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\title{Coupling Light to Higher Order Transverse
Modes of a Near-Concentric Optical Cavity: Supplemental Document}
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\begin{abstract}
In this supplemental document, we provide a more thorough calculation of the atom-cavity coupling constant $g_{ac}$ for radial transverse modes (LG modes with $l=0$). We show that, at a particular cavity length $L$ or critical distance $d$, all the radial transverse modes have the same mode volume $V_m$ and hence the same coupling constant $g_{ac}$. This would allow an atom to have similar interaction strengths with all of the radial transverse modes of the cavity. 
\end{abstract}

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\begin{document}

\maketitle

\section{Interaction Hamiltonian of An Atom-Cavity System}

An electric field operator with a complex mode amplitude profile $U_j(\mathbf{r})$ for configuration parameter $j$ can be expressed as~\cite{loudon2000quantum}
\begin{equation}
\hat{\mathbf{E}}_j(\mathbf{r}) = i \mathbf{e}_{\lambda} N_j \left[U_j(\mathbf{r}) \hat{a}_j - U_j^*(\mathbf{r}) \hat{a}^{\dagger}_j \right] ,
\end{equation}
where $\mathbf{e}_{\lambda}$ is the electric field polarization unit vector, $N_j$ is a normalization coefficient, and $\hat{a}_j$ and $\hat{a}^{\dagger}_j$ are the destruction and creation mode operators. The free field Hamiltonian of the electromagnetic mode inside a cavity with volume $V$ has a similar structure with a harmonic oscillator,
\begin{equation}
\hat{H}_0 = \frac{\epsilon_0}{2}\int{dV} \left[
\hat{\mathbf{E}}_j(\mathbf{r})^2 + c^2\hat{\mathbf{B}}_j(\mathbf{r})^2
\right] = \hbar \omega_j  \left(\hat{a}^{\dagger}_j \hat{a}_j +\frac{1}{2} \right)\,,
\end{equation}
with $\hat{\mathbf{B}}_j(\mathbf{r})$ as the corresponding magnetic field operator. By imposing the normalization condition
\begin{equation}
    \int dV |U_j(\mathbf{r})|^2 = 1 \, ,
\label{eq:normalization}
\end{equation} 
we obtain the normalization coefficient $N_j = (\hbar \omega_j/2 \epsilon_0)^{1/2}$ .

The interaction Hamitonian between an electromagnetic field and an atomic dipole, in the electric-dipole approximation, is given by
\begin{equation}
\hat{H}_I(\mathbf{r}) = -\hat{\mathbf{E}}_j(\mathbf{r}) \cdot \hat{\mathbf{d}}\,,  
\end{equation}
where $\hat{\mathbf{d}}=d_a \left( \hat{\pi}^{\dagger}+\hat{\pi} \right) \mathbf{e}_{d}$ is the atomic electric-dipole operator with $d_a$ as the dipole moment, $\hat{\pi}^{\dagger}=|e\rangle\langle g|$ and $\hat{\pi}=|g\rangle\langle e|$ as the transition operators between ground and excited atomic states, and $\mathbf{e}_{d}$ as the dipole unit vector. Assuming that the atomic dipole and electric field polarization vectors point to the same direction, $\mathbf{e}_{\lambda}=\mathbf{e}_{d}$, we obtain
\begin{equation}
\hat{H}_I(\mathbf{r}) = -i d_a \sqrt{\frac{\hbar \omega_j}{2 \epsilon_0}} \left[U_j(\mathbf{r})\hat{a}_j - U_j^{*}(\mathbf{r})\hat{a}_j^{\dagger} \right] \left[ \hat{\pi}^{\dagger}+\hat{\pi} \right] \, ,
\label{eq:interaction}
\end{equation} 
with an explicit dependence of the atom position $\mathbf{r}$ on the mode amplitude $U_j(\mathbf{r})$. 

\section{Atom-Cavity Coupling Constant For Radial Transverse Modes}
The complex amplitude of radial transverse modes can be obtained by setting $l=0$ to the LG mode amplitude (Eq.~1 of the main text), 
\begin{equation}
U_{p,0}(\rho,\phi,z)= A_{p,0} \frac{w_0}{w(z)}  \mathcal{L}^0_p \left(\frac{2\rho^2}{w^2(z)}\right) \exp \left(-\frac{\rho^2}{w^2(z)}\right) \exp\left(i\psi_{p,0}(\rho,\phi,z) \right) \, .
\label{eq:mode}
\end{equation}
The radial transverse modes of a near-concentric cavity have spatial dependencies on both the axial and radial direction~\cite{aljunid2011interaction}. Here, we assume that the atom is located at the center of the cavity mode ($\mathbf{r}=\mathbf{0}$), such that $\rho=0$ and $z=0$. The amplitude of the radial transverse mode at this center point is
\begin{equation}
U_{p,0}(\mathbf{r}=\mathbf{0})= A_{p,0} \ ,
\end{equation}
where we have evaluated $w(0)=w_0$, $\mathcal{L}^0_p(0)=1$, and $\exp\left(i\psi_{p,0}(0,\phi,0) \right) = 1$.

The prefactor $A_{p,0}$ can be determined from the normalization condition (Eq.~\ref{eq:normalization}), and has a physical significance as follows. If we assume a theoretical cavity with a uniform distribution of mode amplitude $U_{p,0}=A_{p,0}\,$, the normalization condition yields
\begin{equation}
    A_{p,0} = \frac{1}{\sqrt{V_{m,p}}} \ ,
\end{equation}
where $V_{m,p}$ is the mode volume of such theoretical cavity. 

Using Eq.~\ref{eq:interaction}, we evaluate the interaction Hamiltonian of an atom placed at the centre of a radial transverse cavity mode,
\begin{equation}
\hat{H}_I(\mathbf{r}=\mathbf{0}) = -i \hbar 
g_{ac} \left[\hat{a}_j - \hat{a}_j^{\dagger} \right] \left[ \hat{\pi}^{\dagger}+\hat{\pi} \right] \, ,
\end{equation} 
with the atom-cavity coupling constant
\begin{equation}
g_{ac} = \sqrt{\frac{\omega_j d_a^2}{2 \hbar \epsilon_0 V_{m,p}}} \ .
\end{equation}
It is interesting to note that the coupling constant $g_{ac}\propto 1/\sqrt{V_{m,p}}$ only depends on the radial mode number $p$ through the mode volume $V_{m,p}$ . This is not necessarily true for other classes of transverse modes. For example, non-radial transverse modes ($l \neq 0$) has zero mode amplitude at the centre, i.e. $U_{p,l\neq 0}(\mathbf{r}=\mathbf{0})= 0$ , which complicates the interpretation of such mode volumes. 

\section{Mode Volume Calculation OF Radial Transverse Modes}

In this part, we show that the mode volume of radial transverse modes $V_{m,p}$ only depends on the beam waist $w_0$ and cavity length $L$, and does not depend on the radial mode number $p$.
Using the normalization condition (Eq.~\ref{eq:normalization}),
\begin{align}
    \int dV |U_{p,0}(\rho,\phi,z)|^2 &= \int dz \int \rho d\rho \int d\phi \, \frac{1}{V_{m,p}} \left[\frac{w_0}{w(z)} 
    \mathcal{L}^0_p \left(\frac{2\rho^2}{w^2(z)}\right)\right]^2
    \exp \left(-\frac{2\rho^2}{w^2(z)}\right)
    \\
    1 &= \frac{1}{V_{m,p}}\frac{\pi w_0^2 L}{2}  \int_0^{\infty} du\, e^{-u} \mathcal{L}^0_p(u)^2 \ ,
\end{align}
where we have used the substitution $u=2\rho^2/w^2(z)$. The integration can be calculated using the orthonormal property of Laguerre polynomials~\cite{arfken1999mathematical},
\begin{equation}
    \int_0^{\infty} dx\, e^{-x} \mathcal{L}^0_m (x) \mathcal{L}^0_n (x) = \delta_{m,n} \ .
\end{equation}
The mode volume of radial transverse modes is thus calculated to be
\begin{equation}
    V_{m,p} = \frac{1}{2}\pi w_0^2 L \ ,
\end{equation}
independent of the radial mode number $p$. At a particular cavity length $L$, the beam waist parameter $w_0$ is evaluated from the cavity boundary conditions, and is also independent on the radial mode number $p$. Hence, the atom-cavity coupling constant $g_{ac}$ is maintained over all the radial transverse modes,
\begin{equation}
g_{ac} = \sqrt{\frac{\omega_j d_a^2}{\pi \hbar \epsilon_0 w_0^2 L}} \ .
\end{equation}

As a completing remark, we note that the mode of a physical cavity forms a standing wave on the longitudinal direction. Hence, the complex term $\exp(i\psi_{p,0})$ in Eq.~\ref{eq:mode} should be replaced with $\cos(\psi_{p,0}+\delta)$ with $\delta$ determined by the boundary condition and longitudinal mode number. Assuming the atom is located at the anti-node of such standing wave, the evaluation of the atom-cavity coupling is similar as above, but with the mode volume reduced to half,
\begin{equation}
    V_{m,p}^{sw}=\frac{1}{4}\pi w_0^2 L \ .
\end{equation}




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