
\section{Numerical simulation of the atomic motion in the BoB trap}

To assess the quality of the BoB traps, we performed classical Monte-Carlo simulations of the atomic motion in the trapping potential. This allowed us to compute the recapture probability obtained in a release and recapture experiment (see Fig. 3a of the main text for the sequence) and compare it with experimental data. We simulate the full experimental sequence, including the two free-flight stages during Rydberg excitation and de-excitation, and the atomic motion in the BoB trap for a time $\tau$. To determine if an atom is recaptured at the end of the sequence, we compare its kinetic energy to the potential energy of the ground state trap at the final position of the atom. 

We account for the finite Rydberg lifetime at 300~K by introducing, for each Rydberg state, a single exponential probability distribution function in the dynamics. This approximation is justified because the two processes contributing to the Rydberg lifetime (decay to low-lying states and blackbody radiation-induced emission to other Rydberg states) produce the same outcome in this experiment: a loss of the atom in the final fluorescence image. Atoms that are not successfully excited by the STIRAP pulses, or Rydberg atoms that spontaneously decay to the ground state, are quickly repelled from the trapping region by the BoB potential and not recaptured at the end of the sequence. On the contrary, Rydberg atoms transferred to other Rydberg states by black-body radiation can still remain trapped in the BoB, but are not projected to the ground state by the de-excitation pulse. As a consequence, they are repelled by the repulsive potential of the standard optical tweezers before imaging, and finally lost.

For each simulation we use the same parameters as in the experiment (namely, a total power of 400 mW, the recorded three-dimensional trap intensity distribution shown in Fig. 2a of the main text, and the measured STIRAP excitation efficiency). The final result of the simulation is obtained by averaging over $\sim10^4$ realizations, starting from random atom positions (with rms values $\sigma_r = \sqrt{k_B T/m \omega_r^2}\sim 230\, \rm{nm}$; $\sigma_z = \sqrt{k_B T/m \omega_z^2}\sim 1.15\,\mu\rm{m}$) and velocities (rms value $\sigma_v = \sqrt{k_B T/m}\sim 110\, \rm{nm}/ \mu\rm{s}$),  according to a thermal atomic distribution in the standard optical tweezers at $T=130\,\mu$K, with radial and longitudinal trapping frequencies $\omega_r / (2\pi) = 75\, \rm{kHz}$, and $\omega_z / (2\pi) = 15\, \rm{kHz}$, respectively.



\begin{figure*}[b]
\centering
\includegraphics[width=\textwidth]{fig_S1.pdf}
\caption{Simulated recapture probability as a function of the trapping time $\tau$ (solid lines) for the Rydberg states $75S_{1/2}$ (a) and $84S_{1/2}$ (b). The experimental data (disks) are shown for comparison. Insets are zooms on the short-time behavior (dashed lines) showing, in linear scale, a fast atom loss at short times.}
\label{fig:S1}
\end{figure*}

Figure S\ref{fig:S1} shows the results of the simulation for two different Rydberg states, $75S_{1/2}$, and $84S_{1/2}$. The agreement between the experimental data and the simulation with no free parameters is very good over the full time span. At short times ($\tau < 50 \,\mu\rm{s}$) we observe a fast decrease in the recapture probability (see inset) both in the experimental data and in the simulation. In the simulation, this initial drop has two contributions: (i) about $10-15\%$ of this decrease (depending on the dataset) corresponds to the loss of atoms that remain in the ground state after the STIRAP excitation pulses; (ii) the other $10\%$ is due to energetic atoms having enough kinetic energy to overcome the potential barrier of the BoB trap. 

\section{Analysis of the mechanical losses in the BoB trap}

In order to investigate the (small) initial  atom loss and to distinguish ``mechanical losses'' due to inefficient or unstable trapping from atom losses due to Rydberg state decay, we performed the same simulation as described in the previous section, but now assuming an infinite Rydberg lifetime. The results are shown in Fig. S\ref{fig:S2}a.

\begin{figure*}[t]
	\centering
	\includegraphics[width=17cm]{fig_S2.pdf}
	\caption{(a) Simulated recapture probability as a function of the trapping time for different Rydberg states assuming an infinite Rydberg lifetime. (b) Classical trajectories of Rydberg atoms evolving in the BoB potential. For the $60S_{1/2}$ state mechanical losses occur close to the saddle points (`escape cones'), whereas for $100S_{1/2}$ atoms leave the trap uniformly in the radial direction.}
	\label{fig:S2}
\end{figure*}

For $n\le 100$, we observe again the fast initial decrease in the recapture probability in a time scale of $\sim 50\, \mu\rm{s}$, before reaching an almost stationary value at around $\tau = 1$~ms. For times longer than $\tau\sim 50\,\mu\rm{s}$ the decay rate due to mechanical losses is comparatively much lower than the inverse of the Rydberg state lifetime. The fact that extra losses of atoms occur mainly at short times allows us to measure Rydberg trapping times close to the theoretical Rydberg state lifetimes. For $n=120$, the potential does not have a local minimum (see Fig. 2 of the main text), and Rydberg atoms are quickly repelled from the trapping region.

The atom trajectories shown in Fig. S\ref{fig:S2}b give us information about the escape regions in the trapping potential. For the $60S_{1/2}$ atoms escape mainly though the saddle points. For higher Rydberg states, the effect of the convolution is to smooth the saddle cones at the expense of decreasing the effective potential barrier, and atoms escape the trap in the radial direction in a more homogeneous manner.



\section{Oscillation frequency of the atoms in the BoB trap}

\begin{figure*}[b]
\centering
\includegraphics[width=17cm]{fig_S3.pdf}
\caption{(a) Sketch of the experimental sequence to measure the oscillation frequency of the atoms in the BoB trap. (b) Experimental data (disks) and classical simulation (solid curve) of the 'release and recapture' experiment performed using the sequence in (a) for a trap depth of 0.6 mK.}
\label{fig:S3}
\end{figure*}

We present in this section an experiment to measure the typical radial  trapping frequency in the BoB trap. The experimental sequence is shown in Fig. S\ref{fig:S3}a. We first transfer the Rydberg atom to the BoB trap. We then wait for $30\,\mu\rm{s}$. During this preliminary waiting time, the hottest atoms escape, as explained in the previous section. Then, we follow the same approach as in \cite{Tuchendler2008}, replacing the standard optical tweezers by our BoB trap. We switch off the trap during $4\,\mu\rm{s}$ to excite the radial breathing mode. Then, atoms oscillate in the BoB trap for a varying duration $\tau$. We release the atoms for $10\,\mu\rm{s}$ and finally de-excite and recapture them in the standard optical tweezers. For a harmonic trap, the recapture probability is expected to oscillate at $2\omega$, with $\omega$ the trapping frequency.

Fig. S\ref{fig:S3}b shows the measured recapture probability for a trap depth of 0.6 mK. From the observed oscillations we extract a trapping frequency, $\omega_{\rm{BoB}} / (2\pi) = 59\pm 3\,\rm{kHz}$. The observed damping of the oscillations is most likely due to the finite atom temperature and the anharmonicity of the trapping potential. In the simulation, we follow the same procedure described above to numerically compute the recapture probability with no adjustable parameters. The result (solid line) is in reasonable agreement with the experimental data, as we find a trapping frequency $\omega_{\rm{BoB}} = 2\pi\times 59\,\rm{kHz}$.



