximum single-photon vacuum Rabi frequency: $\mathcal{G}_{0}=\mathbf{d}_{ge}\cdot\boldsymbol{\epsilon}_{c}\mathcal{E}_{0}/\hbar$
$\mathcal{G}(\mathbf{r})$ |  position-dependent vacuum Rabi frequency: $\mathcal{G}(\mathbf{r})=\mathcal{G}_{0}\mathcal{E}(\mathbf{r})/\mathcal{E}_{0}$
$\hbar U_{0}$ |  maximum cavity-lattice depth per photon (or maximum intra-cavity light frequency shift per atom): $\hbar U_{0}=\hbar\mathcal{G}_{0}^{2}/\Delta_{a}$
$\hbar U(\mathbf{r})$ |  position-dependent cavity lattice per photon: $\hbar U(\mathbf{r})=\hbar\mathcal{G}^{2}(\mathbf{r})/\Delta_{a}$
$\kappa$ |  decay rate of the cavity field through cavity mirrors
$\Delta\nu$ |  full width at half maximum of cavity resonance peak: $\Delta\nu=2\kappa/2\pi$
$\mathcal{F}$ |  Finesse of the cavity: $\mathcal{F}=\pi\nu_{\mathrm{FSR}}/\kappa$
$C$ |  cooperativity parameter: $C=\mathcal{G}_{0}^{2}/\kappa\gamma$

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