\section*{\large Supplementary Materials}
\setcounter{figure}{0}
\renewcommand{\figurename}{Fig.\,S}
\renewcommand{\tablename}{Tab.\,S}




\section{Experimental Design} 
\subsection{Preparing the Sample}
We use the three lowest Zeeman sublevels of the hyperfine ground state of $^6$Li - labeled $\ket{1},\ket{2}$ and $\ket{3}$, as shown in Fig.\,S \ref{fig:S2_RFSpectroscopy} A and B. The starting point for our experiments is an almost pure molecular Bose-Einstein condensate (mBEC) of  $\sim 5\times10^4$ $\ket{1}\ket{2}$ or $\ket{1}\ket{3}$ dimers produced by optical evaporation at magnetic offset fields of $B=\unit[690]{G}$ or $B=\unit[795]{G}$ respectively. The sample is then transferred into an optical standing wave trap (SWT)  (\textit{8}), where we perform additional evaporation to reach the quasi-2D regime with $\sim 3\times 10^4$ atoms in each spin state. At $B=\unit[795]{G}$, the trapping frequencies for the central layers of the SWT are $\omega_\mathrm{r} \approx 2\pi \times \unit[22]{Hz}$ and $\omega_\mathrm{z} \approx 2\pi \times \unit[6.95]{kHz}$. This results in an aspect ratio of $\omega_\mathrm{r}:\omega_\mathrm{z}\approx 1:300$. Here the system is in the kinematic quasi-2D regime, which we experimentally verified in \cite{Ries2015}.

\subsection{Creating a Spin-Imbalanced Mixture}

After evaporation we introduce imbalance by a sequence of Landau-Zener passages at $B = \unit[1000]{G}$ where the interaction strength is relatively weak (see Fig.\,S \ref{fig:S2_RFSpectroscopy} C). In the case of the $\ket{1}\ket{3}$-mixture, we transfer a small fraction of atoms from state $\ket{1}$ to state $\ket{2}$. The ensuing three-body collisions lead to an imbalance between state $\ket{1}$ and state $\ket{3}$ with a majority of the atoms in state $\ket{3}$. For the $\ket{1}\ket{2}$-mixture, we transfer atoms from state $\ket{2}$ to state $\ket{3}$ and invert the resulting imbalance with an additional Landau-Zener passage to end up with the majority in state $\ket{2}$.\\ 
Radial profiles for both the majority and minority component are shown in the main text in Fig. 2 for a $\ket{1}\ket{3}$-mixture. We work with a local polarization $p_\mathrm{loc} = \frac{n_{\ket{3}}-n_{\ket{1}}}{n_{\ket{3}}+n_{\ket{1}}}\lesssim \unit[15]{\%}$. As we operate in the normal phase, we do not observe a phase separation into a balanced core and a polarized wing as one observes in a superfluid \cite{Schirotzek2008,Mitra2016} but rather see a local polarization that varies only weakly over the sample. The heat introduced due to the three-body losses in this imbalance scheme limits the achievable temperatures to about $T/T_\mathrm{F}\gtrsim 0.4$.

\subsection{Radio-Frequency Spectroscopy}
Fig.\,S \ref{fig:S2_RFSpectroscopy} B schematically shows the RF transitions accessed in our experiments. In the case of a $\ket{1}\ket{3}$-mixture as depicted in the upper panel, we drive the transition from state $\ket{3}$ to $\ket{2}$. Without interactions, the resonance energy for the transition of a free atom is given by $E_\mathrm{free-free}$ which is measured in a spin-polarized, non-interacting sample. If the atom in state $\ket{3}$ is paired with an atom in state $\ket{2}$, the energy level is shifted by the binding energy resulting in a reduced resonance energy $E_\mathrm{bound-free}$. Thus $E_\mathrm{bound-free} < E_\mathrm{free-free}$ and the paired branch is at negative frequencies relative to $E_\mathrm{free-free}$. For the $\ket{1}\ket{2}$-mixture as depicted in the lower panel, the situation is similar. However, here $E_\mathrm{bound-free} > E_\mathrm{free-free}$ and hence the paired branch is at positive RF frequency offsets. For the sake of brevity, we will attribute pairing with positive frequency shifts independent of the mixture throughout this work.  \\
\begin{figure*}[t!]
	\centering
	\includegraphics{S1_RFLevels.pdf}
	\caption{\textbf{A:} Zeeman sublevels of the $^6\mathrm{Li}$ $2^2S_{1/2}$ ground state. Indicated in color are the sublevels experimentally employed. \textbf{B:} Schematic illustration of RF spectroscopy in $^6$Li for both the $\ket{1}\ket{3}$-mixture in the upper panel and the $\ket{1}\ket{2}$-mixture in the lower panel. \textbf{C:} Upper panel: The 3D scattering length $a_\mathrm{3D}$ in units of the Bohr radius $a_0$ is plotted versus magnetic field for the $\ket{1}\ket{2}$-(blue) and $\ket{1}\ket{3}$-mixture (green). Lower panel: The corresponding 2D interaction strength $\ln\left(k_\mathrm{F}a_\mathrm{2D}\right)$ is plotted as a function of magnetic field for a typical Fermi momentum $k_\mathrm{F}=\unit[3.5]{\mu m^{-1}}$.}
	\label{fig:S2_RFSpectroscopy}
\end{figure*}\\
\noindent
\textbf{Final state effects:} 
In an interacting system, the non-interacting free-free transition is not a suitable reference anymore, as interactions - both in the initial and final state - lead to density-dependent mean-field shifts. We address this question, as explained in the main text, by considering only energy differences. In addition, the final state interactions lead to a broadening of the RF transitions which reduces the resolution. This effect can be significant for the $\ket{1}\ket{3}$-mixture where the final state is most likely a short-lived repulsive polaron \cite{Koschorreck2012}. Therefore we make use of the $\ket{1}\ket{3}$- as well as the $\ket{1}\ket{2}$-mixture to minimize the effect of final state interactions. A plot of the broad Feshbach resonances which have considerable overlap between the different mixtures can be seen in Fig.\,S \ref{fig:S2_RFSpectroscopy} C, together with typical values for $\ln\left(k_\mathrm{F}a_\mathrm{2D}\right)$. 
We use the $\ket{1}\ket{3}$-mixture for magnetic fields below the $\ket{1}\ket{3}$-resonance where the final state interaction strength has values of $\ln\left(k_\mathrm{F}a_\mathrm{2D}\right)<-7$, and the $\ket{1}\ket{2}$ mixture for larger fields where the final state  $\ln\left(k_\mathrm{F}a_\mathrm{2D}\right)>4.5$.
The pairing energies measured with both mixtures in the crossover regime show consistent behaviour independent of the mixture used as shown in Fig. 4 B in the main text.\\

\noindent
\textbf{Spatially resolved spectroscopy:} To avoid diffusion of the transferred density, we have to use RF pulses which are short compared to the trapping period and image the sample directly after the pulse application. However, short RF pulses lead to a Fourier limited frequency resolution and therefore we have to find the optimum trade-off. To investigate this experimentally, we prepare a $\ket{1}\ket{2}$-mixture at $B = \unit[854]{G}$ and record the RF spectrum for different pulse durations $\tau_\mathrm{RF}$ as can be seen exemplary in Fig. S \ref{fig:S3_RFPulsedur} A. We apply rectangular RF pulses which lead to a Fourier limited width $\Delta \nu \geq 0.886/\Delta t$. For all applied RF pulse durations, the density dependence of the bound-free transition is readily visible. For each pulse duration we adjust the RF power such that a similar fraction of atoms is transferred. This leads to Rabi frequencies between $\unit[60]{Hz}$ for the longest and $\unit[240]{Hz}$ for the shortest pulse durations and hence the effect of power broadening is small compared to the Fourier limit. Binning the spectra over two pixels and taking a cut at a fixed radius, we can then compare the measured widths of the bound-free transition. This is shown in Fig.\,S \ref{fig:S3_RFPulsedur} B for the FWHM of the peak. For short RF pulses on the order of $\unit[1]{ms}$, the width increases as the Fourier limit there is on the order of $\unit[1]{kHz}$. Although the Fourier limit decreases for longer pulse durations, the FWHM increases which we attribute to diffusion of particles during the application of the RF pulse. At around $\tau_\mathrm{RF}\approx\unit[4]{ms}$ the minimal FWHM occurs, which is the pulse length we used throughout this paper. This results in a Fourier limited RF resolution of $\Delta \nu = \unit[222]{Hz}$ with a typical Rabi frequency on the order of $\Omega=\unit[150]{Hz}$.
\begin{figure}[b!]
	\centering
	\includegraphics{S2_RFPulsedur.pdf}
	\caption{Influence of the RF pulse duration on the measured response spectra at $\unit[854]{G}$. In \textbf{A} the spectra are shown for pulse durations of $\unit[2]{ms}$, $\unit[4]{ms}$ and $\unit[10]{ms}$. Along the cuts marked as the shaded area in the spectra, the FWHM is obtained from fits and shown in \textbf{B}. A pulse duration of $\tau_\mathrm{RF}=\unit[4]{ms}$ gives the optimum trade-off between frequency resolution and the effect of diffusion in the sample.}
	\label{fig:S3_RFPulsedur}
\end{figure}

\section{Data Analysis}

\subsection{Fitting Procedure}

\begin{figure*}[t!]
	\centering
	\includegraphics[width=0.62\textwidth]{S3_cuts}
	\caption{Example cuts through the RF response map at magnetic fields of 672G in \textbf{A}, 692G in \textbf{B} (both in the $\ket{1}\ket{3}$ mixture) and 854G in \textbf{C} (in the $\ket{1}\ket{2}$ mixture). The cuts are chosen such that the local temperature is $T/T_F \approx 0.5$ (c.f. Fig. 4 B of the main text). Shown are the experimental data points (blue circles), a fit of the model function (red dashed line) and the frequency region excluded in the fit (gray area). For the features of interest we see good agreement with the model function at all fields.}
	\label{fig:S5_Fit}
\end{figure*}

We model the RF response at a fixed density and interaction strength by a combination of a symmetric Gaussian profile accounting for the quasi-particle peak and an asymmetric threshold function describing the bound-free peak. For the case of dimers as well as zero temperature BCS theory, the line shape $\Gamma$ of the bound-free peak can be calculated from Fermi\textsc{\char13}s golden rule and is given in terms of the RF frequency $\omega$ by (see Section 3.1) 
\begin{equation} 
\Gamma(\omega) \propto \frac{\theta(\hbar \omega - E_{th})}{\omega^2},
\end{equation}
where $E_{th}=E_B$ denotes the threshold energy for pair breaking, i.e. the binding energy. Due to the $\frac{1}{\omega^2}$ factor, the formula crucially depends on the exact knowledge of the reference frequency (the free-free transition which is set to zero here). This frequency can be affected by mean field shifts, however, which we account for by including a free parameter $\eta$ and replacing $\frac{1}{\omega^2} \rightarrow \frac{1}{(\omega-\eta)^2}$. It is important to note that the maximum of the experimentally recorded RF response does not necessarily coincide with $E_{th}$. The peak position is shifted towards higher energies due to symmetric broadening of the asymmetric response function, as given in particular by the Fourier broadening, as well as final state effects. Similar to what has been presented in \cite{Sommer2012, Langmack2012}, we account for these effects by including an additional factor $\frac{\ln^2(E_{th}/E^f_B)}{\ln^2((\hbar\omega-E_{th})/E^f_B)+\pi^2}$ in terms of the final state two-body binding energy $E^f_B$. This also accounts for the 2D nature of the scattering process. In addition, we convolve the resulting lineshape with a Gaussian profile to account for the aforementioned broadening effects. Thus, our fitting function for the bound-free peak is given by

\begin{align}
\Gamma(\omega) = {\underbrace{\vphantom{\frac{1}{\sqrt{2 \pi\sigma^2_\mathrm{broad}}}}\left( A_{bound}\frac{\theta(\hbar \omega - E_{th})}{(\omega-\eta)^2}\frac{\ln^2(E_{th}/E^f_B)}{\ln^2((\hbar\omega-E_{th})/E^f_B)+\pi^2} \right)}_\text{Threshold function}} \nonumber \\
* {\underbrace{\left( \frac{1}{\sqrt{2 \pi\sigma^2_\mathrm{broad}}}e^{-\frac{\omega^2}{2\sigma_\mathrm{broad}^2}}\right)}_\text{Gaussian broadening}}
\end{align}

where the threshold energy $E_{th}$ relative to the non-interacting free-free transition energy, the amplitude of the bound peak $A_{bound}$, the frequency offset $\eta$ and the width $\sigma_{broad}$ of the Gaussian broadening are kept as free parameters, while the final state binding energy $E^f_B$ is fixed for each magnetic field.\\

\noindent
We observe the broadening to be mostly given by the Fourier limit when using the $\ket{1}\ket{2}$-mixture. However, there is additional broadening in the $\ket{1}\ket{3}$-mixture, possibly due to a short lived repulsive polaron final state \cite{Koschorreck2012}. The model describes the RF response reasonably well in the full crossover, as presented in Fig.\,S \ref{fig:S5_Fit}, and gives a consistent estimate for the onset of pairing throughout the crossover. However, for an exact quantitative determination of the paring gap, a model capturing the spectra of the many-body system is needed, but is currently not available.\\
The observed quasi-particle peak is mostly symmetric and we model it by a Gaussian profile with the peak position, the amplitude and the width as free parameters. We interpret the peak position as the quasiparticle energy. In addition to the signal stemming from bound particles and quasi-particle excitations in the central layer of our 2D trap, we observe additional features at intermediate frequencies which we attribute partly to a small fraction of the atoms being in adjacent layers of the SWT. Therefore, we exclude the corresponding frequency range from the fits. 


\subsection{Thermometry}
We extract the temperature of the system with the method established in \cite{Boettcher2016} by fitting reference equations of state (EOS) to the density distribution. Here we slightly modify the thermometry scheme by including an effective mass interpolation in the Boltzmann EOS: 
\begin{equation}\label{eqn:boltz}
n_{ 0 }= \frac{ \alpha }{ \lambda_T^2 } e^{ \alpha\beta\mu},
\end{equation}
where $\lambda_T$ is the thermal de Broglie wavelength and $\beta=\frac{1}{k_\mathrm{B}T}$.
Depending on the magnetic field $B$, the effective temperature exponent $\alpha$ specifies whether the particles are treated as dimers or as free atoms. For $\ln\left(k_\mathrm{F}a_\mathrm{2D}\right) \leq 0$ we assume the system to be molecular and use $\alpha=2$ whereas for $\ln\left(k_\mathrm{F}a_\mathrm{2D}\right) \geq 2$ the system is fermionic with $\alpha=1$. Between these two limits we interpolate $\alpha$ linearly as described in \cite{Ries2015}. Typical statistical errors on the extracted temperatures are on the order of \unit[1-2]{\%}.\\

\noindent
\textbf{Local density approximation:} Due to a spatially inhomogeneous density distribution the thermodynamic quantities $T/T_\mathrm{F}$ and $\ln(k_\mathrm{F}a_\mathrm{2D})$ vary from the centre to the outsides of the cloud. We relate these local quantities to those of a homogeneous system by applying the local density approximation (LDA). This approach is justified as long as the external potential varies weakly over the correlation length of the corresponding homogeneous system. In the normal phase this translates into the condition $k_\mathrm{F}(r)R\gg1$, where $R$ is the typical spatial extent of the cloud which in turn accounts for the scale of variation of the potential. For our typical experimental parameters, we can reliably apply the LDA up to around $T/T_F\approx 1.5$. Consequently we restrict our analysis of the pairing gap in Fig. 4 of the main text to this regime.

\begin{table*}[ht!]
	\centering
	\begin{tabular}{c|c|c|c|c|c|c}
		\parbox[c]{1.7cm}{Magnetic\\field (G)}& Mixture &\parbox[c]{1.7cm}{central\\$\ln(k_F a_{2D})$} &\parbox[c]{1.3cm}{central\\$T/T_F$}&\parbox{2.1cm}{two-body\\$E_B/h$ (kHz)}& \parbox[c]{2.1cm}{central\\$E_F/h$ (kHz)} &\parbox[c]{1.7cm}{central\\$\ln(k_F a_{2D})$\\ final state} \\
		\hline
		\hline
		672&$\ket{1}\ket{3}$&-0.16&0.50&9.31&7.61&-9.54\\
		682&$\ket{1}\ket{3}$&0.57&0.45&3.62&7.89 &-8.15\\
		687&$\ket{1}\ket{3}$&0.88&0.39&2.20&7.95&-7.53\\
		692&$\ket{1}\ket{3}$&1.06&0.44&1.37&6.56&-7.05\\
		822&$\ket{1}\ket{2}$&0.96&0.27&2.74&12.12&4.71\\
		834&$\ket{1}\ket{2}$&1.12&0.46&1.52&8.37&4.66\\
		854&$\ket{1}\ket{2}$&1.81&0.14&0.60&11.70&5.02\\
	\end{tabular}
	\caption{Experimental parameters for the datasets used in the main text.}
	\label{tab:T1_parameters}
\end{table*}


\subsection{Systematic Effects}
\textbf{Absorption imaging:} To determine the density $n_\mathrm{2D}\left(r\right)$ we use high intensity absorption imaging at intensities approximately equal to the saturation intensity. To compensate for the Doppler shift the atoms experience during the imaging, we perform a linear ramp of the laser frequency as described in \cite{Ries2015}. As the imaging transitions for state $\ket{1}$ and state $\ket{2}$ are not fully closed, especially at lower magnetic fields, we correct our obtained densities for dark state losses. Overall, our imaging calibration results in an uncertainty of $\sim\unit[7]{\%}$ in the density determination.\\

\noindent
\textbf{Atoms in adjacent layers of optical trap:} From tomographic RF measurements \cite{Ries2015} we can estimate that a small fraction  $ \lesssim \unit[10]{\%}$ of atoms is transferred into adjacent layers of the SWT. This results in an overestimation of the density and thus of our local Fermi momentum and temperature. We expect small additional contributions to the RF signal in the central region at intermediate frequencies due to the weakly populated layers.


\subsection{Experimental Parameters}
We quantify the different regimes of the BEC-BCS crossover by the local parameters $T/T_F$ and $\ln(k_Fa_{2D})$. The temperature $T$ of the sample is determined by the fitting procedure described above. The Fermi momentum and temperature are directly related to the measured local density via $k_F=\sqrt{4\pi n}$ and $T_F=\frac{\hbar^2 k_F^2}{2 m k_B}$, where $n$ is the atomic density of a single component. In addition, the 2D scattering length is obtained from the harmonic oscillator length $l_z=\sqrt{\hbar/m\omega_z}$ in z-direction and the 3D scattering length $a_{3D}$ following (\textit{7}) as
\begin{equation}
a_{2D}=l_z\sqrt{\frac{\pi}{A}}\exp\left(-\sqrt{\frac{\pi}{2}}\frac{l_z}{a_{3D}}\right),
\end{equation}
where $A\approx0.905$. We set the scattering length by adjusting the magnetic offset field in the vicinity of the broad Feshbach resonance at 690 G (832 G) in the $\ket{1}\ket{3}$ ($\ket{1}\ket{2}$) mixture as illustrated in Fig.\,S \ref{fig:S2_RFSpectroscopy} C. The magnetic field is calibrated by comparing the experimentally obtained free-free transition energies $E_\mathrm{free-free}$ in a spin-polarized and thus non-interacting sample to theoretical predictions using the Breit-Rabi formula.\

For the interpretation of our data, we compare the experimentally obtained pairing gap to the two-body bound state energy $E_\mathrm{B}$. It is calculated from the transcendental equation \cite{Levinsen2015}
\begin{equation}
\frac{l_z}{a}=\int_{0}^{\infty}\frac{du}{\sqrt{4\pi u^3}}\left(1-\frac{\exp(\frac{-E_B}{\hbar\omega_z}u)}{\sqrt{\frac{1}{2u}(1-\exp(-2u))}}\right).
\end{equation}
An overview over the experimental parameters can be found in Tab. S 1.\\





\section{Supplementary Information}

\subsection{Theoretical Background on Pairing in the BEC and BCS Limits}
In this section, we provide a brief account of the theoretical aspects of pairing in the well-understood BEC and BCS limits. The treatment presented here has been described in detail in the references \cite{Barth2014,Fischer2014}.\\
The BCS quasiparticle dispersion for an attractive Fermi gas is $E_k = \sqrt{\Delta^2+\xi_k^2}$, where $\Delta$ denotes the superconducting gap, and $\xi_k=E_k-\mu$ is the free dispersion relation $E_k=\hbar^2k^2/2m$ measured from the chemical potential $\mu$.  The spectral function has the form

\begin{equation}
\label{eq:ABCS}
A_\text{BCS}(k,E)
= \underbrace{v_k^2 \delta(E+E_k-\mu)}_\text{bound} + \underbrace{u_k^2 \delta(E-E_k-\mu)}_\text{free}
\end{equation}
for the bound and the free branches, with coherence factors $v_k^2 = (1-\xi_k/E_k)/2$ and $u_k^2 = (1+\xi_k/E_k)/2$.  Within 2D BCS theory the chemical potential at zero temperature $\mu={E_\mathrm{F}} - {E_\mathrm{B}}/2$
and the gap $\Delta = \sqrt{2{E_\mathrm{F}}{E_\mathrm{B}}} = 2{E_\mathrm{F}} e^{-\ln({k_\text{F}a_\text{2D}})}$, while $T_c = e^\gamma\Delta/\pi = 1.13\,{E_\mathrm{F}}\, e^{-\ln({k_\text{F}a_\text{2D}})}$.\\

\noindent
On the BCS side ${E_\mathrm{B}}\ll{E_\mathrm{F}}$, $\mu>0$ and the dispersion reaches a
minimum gap $\Delta=E_{k_0}$ at wavevector $k_0 = \sqrt{2m\mu}/\hbar \approx k_F$, cf.\ Fig.~2C.  In the BEC limit ${E_\mathrm{B}}\gg{E_\mathrm{F}}$ one has $\mu\to-{E_\mathrm{B}}/2$, and the coherence factor $v_k^2\to Z_k=k_F^2a_\text{2D}^2/(1+k^2a_\text{2D}^2)^2$ approaches the square of the bound-state wave function; thus the BCS spectral function asymptotically crosses over into BEC spectral function \cite{Barth2014}

\begin{equation}
\label{eq:ABEC}
A_\text{BEC}(k,E)
= \underbrace{Z_k \delta(E+\xi_k-\mu)}_\text{bound} + \underbrace{(1-Z_k)\delta(E-\xi_k-\mu)}_\text{free},
\end{equation}\\
\noindent
for the bound and the free branch.  The two branches bend outwards in Fig.\ref{fig:imbalance} B, with a minimal distance ${E_\mathrm{B}}$ at $k=0$. \\

\noindent
The RF response in the absence of final-state interactions involving species $\ket c$ takes the form \cite{Fischer2014}

\begin{align}
\label{eq:rf1}
\Gamma_\text{BCS}(\omega)
= \pi\Omega^2 \sum_k A_\text{BCS}(k,E_k-\omega) f(\xi_k-\omega) \nonumber \\
= \frac{\pi\Omega^2N_b}{2{E_\mathrm{F}}}\, f(\xi(\omega)-\omega)
\frac{\Delta^2}{\omega^2} \Theta(\xi(\omega)+\mu),
\end{align}
\\
\noindent
where $\Omega$ is the Rabi frequency, $f(E)$ the Fermi function, and $\xi(\omega)=(\omega^2-\Delta^2)/2\omega$.  $N_b$ denotes the number of particles in state $\ket b$, and we have used the constant density of states $\rho(E) = m/(2\pi\hbar^2)$ in 2D.  The $\Theta$ step function constrains the response to two branches: first, bound-free transitions from the lower branch occur only above a threshold frequency $\omega > \omega_\text{th} = \sqrt{\mu^2+\Delta^2}-\mu = {E_\mathrm{B}} > 0$. Second, free-free transitions from the thermally excited upper branch appear at $\omega<0$, but are too weak to be observed at low temperature, cf.\ Fig.~1D.  Instead, for small spin imbalance a region around the minimum of the upper quasiparticle branch becomes occupied, and a new RF peak appears at $E_\text{free} = -\Delta$ \cite{Schirotzek2008}, cf.\ Fig.~ \ref{fig:imbalance} C,E.  Furthermore, if the interacting species $\ket b$ experiences a mean-field shift $U<0$, the
overall spectrum shifts to $E_\text{th} = {E_\mathrm{B}}-U$ and $E_\text{free} = -\Delta-U$.  At zero temperature we thus obtain 

\begin{align}
\label{eq:13}
\Gamma_\text{BCS}(\omega)
= \frac{\pi\Omega^2N_b}{2{E_\mathrm{F}}}\, \frac{\Delta^2}{\omega^2}
[ \Theta(\omega-{E_\mathrm{B}}+U) \nonumber \\
 + \delta_\text{free}(\omega+\Delta+U) ].
\end{align}
\\
\noindent
{It is a remarkable consequence of pairing in 2D that, within the BCS theory, the RF threshold remains $\mathrm{E_\mathrm{th} = {{E_\mathrm{B}}-U}}$ throughout the crossover. Hence, one cannot distinguish between two-body and many-body pairing by measuring the bound-free transition alone.}  Only by occupying the upper branch and measuring the free peak at $E_\text{free}=-\Delta-U$ can one determine the pairing energy $\Delta E=E_\text{th} - E_\text{free} = {E_\mathrm{B}}+\Delta$ in a way that requires no knowledge of the mean-field shift $U$ and quantifies the many-body gap $\Delta$.




\subsection{Onset of Pairing}

\begin{figure*}[t!]
	\centering
	\includegraphics[width=0.6\textwidth]{S4_Weights}
	\caption{The onset of pairing in a balanced sample. \textbf{A}  and \textbf{C} show, for different temperatures regimes (T=\unit[124]{nK} and T=\unit[295]{nK}), the RF spectra in the $\ket{1}\ket{2}$-mixture at \unit[844]{G} plotted against RF frequency offset and radius, where the relative temperature and interaction parameter are indicated for a selection of radii.
		The vertical line in \textbf{A} marks the prediction for \unit[35]{\%} free particle fraction according to the Saha formula. The red curve in \textbf{B} represents the ${T/T_\textrm{F}}$-dependence of the free-particle fraction for the parameters used in \textbf{A}. We see no significant free fraction at this radius. \textbf{D} shows a cut along the dotted line in \textbf{C} corresponding to $T/T_\mathrm{F} \approx 2.5$ where we observe a free particle fraction of $\sim\unit[35]{\%}$. Here, the Saha formula (blue curve in \textbf{B}) predicts more than \unit[60]{\%} unpaired particles.}
	\label{fig:S4_Weights}
\end{figure*}

The measurement of the pairing gap presented in the main text relies crucially on a sufficiently large population in both the paired and the quasi-particle branch. At finite temperature, quasiparticles are excited either thermally or by introducing a small imbalance. For the studied temperature range of $T/T_F \lesssim 1.5$ we do not observe a significant thermal occupation and thus we work with slight imbalance. This finding is illustrated in Fig.\,S \ref{fig:S4_Weights} A where the RF response of a spin-balanced sample is shown for an experimental setting corresponding to a central $\ln(k_Fa_{2D})\approx 1.4$. Indeed, there is no pronounced quasiparticle peak visible for $T/T_F \lesssim 1.5$. Nevertheless, as the phase-space-density decreases towards the wings of the cloud, a regime of mostly unpaired particles is eventually reached.

\noindent
For a non-interacting gas consisting of dimers and free atoms in thermal equilibrium, the unpaired fraction is given by the Saha formula \cite{Barth2014}:
\begin{equation}
\frac{n^2_f}{n_d}=\frac{mk_BT}{4\pi \hbar^2}e^{-\frac{E_B}{k_B T}},
\end{equation}
where $n_f$ ($n_d$) denotes the free (bound) density of particles. Indicated in Fig.\,S \ref{fig:S4_Weights} A is the relative temperature $T/T_F \approx 0.54$, where the Saha formula predicts a significant free particle fraction of \unit[35]{\%}. We observe an almost fully paired system at this temperature, however. To further study the onset of pairing in a density regime LDA is still valid, we heat the system by trap depth modulation. The corresponding RF spectrum is shown in Fig.\,S \ref{fig:S4_Weights} C. We observe a free particle fraction of $\sim \unit[35]{\%}$ only at around $T/T_F\approx2.5$.
While a quantitative analysis of the onset of pairing remains subject to further studies, the qualitative finding of significantly increased paired fractions compared to expectations from two-body theory further strengthens our interpretation of the normal phase above $T_c$ being strongly influenced by many-body correlations.  


\subsection{Spectra in the Weakly Interacting Limit}

\begin{figure}[b!]
	\centering
	\includegraphics[width=0.4\textwidth]{S5_1000G}
	\caption{Spatially resolved spectra at $\ln{k_F a_{2D}} \sim 3.5$. The Fermi energy $E_\textrm{F}/h \approx 7\,$kHz at the center of the cloud. The fact that we see a single branch suggests the absence of many-body pairing in this regime.}
	\label{fig:S5_Highfield}
\end{figure}

In this section, we show the spatially resolved response function $I(r, \omega_\textrm{RF})$ measured at 1000 G with a $\ket{1}\ket{2}$-mixture, which corresponds to $\ln(k_\textrm{F} a_\textrm{2D}) \sim 3.5$ and therefore represents the weakly interacting limit of the 2D BEC-BCS crossover. At this interaction strength, the binding energy is ${E_\mathrm{B}}/h \approx 7\,$Hz, which is much smaller than all  thermodynamic scales and our experimental RF resolution. In  Fig.\,S \ref{fig:S5_Highfield}, we observe only a single branch in the spectrum. This branch appears close to $ \omega_\textrm{RF} \sim 0$ and has a maximum value of approximately 500 Hz at the center where the Fermi energy (${E_\mathrm{F}}/h$) is about $\unit[7]{kHz}$. The response also exhibits a density-dependent mean-field shift as expected from an interacting system of fermions. \\
Although it is difficult to make a reliable statement about pairing in this regime due to our finite resolution, the presence of a single branch does suggest the absence of a gap that is larger than the scale of our resolution, or in other words $\Delta E \lesssim 0.05\,E_\textrm{F}$ at the center of the trap. This should be contrasted with the spectra at $\ln(k_\textrm{F} a_\textrm{2D}) \sim 1$, where the measured $\Delta E \approx 0.6\,E_\textrm{F}$.



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