*Appendix B.1. Self-Consistent Theory*

In this section, we shortly review the description of the mesoscopic capacitor in the seminal works by Büttiker, Thomas, and Prêtre [21,51–53], based on a self-consistent extension of the Landauer–Büttiker scattering formalism [45–47]. The following discussion is also inspired from Refs. [20,241,242].

Figure A2 illustrates the intuition behind the interpretation of the mesoscopic capacitor as a quantum analog of a classical *RC* circuit. The top metallic gate (a classical metal) and the quantum dot cannot exchange electrons. These two components make up the two plates of a capacitor on which electrons accumulate according to variations of the gate potential *<sup>V</sup>*g. The value of the capacitance depends on the geometry of the contact and, in the experiment of Ref. [20], the *geometrical capacitance C*g was estimated ∼10–100 fF. This capacitance is in series with a quantum point contact. As mentioned in the main text, direct transport measurements [160,161] consider QPCs as resistive elements of resistance *R*DC = *h*/*e*2*D*, where *D* is the QPC transparency. The above considerations sugges<sup>t</sup> the interpretation of the device in Figure A2 as an *RC* circuit. The admittance of a classical *RC* circuit reads:

$$\mathcal{A}(\omega) = \frac{-i\omega \mathbb{C}}{1 - i\omega \mathbb{R}\mathbb{C}} = -i\omega \mathbb{C} \left( 1 + i\omega \mathbb{C} \mathbb{R} \right) + \mathcal{O} \left( \omega^3 \right). \tag{A35}$$

The admittance (A35) can be calculated with the scattering formalism [45–47], which requires to be adapted to describe the mesoscopic capacitor in Figure A2. The main problem is the "mixed" nature of the quantum *RC* device: The mesoscopic capacitor is composed of a phase-coherent part (two-dimensional electron gas + quantum dot) in contact to an incoherent top metallic gate. As these constituents do not exchange electrons, preventing a direct current, electron transport is only possible by driving the system. We focus on the case of a gate potential oscillating periodically, as Equation (35) in the main text,

$$V\_{\mathfrak{F}}(t) = V\_{\mathfrak{F}} + \varepsilon\_{\omega^\circ} \cos(\omega \, t) \,. \tag{A36}$$

For small oscillation amplitudes *εω*, the Landauer–Büttiker formalism allows to derive the circuit admittance within linear response theory. In the case of a single conduction mode, the admittance reads [51]:

$$a(\omega) = \frac{e^2}{\hbar} \int d\varepsilon \text{Tr} \left[ 1 - S^\dagger(\varepsilon) S(\varepsilon + \hbar \omega) \right] \cdot \frac{f(\varepsilon) - f(\varepsilon + \hbar \omega)}{\hbar \omega},\tag{A37}$$

in which *f*(*ε*) is the Fermi distribution function, in which we fix to zero the value of the Fermi energy. For one channel, the elastic scattering assumption implies that the matrix *<sup>S</sup>*(*ε*) reduces to a pure phase *<sup>S</sup>*(*ε*) = *<sup>e</sup>*2*iδε*, as electrons entering the dot return to the lead with unit probability. The phase-shift *δε* is related to the dot electron occupation, via the Friedel sum rule (A14). Additionally, this phase is also related to the *dwell-time* that electrons spend in the quantum dot, or Wigner–Smith delay time [89,243]:

$$\frac{\pi(\varepsilon)}{h} = \frac{1}{2\pi i} S^\dagger(\varepsilon) \frac{dS(\varepsilon)}{d\varepsilon} = \frac{1}{\pi} \frac{d\delta\_\varepsilon}{d\varepsilon} \,. \tag{A38}$$

The interpretation of *τ* as a dwell-time is illustrated in Section 4.2. In the limits *T* → 0 and *h*¯ *ω* → 0, Equation (A37) becomes:

$$a(\omega) = -i\omega \frac{e^2}{\hbar} \left[ \tau + \frac{1}{2} i\omega \tau^2 + O(\omega^2) \right]. \tag{A39}$$

The dwell-time *τ* = *τ*(0) is considered at the Fermi energy. Notice that this expression has the same frequency expansion as Equation (A35). Matching term by term, one finds:

$$\mathbf{C\_{q}} = \frac{e^{2}}{h} \mathbf{r}\_{\prime} \tag{A} \tag{A\_{\parallel}} \\ = \frac{h}{2e^{2}} \,. \tag{A40}$$

Such relation between time-delays and circuit elements comes from the fact that electrons arriving on the dot at different times are differently phase-shifted, because of the variations in time of the gate potential *<sup>V</sup>*g. This effect causes a local accumulation of charges, which is responsible for the emergence of quantum capacitive effects, corresponding to *<sup>C</sup>*q. The time delay of the electron phase *δ* with respect to the driving potential *U*(*t*) is responsible for energy dissipation, controlled by *<sup>R</sup>*q. The characteristic time that an electron spends in the quantum dot is given by *τ* = <sup>2</sup>*R*q*C*q, twice the *RC* time because it includes the charging and relaxation time of the *RC* circuit. Notice the emergence of a universally quantized relaxation resistance, regardless of any microscopic detail of the quantum *RC* circuit, in contrast with the resistance *R*DC = *h*/*e*2*D*, sensitive to the transparency *D* of the QPC and which would be measured in a DC experiment, see also Equation (43) in the main text.

**Figure A2.** (**<sup>a</sup>**,**b**) Schematic representation of the quantum *RC* circuit. Electrons in the edge states of a two-dimensional electron gas (2DEG) in the integer quantum Hall regime tunnel inside a quantum dot through a QPC. The dot is driven by a top metallic gate. The dot and the gate are separated by an insulator and cannot exchange electrons, thus forming the two plates of a capacitor *C*. The QPC is a resistive element of resistance *R*. These two circuit elements are in series and define a quantum coherent *RC* circuit. (**<sup>c</sup>**,**d**) The blue region of the two-dimensional gas is phase coherent. The top metallic gate is an incoherent metal, driven by a time-dependent gate potential *Vg*(*t*), which induces an unknown uniform potential *U*(*t*) on the dot. The classical circuit analogy in (**d**) is made possible by shifting all energies by − *<sup>U</sup>*(*t*). The whole device behaves as a charge relaxation resistance *<sup>R</sup>*q in series to a total capacitance *C*: series of a quantum and geometrical capacitance *C*q and *<sup>C</sup>*g.

Notice that the geometrical capacitance *C*g does not appear in the previous discussion. The admittance Equation (A37) has been derived by applying linear response theory for the driving potential *U*(*t*) in the quantum dot, see Figure A2. The potential *U*(*t*) does not coincide with the actual driving gate-potential *Vg*(*t*). The situation is pictured in Figure A2: The geometric capacitance *C*g leads to a potential drop between gate and dot. In a mean-field/Hartree–Fock treatment, the potential *U*(*t*) on the dot is assumed to be uniform for each electron. This assumption is equivalent to a Random Phase Approximation (RPA), valid for weak interactions or to leading order in a 1/*M* expansion, with *M* the number of channels connected to the dot [78]. The potential *U* can then be determined self-consistently from the constraint of charge/current conservation in the whole device. The current *I*dev flowing in the coherent part of the device has to equal the current *<sup>I</sup>*gate flowing in the incoherent metallic gate:

*I* = *I*dev = *<sup>I</sup>*gate . (A41)

As all potentials are defined with respect to an arbitrary energy, they can be shifted by <sup>−</sup>*eU*(*t*), setting the potential to zero in the quantum dot. Thus, the currents in the device and in the metallic gate read:

$$I\_{\rm dev} = -lI(\omega)g(\omega) \,, \qquad \qquad \qquad I\_{\rm gate} = -i\mathbb{C}\_{\rm g}\omega \left[lI(\omega) - V\_{\mathcal{J}}(\omega)\right] \,. \tag{A42}$$

Applying the current conservation condition (A41), the potential *U* can be eliminated, and the admittance of the total device is derived:

$$\mathcal{A}(\omega) = -\frac{I}{V\_{\mathcal{S}}} = \frac{1}{\frac{1}{a(\omega)} + \frac{1}{-\omega \omega \mathbb{C}\_{\mathbb{B}}}} \,. \tag{A43}$$

Recalling the low frequency behavior of *a*(*ω*) in Equation (A39), the above expression shows that the whole device behaves as a RC circuit. Albeit with two capacitances in series, Equation (A43) still gives a universally quantized *<sup>R</sup>*q = *h*/2*e*2. The series of *C*q and *C*g gives the total capacitance *C*, originally denoted as *electro-chemical capacitance* [21,51–53].

We can consider a simple model for the mesoscopic capacitor to estimate the behavior of the dwell-times *τ* setting the quantum capacitance (A40). We consider the case of Figure A2, in which electrons propagate in the integer quantum Hall edges inside the quantum dot, see also Figure 7 in the main text. We label - the length of the edge state in the quantum dot and *vF* the Fermi velocity of the electron. The dwell-time for electrons of velocity *vF* inside the dot is *τ*f = -/*vF*. An electric wave of energy *ε* acquires a phase *φ*(*ε*)=(*ε* − *eU*)*τ*f/¯*h*, when making a tour of the dot. Notice that we had to shift the energy *ε* of the electron by −*eU* because of the potential shift schematized in Figure A2. The chiral nature of the edge states allows for a one-dimensional representation of the problem, pictured in the right-top of Figure 7. For a quantum well of size -/2, close to the Fermi energy, the level spacing Δ = *hvF*/- is constant. Thus *τ*f = *h*/Δ and, substituting in Equation (A40), leads to the uniform quantum capacitance *C*q = *<sup>e</sup>*2*τ*f/*h*, derived heuristically in the main text, see Equation (41). If the reflection amplitude at the entrance of the dot is *r* and *D* = 1 − |*r*| 2 the transmission probability, the dot can be viewed as a Fabry–Perot cavity and the phase of the out-coming electron is:

$$S(\varepsilon) = r - De^{i\phi(\varepsilon)} \sum\_{q=0}^{\infty} r^q e^{i\phi\phi(\varepsilon)} = \frac{r - e^{i\phi(\varepsilon)}}{1 - re^{i\phi(\varepsilon)}} = e^{i2\delta\_{\varepsilon}}.\tag{A44}$$

Applying Equation (A38) we obtain the local density of states:

$$\mathcal{N}(\varepsilon) = \frac{\tau\_{\rm f}}{h} \frac{1 - r^2}{1 - 2r \cos\left[\frac{2\pi}{h} \left(\varepsilon - \varepsilon lI\right)\tau\_{\rm f}\right] + r^2}. \tag{A45}$$

This quantity is plotted in Figure A3 and reproduces the oscillatory behavior of the capacitance in Figure 8. In the limit of small transmission (*D* 1 and *r* ≈ 1), Equation (A45) reduces to a sum of Lorentzian peaks of width ¯*hγ*, *γ* = *D*/*<sup>τ</sup>*f:

$$\mathcal{N}(\varepsilon) = \frac{2}{\pi \hbar \gamma} \sum\_{n} \frac{1}{1 + \left(\frac{\varepsilon - cL - n\Delta}{\hbar \gamma/2}\right)^2}. \tag{A46}$$

These peaks are the discrete spectrum of the dot energy levels.

The above arguments can be readily generalized to the case with *M* channels in the lead and in the cavity. In this case, every single channel *σ* can be considered independently. The admittance (A39) can be then cast in the form:

$$\mathfrak{a}(\omega) = -i\omega \frac{e^2}{\hbar} \sum\_{\sigma=1}^{M} \left[ \mathfrak{r}\_{\sigma} + \frac{1}{2} i\omega \mathfrak{r}\_{\sigma}^2 + \mathcal{O}\left(\omega^2\right) \right]. \tag{A47}$$

The low-frequency expansion of the RC circuit admittance (A35) is then recovered by defining:

$$\mathbf{C\_{q}} = \frac{\mathbf{c^{2}}}{h} \sum\_{\sigma=1}^{M} \mathbf{r\_{\sigma}} \, , \tag{4.48} \\ \mathbf{R\_{q}} = \frac{h}{2\epsilon^{2}} \frac{\sum\_{\sigma} \mathbf{r\_{\sigma}^{2}}}{\left(\sum\_{\sigma} \mathbf{r\_{\sigma}}\right)^{2}} \, , \tag{A48}$$

in which *τσ* are the dwell-times in the quantum dot of electrons in the *σ*-th mode.

### *Appendix B.2. Hamiltonian Description of the Quantum RC Circuit with a Resonant Level Model*

In this appendix, we study the resonant level model (A16) as a RC circuit. The generalization of the following calculations to the many-channel case is straightforward and, in particular, we extend to the multi-level case. We thus recover the self-consistent scattering theory analysis discussed in Appendix B.1, corresponding to the limit *C*g → ∞ (non-interacting limit). Our aim is the calculation of the dynamical charge susceptibility:

$$
\chi\_c(t - t') = \frac{i}{\hbar} \theta(t - t') \left< \left[ N(t), N(t') \right] \right>\_{0 \text{ \textquotedbl{}}} \tag{A49}
$$

which leads to the admittance of the circuit A(*ω*) = <sup>−</sup>*iωe*2*χ*c(*ω*), see also Equation (36) in the main text. We make use of the path integral formalism as in Appendix A.4. The partition function corresponding to Equation (A16) reads:

$$\mathcal{Z} = \int \mathcal{D}\left[c, c^{\dagger}, d, d^{\dagger}\right] e^{-\mathcal{S}\_{\text{Ran}}\left[c, c^{\dagger}, d, d^{\dagger}\right]},\tag{A50}$$

where S *c*, *c*†, *d*, *d*† is the action of the system, which reads

$$\mathcal{S}\_{\text{Res}} = \int\_0^\beta d\tau \left\{ -\sum\_k c\_k^\dagger(\tau) G\_k^{-1}(\tau) c\_k(\tau) - d^\dagger(\tau) D^{-1}(\tau) d(\tau) + t \sum\_k \left[ c\_k^\dagger(\tau) d(\tau) + \text{c.c.} \right] \right\},\tag{A51}$$

with the free propagators:

$$G\_k^{-1}(\tau) = -\partial\_\tau - \varepsilon\_k, \qquad \qquad D^{-1}(\tau) = -\partial\_\tau - \varepsilon\_d,\tag{A52}$$

in which *ck* and *dl* are Grassmann variables. It is practical to switch to the Matsubara frequency representation, which leads to:

$$\begin{split} \mathbf{S}\_{\text{Res}} = \sum\_{i\omega\_{n}} \left\{ -\sum\_{k} c\_{k}^{\dagger}(i\omega\_{n}) \mathbf{G}\_{k}^{-1}(i\omega\_{n}) \mathbf{c}\_{k}(i\omega\_{n}) - d^{\dagger}(i\omega\_{n}) D^{-1}(i\omega\_{n}) d(i\omega\_{n}) \\ \\ + t \sum\_{k} \left[ c\_{k}^{\dagger}(i\omega\_{n}) d(i\omega\_{n}) + d^{\dagger}(i\omega\_{n}) c\_{k}(i\omega\_{n}) \right] \right\}, \quad \text{(A53)} \end{split} \tag{A53}$$

with *G*−<sup>1</sup> *k* (*<sup>i</sup><sup>ω</sup>n*) = *i<sup>ω</sup>n* − *εk* and *<sup>D</sup>*−<sup>1</sup>(*<sup>i</sup><sup>ω</sup>n*) = *i<sup>ω</sup>n* − *εd*. Reestablishing dimensions, the Fourier transform of Equation (A49) reads:

$$\chi\_{\mathfrak{C}}(i\nu\_{n}) = \frac{1}{\hbar} \int\_{0}^{\hbar \beta} d(\tau - \tau') e^{i\nu\_{n}(\tau - \tau')} \left< N(\tau)N(\tau') \right>\_{0} \tag{A54}$$

and *<sup>χ</sup>*c(*ω*) is recovered by performing the analytical continuation *iν* → *ω* + *i*0+. This function is periodic in imaginary time and its Fourier transform is a function of the bosonic Matsubara frequencies *i<sup>ν</sup>n* = <sup>2</sup>*n<sup>π</sup>*/*β*. *<sup>N</sup>*(*τ*) = *<sup>d</sup>*†(*τ*)*d*(*τ*) counts the number of charges on the dot. The cyclic invariance property of the trace implies that *N*(*τ*)*N*(*τ*)0 = *f*(*τ* − *<sup>τ</sup>*), allowing us to recast (A54) in the form:

$$\chi\_{\mathbf{c}}(i\nu\_{n}) = \frac{1}{\mathcal{\mathcal{S}}} \sum\_{i\omega\_{1,2}} \left\langle d^{\dagger}(i\omega\_{1})d(i\omega\_{1} + i\nu\_{n})d^{\dagger}(i\omega\_{2})d(i\omega\_{2} - i\nu\_{n})\right\rangle \,. \tag{A55}$$

To calculate this expression, we first perform the Gaussian integral of the lead modes in Equation (A53), leading to the effective action SResof the resonant level model:

$$\mathcal{L}\_{\text{Res}}^{\prime} = -\sum\_{i\omega\_{n}} d^{\dagger}(\text{i}\omega\_{n}) \mathcal{D}(\text{i}\omega\_{n}) d(\text{i}\omega\_{n}) \,, \qquad \mathcal{D}^{-1}(\text{i}\omega\_{n}) = \text{i}\omega\_{n} - \varepsilon\_{d} - t^{2} \sum\_{k} \text{G}\_{k}(\text{i}\omega\_{n}) \,. \tag{A56}$$

In the wide-band approximation the propagator can be written as D−<sup>1</sup>(*<sup>i</sup><sup>ω</sup>n*) = *i<sup>ω</sup>n* − *εd* + *<sup>i</sup>*Γsgn(*<sup>ω</sup>n*), where we introduced the hybridization constant Γ = *πν*0*t*2. The action (A56) is quadratic and the application of Wick's theorem [83] in Equation (A55) leads to:

$$\chi\_{\mathbf{c}}(i\nu\_{\mathrm{n}}) = -\frac{1}{\beta} \sum\_{\mathrm{i}\omega\_{\mathrm{n}}} \mathcal{D}(\mathrm{i}\omega\_{\mathrm{n}}) \mathcal{D}(\mathrm{i}\omega\_{\mathrm{n}} + i\nu\_{\mathrm{n}}) \to -\frac{1}{\pi\Gamma} \int\_{-\infty}^{\infty} d\mathbf{x} f(\Gamma \mathbf{x} + \varepsilon\_{d}) \frac{2\mathbf{x}}{(\mathbf{x}^{2} + 1)[\mathbf{x}^{2} - (\omega/\Gamma + i)^{2}]},\tag{A57}$$

where the analytical continuation *iν* → *ω* + *i*0<sup>+</sup> has been performed. At zero temperature, the integral can be calculated analytically, leading to:

$$\chi\_{\mathbb{C}}(\omega) = \frac{1}{\pi \Gamma} \frac{1}{\frac{\omega}{\Gamma} \left(\frac{\omega}{\Gamma} + 2i\right)} \ln \frac{\varepsilon\_d^2 + \Gamma^2}{\varepsilon\_d^2 - (\omega + i\Gamma)^2}. \tag{A58}$$

The low frequency expansion of this expression matches the one of a classical RC circuit (A35). Reestablishing correct dimensions *ω* → *h*¯ *ω*, the result recovers Equation (A40) obtained within scattering theory:

$$\mathcal{C}\_0 = \frac{\varepsilon^2}{h} \nu(\varepsilon\_d) \,, \tag{A} \\ \qquad \qquad \qquad \qquad R\_{\mathbf{q}} = \frac{h}{2c^2} \, , \tag{A59}$$

where *<sup>ν</sup>*(*<sup>ε</sup>d*) is the density of states associated to the single orbital *εd*

$$\nu(\varepsilon\_d) = \frac{1}{\pi} \frac{\Gamma}{\varepsilon\_d^2 + \Gamma^2} \,. \tag{A60}$$

The extension to *M* channels is straightforward. One can consider a Hamiltonian of the form:

$$H\_{\rm Res-Mch} = \sum\_{kr} \varepsilon\_k \boldsymbol{c}\_{kr}^{\dagger} \boldsymbol{c}\_{kr} + t \sum\_{kr} \left( \boldsymbol{c}\_{kr}^{\dagger} \boldsymbol{d}\_{\sigma} + \boldsymbol{d}\_{\sigma}^{\dagger} \boldsymbol{c}\_{kr} \right) + \sum\_{\sigma} \varepsilon\_{\sigma} \boldsymbol{d}\_{\sigma}^{\dagger} \boldsymbol{d}\_{\sigma} \,\,\,\tag{A61}$$

with *σ* = 1, ... , *M* the number of channels. In this model, each channel can be treated independently and one finds a a generalization of Equation (A58):

$$\chi\_{\mathbb{C}}(\omega) = \sum\_{\sigma=1}^{M} \frac{1}{\pi \Gamma} \frac{1}{\#\left(\Psi + 2i\right)} \ln \frac{\varepsilon\_{\sigma}^{2} + \Gamma^{2}}{\varepsilon\_{\sigma}^{2} - (\omega + i\Gamma)^{2}}.\tag{A62}$$

This expression, when expanded to low frequency, has also the form (A47), where the dwell-times are substituted by the density of states of the channels (A60): *τσ* → *νσ* = *<sup>ν</sup>*(*εσ*). One thus finds expressions for the differential capacitance and the charge relaxation resistance analog to Equation (A48):

$$\mathbb{C}\_{0} = \frac{e^{2}}{h} \sum\_{\sigma=1}^{M} \nu\_{\sigma}, \qquad \qquad \qquad R\_{\mathbf{q}} = \frac{h}{2e^{2}} \frac{\sum\_{\sigma} \nu\_{\sigma}^{2}}{\left(\sum\_{\sigma} \nu\_{\sigma}\right)^{2}}. \tag{A63}$$

**Figure A3.** Peaked structure of the local density of states N (*ε*) on the dot as a function of the orbital energy shift controlled by the gate potential *eVg* from Equation (A45). N (*EF*) is plotted for different values of the backscattering amplitude *r*. The progressive opening of the dot drives a transition from a Lorentzian to an oscillatory behavior of *<sup>C</sup>*q, coherent with the experimental measurements illustrated in Figure 8. For a completely transparent dot (*r* = 0) the density of states is uniform, which implies *C*0 = *<sup>e</sup>*2/<sup>Δ</sup>.

Multi-Level Case

In this section we carry out the calculation of the quantum dot density of states in the case of a single channel and an infinite number of equally spaced levels in the quantum dot. The action reads:

$$\mathcal{S} = \sum\_{i\omega\_n} \left\{ -\sum\_k c\_k^\dagger G\_k^{-1} (i\omega\_n) c\_k - \sum\_l d\_l^\dagger D\_l^{-1} (i\omega\_n) d\_l + t \sum\_{kl} \left[ c\_k^\dagger d\_l + d\_l^\dagger c\_k \right] \right\},\tag{A64}$$

with *G*−<sup>1</sup> *k* (*<sup>i</sup><sup>ω</sup>n*) = *i<sup>ω</sup>n* − *εk* and *D*−<sup>1</sup> *l* (*<sup>i</sup><sup>ω</sup>n*) = *i<sup>ω</sup>n* − *εl*. The Gaussian integration of the lead electron modes leads to the effective action:

$$\mathcal{S}' = \sum\_{i\omega\_{\rm in}} \left\{ -\sum\_{l} d\_{l}^{\dagger}(i\omega\_{\rm n}) D\_{l}^{-1}(i\omega\_{\rm n}) d\_{l}(i\omega\_{\rm n}) + t^{2} \sum\_{k} \mathcal{G}\_{k}(i\omega\_{\rm n}) \sum\_{ll'} d\_{l}^{\dagger}(i\omega\_{\rm n}) d\_{l'}(i\omega\_{\rm n}) \right\}. \tag{A65}$$

Applying Wick's theorem [83] the full propagator of the dot electrons is readily obtained:

$$\mathcal{D}\_{ll'}(\dot{\imath}\omega\_{\boldsymbol{n}}) = \delta\_{ll'}D\_l(\dot{\imath}\omega\_{\boldsymbol{n}}) + D\_l(\dot{\imath}\omega\_{\boldsymbol{n}})D\_{l'}(\dot{\imath}\omega\_{\boldsymbol{n}}) \frac{\gamma(\dot{\imath}\omega\_{\boldsymbol{n}})}{1 - \gamma(\dot{\imath}\omega\_{\boldsymbol{n}})\Theta(\dot{\imath}\omega\_{\boldsymbol{n}})},\tag{A66}$$

where we defined:

$$\gamma\left(i\omega\_{\hbar}\right) = \hbar^2 \sum\_{k} \mathcal{G}\_{k}\left(i\omega\_{\hbar}\right) \,, \tag{A6} \\ \left(i\omega\_{\hbar}\right) = \sum\_{l} D\_{l}\left(i\omega\_{\hbar}\right) \,. \tag{A67}$$

In the wide band limit *<sup>γ</sup>*(*<sup>i</sup><sup>ω</sup>n*) = <sup>−</sup>*i*Γsgn(*<sup>i</sup><sup>ω</sup>n*). The charge *Q* = *e* ∑*l d*†*l dl* on the dot is given by:

$$\langle Q \rangle = \frac{\varepsilon}{\mathcal{B}} \sum\_{l, i\omega\_{\rm n}} \varepsilon^{i\omega\_{\rm n}0^{+}} \mathcal{D}\_{\rm ll}(i\omega\_{\rm n}) = \frac{\varepsilon}{2\pi \text{i}} \sum\_{l} \int\_{-\infty}^{\infty} d\varepsilon f(\varepsilon) \left[ \mathcal{D}\_{\rm ll}(\varepsilon + i0^{+}) - \mathcal{D}\_{\rm ll}(\varepsilon - i0^{+}) \right] \,. \tag{A68}$$

We write the energy spectrum on the dot as *εl* = −*eVg* + *l*Δ, with *l* ∈ Z and Δ the level spacing. Equation (A66) is then a function of *ε* + *eVg*. Shifting all energies by *eVg*, the differential capacitance *C*0 = −*∂ Q* /*∂Vg* is readily obtained at zero temperature:

$$\mathcal{L}\_0 = \frac{e^2}{2\pi i} \sum\_{l} \left[ \mathcal{D}\_{ll}(\varepsilon V\_{\mathcal{J}} + i0^+) - \mathcal{D}\_{ll}(\varepsilon V\_{\mathcal{J}} - i0^+) \right],\tag{A69}$$

with

$$\begin{split} \mathcal{D}\_{\mathrm{II}}(\varepsilon V\_{\mathcal{S}} \pm i0^{+}) &= \frac{1}{\varepsilon V\_{\mathcal{S}} - l\Delta} \mp \frac{1}{(\varepsilon V\_{\mathcal{S}} - l\Delta)^{2}} \frac{i\Gamma}{1 \pm i\Gamma\left[\sum\_{p} \frac{1}{\varepsilon V\_{\mathcal{S}} - p\Delta}\right]} \\ &= \frac{1}{\Delta} \left\{ \frac{1}{\mathbf{x} + l} \mp \frac{i\Gamma/\Delta}{(\mathbf{x} + l)^{2}} \frac{1}{1 \pm i\pi\frac{\Gamma}{\Delta}\coth(\pi x)} \right\}, \end{split} \tag{A70}$$

where *x* = *eVg*/Δ and we exploited the fact that ∑*l* 1*x*+*l* = <sup>Ψ</sup>0(<sup>1</sup> − *x*) − <sup>Ψ</sup>0(*x*) = *π* coth(*<sup>π</sup>x*), in which <sup>Ψ</sup>0(*x*) is the digamma function. Substituting this expression in Equation (A69), the sum over levels can be also carried out, leading to:

$$\mathcal{C}\_{0} = \varepsilon^{2} \frac{\pi \Gamma}{2\Delta^{2}} \frac{1}{\sin^{2}\left(\pi \frac{cV\_{\pi}}{\Lambda}\right)} \left[ \frac{1}{1 + i\pi \frac{\Gamma}{\Lambda} \coth\left(\pi \frac{cV\_{\pi}}{\Lambda}\right)} + \frac{1}{1 - i\pi \frac{\Gamma}{\Lambda} \coth\left(\pi \frac{cV\_{\pi}}{\Lambda}\right)} \right],\tag{A71}$$

where we relied on the identity: ∑*l* 1 (*l*+*x*)<sup>2</sup> = <sup>Ψ</sup>1(<sup>1</sup> − *x*) − <sup>Ψ</sup>1(*x*) = *π*<sup>2</sup> sin<sup>2</sup>(*πx*), in which <sup>Ψ</sup>*n*(*x*) is the polygamma function. Some algebra leads to:

$$C\_0 = \frac{e^2}{\Delta} \frac{2}{\frac{\Lambda}{\pi \Gamma} + \frac{\pi \Gamma}{\Delta} - \left(\frac{\Lambda}{\pi \Gamma} - \frac{\pi \Gamma}{\Delta}\right) \cos\left(\frac{2\pi e V\_\xi}{\Delta}\right)}\tag{A72}$$

This quantity is plotted in Figure A3 as a function of the gate potential *Vg* and reproduces the oscillations of the capacitance observed in Figure 8, in the main text. As we did not consider any many-body interaction to derive *C*0, this quantity corresponds to the quantum capacitance *C*q = *e*<sup>2</sup>N (*EF*) corresponding to the density of states at the Fermi level, see also discussion in Section 4.2. Such density of states was also derived within scattering theory in Appendix B.1. Indeed, Equations (A45) and (A72) coincide if one makes the identification (It is useful to recall here that *τ*f = *h*/Δ)

$$\frac{\pi\Gamma}{\Delta} = \frac{(1-r)^2}{1-r^2} = \frac{1-r}{1+r} \qquad\qquad\Leftrightarrow\qquad r = \frac{1-\frac{\pi\Gamma}{\Delta}}{1+\frac{\pi\Gamma}{\Delta}}.\tag{A73}$$

Notice that the fully transparent limit coincides with *π*Γ/Δ = 1, corresponding to a change of sign of the reflection amplitude *r* (remind that we assumed *r* to be a real number). Additionally, if we consider the tunneling limit *π*Γ/Δ 1, we can write *r* = √1 − *D* and, in the low-transparency limit *D* 1 one recovers *π*Γ/Δ = *D*/4, which is consistent with the expectation *D* ∝ *t* 2 in the tunneling limit of the Hamiltonian (A16). Notice also that the relation (A73) implies *r* = −1 in the Γ → ∞ limit, which can be explained by the formation of bonding and anti-bonding states at the junction between electrons in the lead and in the dot, suppressing tunneling in the dot [240]. For a single level and one channel, we recover the universal charge relaxation resistance *<sup>R</sup>*q = *h*/2*e*2.

### **Appendix C. Useful Results of Linear Response Theory**

In this appendix we remind some useful properties of linear response theory following Ref. [244]. In Appendix C.1, we show that the real/imaginary parts of the dynamical charge susceptibility (A49) are respectively even/odd functions of the frequency, leading to:

$$\mathcal{A}(\omega) = -i\omega \mathbf{e}^2 \left\{ \chi\_{\mathbf{c}} + i \text{Im} \left[ \chi\_{\mathbf{c}}(\omega) \right] \right\} + \mathcal{O}(\omega^2) \,, \tag{A74}$$

that is Equation (38) in the main text. In Appendix C.2, we demonstrate that the power dissipated by the quantum *RC* circuit in the linear response regime is given by:

$$\mathcal{P} = \frac{1}{2} \varepsilon\_{\omega}^{2} \omega \text{Im} \chi\_{\mathbb{C}}(\omega) \,, \tag{A75}$$

that is Equation (55) in the main text.

### *Appendix C.1. Parity of the Dynamical Charge Susceptibility*

The Lehman representation [97] of the dynamical charge susceptibility *<sup>χ</sup>*c(*ω*) (A49) makes explicit its real and imaginary parts. This is obtained from the Fourier transform of Equation (A49):

$$\chi\_{\mathbf{c}}(\omega) = \frac{i}{\hbar} \int\_{-\infty}^{\infty} d(t - t') e^{i(\omega + i0^{+})(t - t')} \theta(t - t') \left\langle \left[ N(t), N(t') \right] \right\rangle\_{0} \,\tag{A76}$$

where the factor *i*0<sup>+</sup> is inserted to regularize retarded functions. Inserting the closure relation with the eigenstates |*n* of energy *En* of the time independent Hamiltonian H0, the average can be written as:

$$
\left< N(t)N(t') \right>\_0 = \sum\_{n,m} p\_n e^{i\omega\_{nm}(t-t')} N\_{nm} N\_{mn} \,\,\,\,\,\tag{A77}
$$

where *pn* = *e*<sup>−</sup>*βEn*/*<sup>Z</sup>* is the Boltzmann weight, *h*¯ *ωnm* = *En* − *Em* and *Nnm* = *n*| *N* |*m* the matrix elements of the dot occupation. In this representation, the Fourier transform (A76) reads:

$$\chi\_{\mathbb{C}}(\omega) = -\frac{1}{\hbar} \sum\_{nm} p\_n \mathbf{N}\_{\mathbb{H}m} \mathbf{N}\_{mn} \left( \frac{1}{\omega + i0^+ + \omega\_{nm}} - \frac{1}{\omega + i0^+ - \omega\_{nm}} \right) \,. \tag{A78}$$

Applying the relation 1 *x*±*i*0<sup>+</sup> = P 1 *x* ∓ *<sup>i</sup>πδ*(*x*), with P[ *f*(*x*)] the principal value of the function *f*(*x*), the real and imaginary part of *<sup>χ</sup>c*(*ω*) are readily obtained:

$$\operatorname{Re}\left[\chi\_{\mathbb{C}}(\omega)\right] = -\frac{1}{\hbar}\sum\_{nm} p\_n \mathbf{N}\_{nm} \mathbf{N}\_{nm} \left\{ \mathbf{P}\left[\frac{1}{\omega + \omega\_{nm}}\right] - \mathbf{P}\left[\frac{1}{\omega - \omega\_{nm}}\right] \right\},\tag{A79}$$

$$\operatorname{Im}\left[\chi\_{\mathbb{C}}(\omega)\right] = \frac{i\pi}{\hbar} \sum\_{nm} p\_n \mathcal{N}\_{nm} \mathcal{N}\_{nm} \left\{ \delta\left(\omega + \omega\_{nm}\right) - \delta\left(\omega - \omega\_{nm}\right) \right\},\tag{A80}$$

which are respectively an even and odd function of *ω*. As a consequence, in the low frequency expansion of the dynamical charge susceptibility *<sup>χ</sup>*c(*ω*) = *<sup>χ</sup>*c(0) + *ω∂ωχ*c(*ω*)|*<sup>ω</sup>*=<sup>0</sup> + <sup>O</sup>(*ω*<sup>2</sup>), the linear term in *ω* has to coincide with the imaginary part of Im[*<sup>χ</sup>*c(*ω*)], leading to Equation (A74).

### *Appendix C.2. Energy Dissipation in the Linear Response Regime*

In the situation addressed in Section 4, the time dependence of orbital energies in the dot is given by *<sup>ε</sup>d*(*t*) = *ε*0 *d*+ *εω* cos(*ω<sup>t</sup>*). In the time unit, the systems dissipates the energy:

$$
\delta \mathcal{W} = \delta \left< \mathcal{N} \right> \varepsilon\_{\omega}, \ \cos \left( \omega t \right). \tag{A81}
$$

In the stationary regime, the average power P dissipated by the system during the time period *T* reads:

$$\mathcal{P} = \frac{\varepsilon\_{\omega}}{T} \int\_{0}^{T} dt \, \frac{d \left< N(t) \right>}{dt} \, \cos(\omega t) \,. \tag{A82}$$

Neglecting constant contributions, *N*(*t*) is given by the dynamical charge susceptibility (A49):

$$
\langle N(t) \rangle = \varepsilon\_{\omega} \int\_{-\infty}^{\infty} dt' \chi\_{\mathbb{C}}(t - t') \cos(\omega t) \,. \tag{A83}
$$

Substituting this expression in Equation (A82), we obtain:

$$\mathcal{P} = -i\omega \frac{\varepsilon\_{\omega}^{2}}{4} \left[ \chi\_{\varepsilon}(\omega) - \chi\_{\varepsilon}(-\omega) \right] + i\omega \frac{\varepsilon\_{\omega}^{2}}{T} \int\_{0}^{T} dt \frac{\chi\_{\varepsilon}(-\omega)\varepsilon^{2i\omega t} - \chi\_{\varepsilon}(\omega)\varepsilon^{-2i\omega t}}{4}.\tag{A84}$$

Expressing *<sup>χ</sup>*c(*ω*) = Re [*<sup>χ</sup>*c(*ω*)] + *i*Im [*<sup>χ</sup>*c(*ω*)] as the sum of its real and imaginary part and applying the parity properties demonstrated in Appendix C.1, the first term recovers Equation (A75) for the dissipated power:

$$\mathcal{P} = \frac{1}{2} \varepsilon\_{\omega}^{2} \omega \text{Im} \left[ \chi\_{\text{c}}(\omega) \right] \,, \tag{A85}$$

while the second term in Equation (A84) reduces to vanishing integrals of sin(<sup>2</sup>*ω<sup>t</sup>*) and cos(<sup>2</sup>*ω<sup>t</sup>*) over their period. In the case of Section 4.5, describing the energy dissipated by the LFL effective low-energy theory, we can apply the same considerations by replacing *δ N* in Equation (A81) with the average of the operator *A*, defined in Equation (57). One thus derives Equation (56) in the main text.
