[**SUPPLEMENTARY INFORMATION**]{}

Figure Details
==============

The tip height for the voltage spectra and all of the constant height images was 19 nm $\pm$ 1 nm, with the exception of Fig. 4d-g and Fig. S1 where the height was $\sim 23$ nm. Additional image details are listed in Supplementary Table 1. The acquisition time of the majority of spectra was 15 seconds.

  Fig.    T (K)   $\df/2\pi$ (Hz)   Oscillation Amplitude (nm)   $\Vbias$ (V)   Acquisition Time (min.)
  ------ ------- ----------------- ---------------------------- -------------- -------------------------
  1c       78          -9.4                    1.6                  -0.35                  6
  1d       4.5           –                     0.4                   -8.0                 119
  1e       4.5           –                     0.4                   -8.0                 119
  3a       4.5           –                     0.4                   -9.0                 51
  3b       4.5           –                     0.4                   -7.6                 17
  3c       78          -9.4                    1.6                  -0.35                 14
  3d-e     4.5           –                     0.4                   -8.0                 51
  4a       78          -9.4                    1.6                  -0.35                  9
  4b       78            –                     1.6                   -8.0                 68
  4c       4.5           –                     0.4                   -8.0                 51
  4d       4.5           –                     0.8                   -8.0                  9
  4e       4.5           –                     0.8                   -9.0                  9
  4f       4.5           –                     0.8                   +6.8                  9
  4g       4.5           –                     0.8                   -8.0                  9
  5        4.5           –                     0.8                    –            17 (last image 9)

Details of dissipation with degenerate shells
=============================================

Here we outline the approach used to derive the general expression for the dissipation in Eq. (2). The charging Hamiltonian for small cantilever tip motion may be written $$\begin{aligned}
  \label{eq:Hc}
  H_{\mathrm{C}} &= 
  \sum_N  E_{\text{C}_N}
  \left[ \left( N - \Nvoltage \right)^2  
  + \left( 1 + \frac{C_\text{2DEG}}{C_\text{tip}}  \right) \Nvoltage^2
  \right]
  \ket{N} \bra{N}
  \\
  &= H_{\mathrm{C},0} + \Delta H_\text{osc} - \sum_N 
  A_N N z \ket{N} \bra{N}, \nonumber\end{aligned}$$ where $\ket{N}$ is a state with $N$ electrons on the QD, $\Nvoltage = - \frac{C_\text{tip} \Vbias}{e}$ is the dimensionless gate voltage, $C_\text{tip}$ is the QD-tip capacitance and $C_\text{2DEG}$ is the QD-2DEG capacitance. In the second line, $H_{\mathrm{C},0}$ is the cantilever-independent part of the charging Hamiltonian, $\Delta H_\text{osc}$ describes an electrostatic modification of the cantilever potential, and the QD-cantilever coupling strength for given $N$ is $A_N = -2E_{\text{C}_N} \frac{\Vbias}{e} \left( 1-\alpha \right) 
\frac{\partial C_\mathrm{tip}}{\partial z}$. We emphasize that $\Ec$ and $A$ may be different for each electron added, as indicated by the index $N$. From the second equality in equation (\[eq:Hc\]), $N$ plays the role of a force on the cantilever. As a result, the dissipation and frequency shift may be found from the linear response coefficient $\lambda_N(\omega)$ describing the response of $N$ to changes in $z$ [@Clerk05].

Consider the charge degeneracy point between $N$ and $N+1$ electrons on the QD. This may always be viewed as $\nShell$ or $\nShell+1$ electrons occupying a shell of degeneracy $\nu$ (even for a nondegenerate single level, for which $\nShell = 0$ and $\nu = 1$). Neglecting interactions, the charge state with $\nShell$ ($\nShell+1$) electrons in the shell is $\Dn$-fold ($\Dnn$-fold) degenerate, with $$\Dn = \binom{\nu}{\nShell}
    \quad,\quad
    \Dnn = \binom{\nu}{\nShell + 1},$$ where $\binom{\cdot}{\cdot}$ denotes a binomial coefficient. These arise simply from the different ways to put $\nShell$ or $\nShell + 1$ electrons into $\nu$ single particle states. Let $P_{\nShell,i}$ be the probability to find $\nShell$ electrons occupying the shell in configuration $i$, and $P_{\nShell+1,j}$ be the probability to find $\nShell+1$ electrons occupying the shell in configuration $j$. In general, these probabilities will satisfy the master equations [@Beenakker91] $$\begin{aligned}
    \partial_t P_{\nShell,i} &= \sum_j \left\{
    \Gamma_{j \rightarrow i}  P_{\nShell+1,j} 
    -\Gamma_{i \rightarrow j} P_{\nShell,i} 
    \right\}
    \label{eq:ME1} \\
    \partial_t P_{\nShell+1,j} &=
     \sum_i \left\{
    \Gamma_{i \rightarrow j} P_{\nShell,i} 
    - \Gamma_{j \rightarrow i} P_{\nShell+1,j} \right\},
    \label{eq:ME2}\end{aligned}$$ where $\Gamma_{i \rightarrow j}$ is the rate to add an electron to configuration $i$ producing configuration $j$, and vice versa for $\Gamma_{j \rightarrow i}$ (note that these rates are nonzero only for configurations $i$ and $j$ that differ by the addition or removal of one electron). We calculate the rates using Fermi’s golden rule.

The master equations ( \[eq:ME1\]) and ( \[eq:ME2\]) may be solved for given values of $\nu$ and $\nShell$, but the solutions are cumbersome for highly degenerate shells. To simplify the equations we assume that for a given charge degeneracy point (i.e. a single dissipation peak), the tunneling matrix elements from Fermi’s golden rule are equal for all single particle states within the relevant shell. This is an approximation, since degenerate states may indeed have different wavefunctions leading to different tunneling rates. However, we expect the rates to be similar since the tunnel barrier between the QD and the 2DEG extends over the entire QD area, minimizing the effects of the spatial variations of different wavefunctions. Moreover, we checked that significantly unequal rates lead only to very small corrections in the peak shifts. For example, taking distinct rates for the two degenerate orbital states in the $p$ shell, we find that rates differing by a factor of 2 lead to a correction of 1.5% for the shift of the 3rd dissipation peak (i.e. the 1st peak in the $p$ shell). We thus neglect these possible differences here. Taking the rates to be equal we arrive at the simplified master equation for the total probability to find $\nShell$ electrons in the shell, $$\begin{aligned}
\label{eq:ME}
    \partial_t P_\nShell &= (\nu-\nShell)
    \left[
    \frac{\Dn}{\Dnn}  \Gm (1- P_\nShell) - \Gp P_\nShell
    \right]\end{aligned}$$ where $$\begin{aligned}
    \Gamma_+ = \Gamma f(E)
    \quad,\quad
    \Gamma_- = \Gamma \left[1- f(E)\right]\end{aligned}$$ are the rates to add (+) or remove ($-$) an electron to or from a single particle state, and $f$ is the Fermi function. Note that the master equation for $P_{\nShell+1}$ is not independent in our approximation of equal rates; this is a result of $P_\nShell + P_{\nShell+1} = 1$. The stationary solution of equation (\[eq:ME\]) is $$\begin{aligned}
\label{eq:pStat}
    P_\nShell &= \frac{(\nShell+1)}{\phi}(1-f) \\
    P_{\nShell+1} &= \frac{(\nu-\nShell)}{\phi} f,\end{aligned}$$ where $\phi$ is defined in equation (3).

The quantity we need is the linear response coefficient $\lambda_N(\omega)$. To find this, we assume that the cantilever is oscillating at frequency $\omega$. This causes the chemical potential difference between the QD and the 2DEG to oscillate, $$\label{eq:Eosc}
    E \rightarrow E + \delta e^{-i\omega t},$$ and this leads to a change in the probabilities, $$\begin{aligned}
    P_{\nShell+1} &\rightarrow  P_{\nShell+1} + \lambda_N(\omega) \delta e^{-i\omega t}, \\
    P_{\nShell} &\rightarrow P_\nShell - \lambda_N(\omega) \delta e^{-i\omega t}.
    \label{eq:pChange}\end{aligned}$$ Inserting equations (\[eq:pStat\])–(\[eq:pChange\]) into equation (\[eq:ME\]) and linearizing in $\delta$, we solve for $\lambda_N(\omega)$. Its real and imaginary part yield the dissipative and conservative parts of the electrostatic force from $(k_0/\omega_0^2) \gamma = - A^2 \Im{\left\{ \lambda_N(\omega) \right\}} / \omega $ and $(2k_0/\omega_0) \Delta \omega = A^2 \Re{\left\{ \lambda_N(\omega)\right\}}$. The dissipation for arbitrary degeneracy is given in equation (2). and for the frequency shift we obtain $$\label{eq:freqShiftGen}
    \Delta \omega = -\frac{\omega_0}{2k_0} \frac{A^2 \Gamma^2}{\kBol T}
    \left[
    \frac{(\nShell+1)(\nu-\nShell)}{\omega^2 + (\phi \Gamma)^2} 
    \right]
    f(1-f)  .$$ Note that we recover the single level result \[i.e. equation (1) for the dissipation\] by taking $\nu = 1$ and $\nShell = 0$ as expected. Finally, we point out that the temperature-dependent level repulsion discussed in the paper is contained in a symmetry of equations (2) and (\[eq:freqShiftGen\]), from which we find that taking $\nShell \rightarrow \nu-\nShell -1$ is equivalent to $E \rightarrow -E$.

The peak shifts of $\gamma$ and $\Delta \omega$ are proportional to temperature and we can solve for the coefficients analytically. However, in general the coefficients are complicated and unenlightening. To show how the peak shifts depend on degeneracy, we provide the coefficients in the low and high frequency limits where they are greatly simplified. Note that our experiment is in the intermediate regime $\omega \sim \Gamma$, so the peak shifts measured and calculated in the main text lie between these two limits. For $\gamma$, the peak shifts in the low and high frequency limits are $$\frac{\Delta E_{\gamma,\text{peak}}}{\kBol T} \rightarrow
    \begin{cases}
    \ln{\left( d + \sqrt{d(d+1) +1} \right)} &\text{as } (\omega\rightarrow 0), \\
    \ln{ \sqrt{d+1} } &\text{as } (\omega\rightarrow \infty),
    \end{cases}$$ where $$d = \frac{\nu-\nShell}{\nShell+1} - 1.$$ For a nondegenerate level, $d=0$ and there is no peak shift at any frequency. For $\Delta \omega$ the peak shifts in the same two limits are $$\frac{\Delta E_{\Delta\omega,\text{peak}}}{\kBol T} \rightarrow
    \begin{cases}
    \ln{\left( d + 1\right)} &\text{as } (\omega\rightarrow 0), \\
    0 &\text{as } (\omega\rightarrow \infty).
    \end{cases}$$ Comparing these limits, we see that the shell degeneracy results in a greater peak shift in $\gamma$ than in $\Delta \omega$. This is a direct consequence of equation (4), from which we see that, aside from an energy-independent prefactor, $\Delta\omega$ differs from $\gamma$ by a factor of $\phi$.

We measured the separation between the peak in $\gamma$ and the peak in $\Delta \omega$ for each charge degeneracy point. This is shown in Fig. \[fig:peak3\] for the third peak in Fig. 2a as a function of temperature and compared to theory with no fit parameters. As argued in the main text, this provides strong evidence that the observed peak shifts are indeed a result of shell degeneracy.
