*Appendix A.1. General Definitions*

We aim at describing the general situation of Figure A1, which is reproduced by mesoscopic settings, such as a single resonant level coupled to a chiral edge state, which also describes the mesoscopic capacitor in the non-interacting limit, see also Figure 7 in the main text. Consider a wave packet emitted and detected in the distant past and future, namely *t* = −∞ and *t* = +<sup>∞</sup>, and which enters a scattering region at time *t* = 0. Close to detection and emission, it is assumed that the wave

packet does not feel the presence of the scatterer, whose interaction range is delimited inside the dashed line in Figure A1. The system is described by a single-particle Hamiltonian of the form:

$$
\mathcal{H} = \mathcal{H}\_0 + \mathcal{V},
\tag{A1}
$$

in which H0 describes the free propagation of a wave-packet and V the scatterer. The T-matrix describes the effects of a scatterer on the propagation of a free particle. It is an improper self-energy for the resolvent of the lead electrons, which appears in a modified form of Dyson's equation:

$$\mathbf{G}(z) = \mathbf{G}\_0(z) + \mathbf{G}\_0(z)T(z)\mathbf{G}\_0(z), \qquad \qquad \mathbf{G}(z) = \frac{1}{z - \mathcal{H}}, \tag{A2}$$

in which z is a complex number. *G*0 is the free resolvent describing free electrons:

$$\left|G\_{kk'}^{0}(z) = \left = \frac{\delta\_{kk'}}{z - \varepsilon\_{k}},\tag{A3}$$

the |*k* states being the single particle eigenvectors of the unperturbed Hamiltonian H0. One can readily show that the T-matrix reads:

$$\mathbf{T}(z) = V(\mathbb{I} - G\_0(z)\mathcal{V})^{-1}.\tag{A4}$$

The general definition of the phase-shift, a key quantity within scattering theory [238,239], reads:

$$\delta\_{\varepsilon} = \arg\left[ \mathcal{T}(\varepsilon + i\mathcal{O}^{+}) \right] \tag{A5}$$

**Figure A1.** Top—Illustration of the physical situation described by the scattering formalism. Electron wave packets are emitted in the IN-State Ψ<sup>+</sup> and then measured in the OUT-state <sup>Ψ</sup>− once they have passed through the scattering region. Bottom—Realization of the scattering setup with a quantum Hall chiral edge state tunnel coupled (in a region of size <sup>2</sup>*η*) with a resonant level of energy *<sup>ε</sup>d*.

We define the IN and OUT states |Ψ± as the eigenvectors of energy *ε* of the Hamiltonian of the whole system, including the scattering region, as the states coinciding asymptotically with free plane waves in the past and in the future respectively. The scattering matrix S gives the overlap between these two states:

$$\mathbf{S}\_{kk'}(\varepsilon) = \langle \Psi\_k^- | \Psi\_{k'}^+ \rangle \,\,\,\,\,\,\,\tag{A6}$$

where *k*/*k* are the momenta of the OUT/IN states. The T- and S-matrix are related by the relation [238,239]:

$$\mathbf{S}\_{kk'} = \delta\_{kk'} - 2\pi i \delta(\varepsilon\_k - \varepsilon\_{k'}) \mathbf{T}\_{kk'} \,. \tag{A7}$$

or equivalently, in the energy representation,

$$\mathbf{S}(\varepsilon) = \mathbb{I} - 2\pi i \nu\_0 \mathbf{T}(\varepsilon) \,. \tag{A8}$$

The S-matrix is unitary and in the single channel case it is completely defined by a phase <sup>S</sup>(*ε*) = *<sup>e</sup>*2*iδε*. The phase *δε* is the phase-shift caused by scattering and in general the condition <sup>S</sup>(*ε*) = *e*2*iδε* is always verified if we take the definition of the phase-shift directly from the T-matrix:

$$\mathcal{T}(\varepsilon) = -\frac{1}{\pi \nu\_0} \sin \delta\_\varepsilon e^{i\delta\_\varepsilon}. \tag{A9}$$

There is an interesting connection with the Kondo regime. In Section 3.3.2, we illustrated how particle-hole symmetry enforces that the phase-shift is given by *δ* = *δ*K = *π*/2. The scattering matrix thus equals the identity. This case is also known as the unitary limit of the Kondo model, in which the transmission probability through a Kondo correlated dot is unity.

## *Appendix A.2. The Friedel Sum Rule*

We show here the Friedel sum rule for non-interacting electrons scattering on an elastic impurity. We first consider the total electron occupation N of an electron gas, which reads:

$$
\langle \mathcal{N} \rangle = \sum\_{a} \int\_{-\infty}^{\infty} d\omega \, A\_{a}(\omega) f(\omega) \,, \tag{A10}
$$

the sum on the label *α* running over all the eigenstates of the Hamiltonian (A1). *<sup>A</sup>α*(*ω*) is the spectral function of the state *α*, defined as:

$$A\_{\mathbb{A}}(\omega) = -\frac{1}{\pi} \text{Im} G\_{\text{aa}}(\omega + i0^{+}) = -\frac{1}{\pi} \text{Im} \langle a|G(\omega + i0^{+})|a\rangle \,\tag{A11}$$

*Gαα* being the *retarded* Green's function associated to the state *α* defined in Equation (A2). In the absence of the scatterer, *Gkk*(*ω* + *i*0+) = *<sup>G</sup>*0(*<sup>ω</sup>* + *<sup>i</sup>*0+)=(*ω* + *i*0<sup>+</sup> − *<sup>ε</sup>k*)−<sup>1</sup> and Equation (A10) reduces to a sum over the Fermi function ∑*k f*(*<sup>ε</sup>k*) giving the total number of electrons N in the system. If the chemical potential is fixed, the introduction of the scatterer modifies the total average number of electrons. The difference with the initial one gives the amount of electrons *N* displaced by the scatterer. In the case of a single channel, one obtains:

$$
\langle \langle \mathcal{N} \rangle\_{\text{with scatter}} - \langle \mathcal{N} \rangle\_{\text{without scatter}} = \langle \mathcal{N} \rangle = -\frac{1}{\pi} \text{Im} \int\_{-\infty}^{\infty} d\omega \text{Tr} [\mathcal{G}(\omega) - \mathcal{G}\_0(\omega)] f(\omega) \,. \tag{A12}
$$

Using Equation (A4) and the fact that *ddε*Tr log[*A*(*ε*)] = Tr *<sup>A</sup>*−<sup>1</sup>(*ε*) *ddε <sup>A</sup>*(*ε*) , one finds:

$$\begin{split} \langle N \rangle &= -\frac{i}{2\pi} \int\_{-\infty}^{\infty} d\omega \, f(\omega) \frac{d}{d\omega} \log \det[\mathbb{I} - 2\pi i \nu\_0 \, \mathrm{T}(\omega + i0^+)] \\ &= -\frac{i}{2\pi} \int\_{-\infty}^{\infty} d\omega \, f(\omega) \frac{d}{d\omega} \log \det \, \mathbb{S}(\omega + i0^+) \, . \end{split} \tag{A13}$$

in which we applied the definition (A8) of the S-matrix. This is the general form of the Friedel sum rule. For <sup>S</sup>(*ε*) = *e*2*iδε* and zero temperature it gives a direct relation between the charge displaced by the impurity and the phase-shift at the Fermi energy *EF*:

$$
\langle \mathbf{N} \rangle = \frac{\delta\_{E\_F}}{\pi}.\tag{A14}
$$

The extension to *M* channels requires to add an overall sum over the channel label *σ* and leads to Equation (10) in the main text.

### *Appendix A.3. Illustration on the Resonant Level Model*

We illustrate now the above concepts on a simple situation, sketched in Figure A1, in which the scatterer is a single resonant level tunnel coupled to chiral electrons propagating on an edge state. Such a situation is an effective representation of the mesoscopic capacitor, see Figures 7 and A2, and it is described by the Coulomb Blockade Model (CBM) (12), that we remind here to the reader:

$$\mathcal{H}\_{\rm CRM} = \sum\_{k} \varepsilon\_{k} c\_{k}^{\dagger} c\_{k} + t \sum\_{k,l} \left[ c\_{k}^{\dagger} d\_{l} + d\_{l}^{\dagger} c\_{k} \right] + \sum\_{l} (\varepsilon\_{d} + \varepsilon\_{l}) d\_{l}^{\dagger} d\_{l} + E\_{\rm c} \left( N - \mathcal{N}\_{\rm g} \right)^{2}. \tag{A15}$$

In this section, we neglect the last term, corresponding to interactions, and, for simplicity, we retain only a single fermionic level (annihilated by the fermion operator *d*) for the cavity, with *εl* = 0. One thus obtains the Hamiltonian of a resonant level:

$$\mathcal{H}\_{\text{Res}} = \sum\_{k} \varepsilon\_{k} c\_{k}^{\dagger} c\_{k} + t \sum\_{k} \left( c\_{k}^{\dagger} d + d^{\dagger} c\_{k} \right) + \varepsilon\_{d} d^{\dagger} d \,. \tag{A16}$$

We first calculate the occupation of the cavity by calculating the retarded Green's functions, defined as *Gdk*(*t* − *t* ) = −*iθ*(*<sup>t</sup>* − *t* ) {*d*(*t*), *c*† *k* (*t* )} . They are derived by solving the equations of motion in frequency space:

$$\left(\omega - \varepsilon\_{d}\right) \mathbf{G}\_{dd}(\omega) = 1 + t \sum\_{k} \mathbf{G}\_{kd}(\omega) \,, \qquad \left(\omega - \varepsilon\_{k}\right) \mathbf{G}\_{kk'}(\omega) = \delta\_{\mathbf{k} \mathbf{k'}} + t \mathbf{G}\_{dd'}(\omega) \,, \tag{A17}$$

$$(\omega - \varepsilon\_k) \mathcal{G}\_{\text{kd}}(\omega) = t \mathcal{G}\_{\text{d} \text{d}}(\omega) \,, \tag{4.18} \\ \text{where} \\ \qquad \qquad (\omega - \varepsilon\_d) \mathcal{G}\_{\text{d} \text{k}} = t \sum\_{k'} \mathcal{G}\_{\text{k'} \text{k}}(\omega) \,. \tag{A18}$$

Solving the system, the Green's function for the lead electrons reads:

$$\mathcal{G}\_{kk'}(\omega) = \frac{\delta\_{kk'}}{\omega - \varepsilon\_k} + \frac{1}{\omega - \varepsilon\_k} \mathbf{t}^2 \mathcal{G}\_{dd}(\omega) \frac{1}{\omega - \varepsilon\_{k'}}.\tag{A19}$$

Writing Equation (A19) in the form *G* = *G*<sup>0</sup> + *G*0*TG*0, the T-matrix is found to be:

$$\mathrm{T}(\omega + i0^{+}) = t^{2} \mathrm{G}\_{dd}(\omega + i0^{+}) = \frac{t^{2}}{\sqrt{(\omega - \varepsilon\_{d})^{2} + \Gamma^{2}}} \epsilon^{i\delta\_{\omega}}, \qquad \delta\_{\omega} = \frac{\pi}{2} - \arctan\left(\frac{\varepsilon\_{d} - \omega}{\Gamma}\right) \tag{A20}$$

in which we introduced the hybridization constant:

$$
\Gamma = t^2 \sum\_{k} (\omega + i0^+ - \varepsilon\_k)^{-1} \sim \pi \nu\_0 t^2,\tag{A21}
$$

which corresponds to the width acquired by the resonant level by coupling to the lead and which depends on the density of states of the lead electrons at the Fermi energy *ν*0.

We can now determine the number of displaced charges *N* as given by Equation (A22) and show the validity of the Friedel sum rule (A14) in this example. In the wide-band limit, the contribution from the second term in Equation (A19) can be neglected. The number of displaced electrons is given solely by the quantum dot Green's function *Gdd*:

$$\langle \langle N \rangle\_{\text{with dot}} - \langle N \rangle\_{\text{without dot}} = \langle N \rangle = -\frac{1}{\pi} \text{Im} \int\_{-\infty}^{\infty} d\omega \,\text{G}\_{\text{dil}}(\omega) f(\omega) = \frac{1}{2} - \frac{1}{\pi} \arctan \left( \frac{\varepsilon\_d}{\Gamma} \right), \tag{A22}$$

which is consistent with the Friedel sum rule (A14) with the phase-shift (A20). Equation (A22) is also meaningful because: i) It shows that, in the wide-band approximation, the number of displaced electrons *N* is given by the local Green's function *Gdd*, which can be interpreted as the charge occupation of the quantum dot and ii) the number of displaced electrons depends on the orbital energy *εd*. As a consequence, a time-dependent variation of *εd* drives a current in the system by displacing electrons in the leads.

As an additional illustration, clarifying how the scattering phase-shift appears on single particle wave-functions, we also solve explicitly the same problem in its real-space formulation. We consider chiral fermions <sup>Ψ</sup>(*x*) which tunnel on the resonant level of energy *εd* (and wave-function amplitude Ψ*d*), from a region of size 2*η* centered around *x* = 0, useful to properly regularize the calculation and to be sent to zero at the end [240]. For lead electrons at the Fermi energy, that we set to zero ( *ω* = 0 in Equation (A20)), the Schrödinger equation reads (for *x* ∈ [−*η*, *η*]):

$$0 = -i\hbar v\_F \partial\_\mathbf{x} \Psi(\mathbf{x}) + \frac{\mathbf{t}}{2\eta} \Psi\_d \,. \tag{4.23} \\ = \varepsilon\_d \Psi\_d + \frac{\mathbf{t}}{2\eta} \int\_{-\eta}^{\eta} d\mathbf{x}' \Psi(\mathbf{x}') . \tag{A23}$$

By integrating the first equation in the interval [−*η*, *x* < *η*] and inserting the result for <sup>Ψ</sup>(*x*) in the second one, one finds:

$$\Psi(\mathbf{x}) = \Psi(-\eta) + \frac{t}{2\eta i\hbar v}(\mathbf{x} + \eta)\Psi\_d,\qquad\qquad\qquad\Psi\_d = \frac{-t}{\varepsilon\_d - i\Gamma}\Psi(-\eta)\,.\tag{A24}$$

Notice that *η* does not appear on the last equality and can be sent safely to zero. We consider, as boundary condition for <sup>Ψ</sup>(*x*), incoming scattering states of the form <sup>Ψ</sup>(*x* < 0) = *eikx*/ √<sup>2</sup>*πhv*¯ *F* = √*<sup>ν</sup>*0*eikx*. One thus finds:

$$\left|\Psi\_{d}\right|^{2} = \frac{1}{\pi} \frac{\Gamma}{\varepsilon\_{d}^{2} + \Gamma^{2}}, \qquad \qquad \left|\Psi(0^{+}) = \sqrt{\nu\_{0}} e^{2i\delta\_{0}}, \qquad \delta\_{0} = \frac{\pi}{2} - \arctan\left(\frac{\varepsilon\_{d}}{\Gamma}\right) \,, \tag{A25}$$

in agreemen<sup>t</sup> with Equation (A20). This short calculation illustrates how, just after scattering with the resonant level, the electron wave-packet at the Fermi energy acquires a phase *e*2*iδ*<sup>0</sup> which is fixed by the resonant level occupation via the Friedel sum rule. Notice that, at the resonance condition for the orbital energy (*<sup>ε</sup>d* = 0), *δ*0 = *π*/2 and Ψ(0+) + <sup>Ψ</sup>(<sup>0</sup>−) = 0, as for the unitary limit in the Kondo model, in which *δ*K = *π*/2, see also the discussion in Section 3.3.2.

### *Appendix A.4. T-Matrix in the Potential Scattering Hamiltonian*

We derive now the phase-shift caused by the potential scattering term on lead electrons in a local Fermi liquid (LFL). It is useful to recall here the LFL Hamiltonian (8):

$$\mathcal{H}\_{\rm LFL} = \sum\_{k\sigma} \varepsilon\_k \mathfrak{c}\_{kr}^{\dagger} \mathfrak{c}\_{kr} + \mathcal{W}(\varepsilon\_{d\prime} E\_{\mathbb{C}\prime} \dots) \sum\_{k \neq k^{\prime}\sigma} \mathfrak{c}\_{kr}^{\dagger} \mathfrak{c}\_{k^{\prime}\sigma} \,. \tag{A26}$$

We focus on the the single-channel case for simplicity ( *M* = 1). The generalization to *σ* = 1, ... *M* channels is straightforward. The Hamiltonian (A26) is quadratic and the Green's function of the lead electrons can be readily obtained relying on the path integral formalism [79]. The partition function corresponding to Equation (A26) reads:

$$Z = \int \mathcal{D}\left[c, c^{\dagger}\right] e^{-S\_{\rm LFL}\left[c, c^{\dagger}\right]}\,,\tag{A27}$$

where SLFL *c*, *c*† is the action of the system, which reads:

$$\mathcal{S}\_{\rm LFL} = \int\_0^\beta d\tau \left\{-\sum\_k c\_k^\dagger(\tau) \mathcal{G}\_k^{-1}(\tau) c\_k(\tau) + W \sum\_{k \neq k'} c\_k^\dagger(\tau) c\_{k'}(\tau) \right\},\tag{A28}$$

where we introduced the free propagator *G*−<sup>1</sup> *k* (*τ*) = −*∂τ* − *εk* and in which *ck* is a Grassmann variable. It is practical to switch to the frequency representation *ckσ*(*τ*) = 1*β* ∑*<sup>i</sup><sup>ω</sup>n <sup>e</sup>*<sup>−</sup>*i<sup>ω</sup>n<sup>τ</sup>ckσ*(*<sup>i</sup><sup>ω</sup>n*), where we defined the fermionic Matsubara frequencies *i<sup>ω</sup>n* = (<sup>2</sup>*n* + <sup>1</sup>)*π*/*β*, *n* ∈ Z. They satisfy the anti-periodicity property *c*(*β*) = −*<sup>c</sup>*(0) and lead to:

$$\mathcal{S}\_{\rm LFL} = \sum\_{i\omega\_n} \left\{ -\sum\_k c\_k^\dagger(i\omega\_n) \mathcal{G}\_k^{-1}(i\omega\_n) c\_k(i\omega\_n) + \mathcal{W} \sum\_{k \neq k'} c\_k^\dagger(i\omega\_n) c\_{k'}(i\omega\_n) \right\},\tag{A29}$$

with *G*−<sup>1</sup> *k* (*<sup>i</sup><sup>ω</sup>n*) = *i<sup>ω</sup>n* − *εk*, which recovers the usual retarded/advanced Green's functions by performing the analytical continuation *i<sup>ω</sup>n* → *ω* ± *i*0+. The full Green's function *Gkk*(*<sup>i</sup><sup>ω</sup>n*) = − *ck*(*<sup>i</sup><sup>ω</sup>n*)*c*†*k* (*<sup>i</sup><sup>ω</sup>n*) is derived by expanding the partition function (A27) in the coupling *W* and by applying Wick's theorem [83]. The pertubation expansion of *Gkk*(*<sup>i</sup><sup>ω</sup>n*) has the simple form:

$$G\_{kk'}(i\omega\_n) = \frac{\delta\_{kk'}}{i\omega\_n - \varepsilon\_k} + \frac{1}{i\omega\_n - \varepsilon\_k} \frac{1}{i\omega\_n - \varepsilon\_{k'}} \mathcal{W} \left[ 1 + \Sigma(i\omega\_n) + \Sigma^2(i\omega\_n) + \Sigma^3(i\omega\_n) + \dots \right],\tag{A30}$$

in which we introduced the self-energy:

$$\Sigma(i\omega\_{\rm{H}}) = \sum\_{p} \frac{\mathcal{W}(\varepsilon\_{\rm{d}})}{i\omega\_{\rm{n}} - \varepsilon\_{p}}.\tag{A31}$$

Using the definition (A2), the T-matrix thus reads:

$$\mathcal{T}(z) = \frac{\mathcal{W}}{1 - \Sigma(z)}.\tag{A32}$$

Making the analytical continuation *i<sup>ω</sup>n* → *ω* + *i*0<sup>+</sup> and considering a constant density of states *ν*0 for the lead electrons, we obtain:

$$\mathcal{T}(\omega + i0^{+}) = \frac{\mathcal{W}(\varepsilon\_{d})}{1 + i\pi\nu\_{0}\mathcal{W}} = \frac{\mathcal{W}(\varepsilon\_{d})}{\sqrt{1 + [\pi\nu\_{0}\mathcal{W}(\varepsilon\_{d})]^{2}}} e^{i\delta\nu\_{0}},\tag{A33}$$

in which we introduced the phase-shift:

$$
\delta\_{\mathcal{W}} = -\arctan\left(\pi\nu\_0\mathcal{W}\right) \,. \tag{A34}
$$

Applying the definition of the phase-shift given in Equation (A5), one finds Equation (9) in the main text. As a consistency check, substituting Equation (A33) in Equation (A8), we find that the scattering matrix reads S = *<sup>e</sup>*2*iδ*.

### **Appendix B. Self-Consistent Description- of a 2DEG Quantum** *RC* **Circuit**

In Appendix B.1, we shortly review the self-consistent scattering theory of the mesoscopic capacitor and, in Appendix B.2, we show how equivalent results can be derived with a Hamiltonian formulation.
