**4. The Mesoscopic Capacitor**

The mesoscopic capacitor in Figure 6 plays a central role in the quest to achieve full control of scalable coherent quantum systems [4,129,130]. A mesoscopic capacitor is an electron cavity coupled to a lead via a QPC and capacitively coupled to a metallic gate [51–53]. The interest in this device stems from the absence of DC transport, making possible the investigation and control of the coherent dynamics of single electrons. The first experimental realization of this system was a two-dimensional cavity in the quantum Hall regime [20,21], exchanging electrons with the edge of a bulk two-dimensional electron gas (2DEG). Operated out of equilibrium and in the weak tunneling limit, this system allows the triggered emission of single electrons [22–24], and paved the way to the realization of single-electron quantum optics experiments [131–134], as well as probing electron fractionalization [25,135], accounted by the scattering of charge density waves (plasmons) in the conductor [136–143], and their relaxation [26]. On-demand single-electron sources were also recently realized with real-time switching of tunnel-barriers [27–32], electron sound-wave surfing [14,16,144], generation of levitons [8–11,13], and superconducting turnstiles [18,19]. We direct again the interested reader to Ref. [35] for a comprehensive review of these experiments.

The key question concerning the dynamics of a mesoscopic capacitor is which electronic state, carrying a current I, is emitted from the cavity following a change in the gate voltage *<sup>V</sup>*g. The linear response is characterized by the admittance A(*ω*),

$$\mathcal{L}(\omega) = \mathcal{A}(\omega) V\_{\mathfrak{F}}(\omega) + \mathcal{O}(V\_{\mathfrak{F}}^2). \tag{30}$$

In their seminal work, Büttiker and coworkers showed that the low-frequency admittance of a mesoscopic capacitor reproduces the one of a classical *RC* circuit [51–53],

$$\mathcal{A}(\omega) = -i\omega \mathbb{C}(1 + i\omega R\_{\text{q}} \mathbb{C}) + \mathcal{O}(\omega^3) \, , \tag{31}$$

in which both the capacitance *C* and the *charge relaxation resistance <sup>R</sup>*q probe novel coherent dynamical quantum effects. The capacitance *C* was originally interpreted as an *electro-chemical capacitance* 1/*C* = 1/*C*g + 1/*C*q, series of a *geometric* (*C*g) and a *quantum* (*C*q) contribution [21,51–53]. The geometric contribution is classical and depends on the shape of the capacitive contact between gate and quantum dot. The quantum contribution is a manifestation of the Pauli exclusion principle and was found proportional to the local density of states in the cavity, see Figure 7. Remarkably, the charge relaxation resistance *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> was predicted to be universally equal to half of the resistance quantum in the case of one conducting channel [20], independently of the transparency of the QPC connecting cavity and lead. This result is in striking contrast with the resistance measured in DC experiments and was originally labeled as a *Violation of Kirchhoff's Laws for a Coherent RC Circuit* [20]. Reference [21] extensively reviews the original theoretical predictions and their experimental confirmation, in a non-interacting and self-consistent setting, which we also review and put in relation with their Hamiltonian formulation in Appendix B.

**Figure 6.** Top—First realization of the mesoscopic capacitor [20]: A two-dimensional electron gas (2DEG) in the Quantum Hall regime is coupled to a quantum cavity via a gate-controlled QPC. Bottom—Working principle of a single electron emission [22–26]. A gate potential moves the quantized levels of the cavity above and below the Fermi surface of the coupled reservoir. Electron/hole emission in steps 2 and 3 follows from moving occupied/empty orbitals above/below the Fermi surface.

Below, we discuss how the LFL approach challenges and extends the above studies. In particular:


$$\operatorname{Im}\left[\chi\_{\mathfrak{c}}(\omega)\right]\vert\_{\omega\to 0} = \omega\hbar\pi\chi\_{\mathfrak{c}}^2(\omega=0)\,,\tag{32}$$

in which *<sup>χ</sup>*c(*ω*) is the Fourier transform of the dynamical charge susceptibility (37);


*4.1. Hamiltonian Description of the Quantum RC Circuit: Differential Capacitance and Korringa–Shiba Relation*

Expanding the square in Equation (1) and neglecting constant contributions, Hc renormalizes the orbital energy *εd* in Equation (12) and adds a quartic term in the annihilation/creation operators *dl*, namely

$$\mathcal{H}\_{\mathbb{C}} = -eV\_{\mathbb{B}}(t)N + E\_{\mathbb{C}}N^2. \tag{33}$$

The driving gate voltage *V*g couples to the charge occupation of the quantum dot *Q* = *<sup>e</sup>N*. In single-electron emitters, one operates on the time dependent voltage drive *<sup>V</sup>*g(*t*) to bring occupied discrete levels above the Fermi surface and then trigger the emission of charge, see Figure 6. The current of the device is a derivative in time of the charge leaving the quantum dot, the admittance reads then, in the Fourier frequency representation,

$$\mathcal{A}(\omega) = -i\omega \frac{\mathcal{Q}(\omega)}{V\_{\mathbb{K}}(\omega)}.\tag{34}$$

We start by considering small oscillations of amplitude *εω* of the gate voltage:

$$V\_{\mathbb{R}}(t) = V\_{\mathbb{R}} + \varepsilon\_{\omega} \cos(\omega \, t) \,. \tag{35}$$

Close to equilibrium, expression (34) is calculated relying on Kubo's linear response theory [149]

$$\mathcal{A}(\omega) = -i\omega \nu^2 \chi\_{\mathbb{C}}(\omega) \,, \tag{36}$$

in which *<sup>χ</sup>*c(*ω*) is the Fourier transform of the dynamical charge susceptibility:

$$
\chi\_c(t - t') = \frac{i}{\hbar} \theta(t - t') \left< \left[ N(t), N(t') \right] \right>\_0 \tag{37}
$$

The notation ·0 refers to quantum averages performed at equilibrium, i.e., without the driving term *<sup>V</sup>*g(*t*) in Equation (33). The low frequency expansion of *<sup>χ</sup>*c(*ω*) reads:

$$\mathcal{A}(\omega) = -i\omega \mathfrak{e}^2 \left\{ \chi\_{\mathbb{C}} + i \text{Im} \left[ \chi\_{\mathbb{C}}(\omega) \right] \right\} + \mathcal{O}(\omega^2) \,. \tag{38}$$

where we relied on the fact that the even/odd part of the response function (37) coincide with its real/imaginary part, see Appendix C. We also introduce the static charge susceptibility *χ*c = *<sup>χ</sup>*c(*<sup>ω</sup>* = <sup>0</sup>). The expansion (38) matches that of a classical RC circuit (31). Identifying term by term, we find the expression of the charge relaxation resistance and, in particular, that the capacitance *C* of the mesoscopic capacitor is actually given by a *differential capacitance C*0:

$$\mathcal{C} = \mathcal{C}\_0 = \varepsilon^2 \chi\_{\mathbb{C}} = -\varepsilon^2 \frac{\partial \left< N \right>}{\partial \varepsilon\_d} = \frac{\partial Q}{\partial V\_{\mathbb{g}}}, \qquad \qquad R\_{\mathbb{q}} = \frac{1}{\varepsilon^2 \chi\_{\mathbb{c}}^2} \left. \frac{\mathrm{Im}\chi\_{\mathbb{c}}(\omega)}{\omega} \right|\_{\omega \to 0}. \tag{39}$$

The differential capacitance is proportional to the density of states of *charge* excitations on the dot, which, as mentioned above, generally differs from the local density of states in the presence of strong correlations. Equation (39) provides also the general condition for the universal quantization of the charge-relaxation resistance *<sup>R</sup>*q = *h*/2*e*2, namely:

$$\left. \mathrm{Im} \chi\_{\mathrm{c}}(\omega) \right|\_{\omega \to 0} = \hbar \pi \omega \chi\_{\mathrm{c}}^2. \tag{40}$$

Such kind of relation is known as a Korringa–Shiba (KS) relation [146]. The KS relation establishes that the imaginary part of the dynamic charge susceptibility, describing dissipation in the system, is controlled by the static charge fluctuations on the dot, *χ*c.

Additionally, we mention that the relation (40) also affects the phase-shift of reflected or transmitted light through a mesoscopic system in the Kondo LFL regime [150–152]. Such situations have been recently realized with quantum-dot devices embedded in circuit-QED architectures [153–159], in which the driving input signal can be modeled by an AC potential of the form (35).

**Figure 7.** Physical origin of the quantum capacitance *<sup>C</sup>*q. Pauli exclusion forces electrons entering the dot to pay an energy price equal to the local level spacing Δ, resulting in a capacitance *C*q = *<sup>e</sup>*2/<sup>Δ</sup>. On the right, a one-dimensional representation of the mesoscopic capacitor, with a dot of size -.

### *4.2. The Origin of the Differential Capacitance as a 'Quantum' Capacitance as far as Interactions are Neglected*

As far as interactions are neglected, the differential capacitance *C*0 is a manifestation of the fermionic statistics of electrons, determined by the Pauli exclusion principle. For this reason it has been originally labeled as a 'quantum' capacitance *C*q [51–53]. When an electron is added to the quantum dot, in which energy levels are spaced by Δ, the Pauli exclusion principle does not allow one to fill an occupied energy state, but requires to pay a further energy price Δ, see Figure 7. The capacitance associated to this process is then *C*q = *δQ*/*δV*. For one electron *δQ* = *e* and *δV* = Δ/*<sup>e</sup>*. Substituting these two expressions, we recover a uniform quantum capacitance:

$$\mathbf{C\_{q}} = \frac{e^{2}}{\Delta}.\tag{41}$$

This expression establishes that the quantum capacitance is proportional to the density of states in the quantum dot at the Fermi energy *C*q = *e*<sup>2</sup>N (*EF*), with N (*EF*) = 1/Δ, to be distinguished from *ν*0, the density of states of the lead electrons.

Additionally, the quantum capacitance is related to the dwell-time spent by electrons in the cavity. The general relation is derived in Appendix B, but it also results from simple estimates. Considering the representation of the mesoscopic capacitor of Figure 7, in the open-dot limit, the time spent by an electron in the cavity coincides with its time of flight *τ*f = -/*vF*: The ratio between the size of the cavity - and its (Fermi) velocity *vF*. The level spacing Δ of an isolated cavity of size - is estimated by linearizing the spectrum close to the Fermi level. The distance in momentum between subsequent levels is *h*/-, corresponding to Δ = *hvF*/- = *h*/*<sup>τ</sup>*f. Substituting in Equation (41) leads to an equivalent expression for the quantum capacitance:

$$\mathbf{C\_{q}} = \frac{\varepsilon^{2}}{h} \mathbf{r\_{f}}.\tag{42}$$

On the experimental side, the level spacing of the quantum dot can be actually estimated and, in the experimental conditions of Ref. [20], it was established to be of the order of Δ ∼ 15 GHz, corresponding to a quantum capacitance *C*q ∼ 1 fF. Experimental measurements of *C*, reported in Figure 8, give an estimate also for *<sup>C</sup>*g, showing that *C*q *<sup>C</sup>*g. This implies that the level spacing Δ was much larger than the charging energy *Ec* = *<sup>e</sup>*2/2*C*g , of the order of fractions of the GHz, apparently justifying the mean-field approach to describe experimental results, with the limitations that we are going to discuss in out-of-equilibrium regimes, see Section 5.

The argumen<sup>t</sup> leading to Equation (41) implicitly assumes the perfect transparency of the QPC, namely that the probability amplitude *r* for a lead electron to be reflected when passing though the QPC to enter the cavity is equal to zero (*r* = 0). In this limit, the density of states on the dot is uniform. Finite reflection *r* = 0 is responsible for resonant tunneling processes, leading to oscillatory behavior of the local density (or the dwell-time) of states as a function of the gate potential *<sup>V</sup>*g, in agreemen<sup>t</sup> with the experimental findings reported in Figure 8. In Appendix B, we provide a quantitative analysis of this effect by explicitly calculating the differential capacitance *C*0, Equation (39), by neglecting the term proportional to *<sup>E</sup>*c*N*<sup>2</sup> in Equation (33).

**Figure 8.** Top—Measurement of the universal charge relaxation resistance from Ref. [20]. The resistance of two different samples (E3 and E1) is given by the real part of their impedance *Z* = 1/A as a function of the QPC potential *V*G, in Figure 6, which also affected the gate potential *<sup>V</sup>*g. Measurements were carried out for *T* = 30*mK* and a magnetic field *B* = 1.3*T* polarizing the electrons, resulting in one conducting channel. Uncertainties are indicated by the hatched areas. Bottom—Measurement of the total capacitance *Cμ* = *C*, through Im(Z) = <sup>−</sup>1/*ωC*, for the same samples. The oscillatory behavior is related to resonances in the density of states of the dot.

### *4.3. The Physical Origin of the Universal Charge Relaxation Resistance*

The universality of *<sup>R</sup>*q was also verified experimentally in Ref. [20], see Figure 8. In the quantum coherent regime the charge relaxation resistance is *universal*: It does not depend on the microscopic details of the circuit (*C*g, *r*, ...), but only on fundamental constants, namely Planck's constant *h* and the electron charge *e*. This is surprising. When applying a DC voltage across a QPC connecting two leads, the QPC behaves as a resistive element of resistance [160,161]:

$$R\_{\rm DC} = \frac{h}{c^2 D} \,'\,\tag{43}$$

where *D* = 1 − |*r*|<sup>2</sup> is the QPC transparency. This quantity depends on the amplitude *r* for electrons to be backscattered when arriving at the QPC, which can be tuned by acting on a gate potential. As we considered spinless electrons, the factor 2 in *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> cannot be related to spin degeneracy and *D* = 1 in Equation (43). It is rather related to the fact that the dot is connected to a single reservoir, in contrast to the source-drain reservoirs present in DC transport experiments [56,162]. In direct transport, each metallic contact is responsible for a quantized contact resistance *R*c = *h*/2*e*2, the Sharvin–Imry resistance [163,164]. In source-drain experiments, Equation (43) could be recast in the form *<sup>R</sup>*QPC + 2*R*c, with *<sup>R</sup>*QPC = *he*2 <sup>1</sup>−*DD* the resistive contribution proper to the QPC. For the

case of a mesoscopic capacitor, there is a single reservoir and one would thus expect for the charge relaxation resistance:

$$R\_{\rm q}^{\rm expected} = R\_{\rm QPC} + R\_{\rm c} = \frac{h}{2c^2} + \frac{h}{c^2} \frac{1 - D}{D} \,. \tag{44}$$

The fact that *<sup>R</sup>*q does not depend on the transparency *D*, assuming the universal value *h*/2*e*2, cannot be attributed to the laws governing DC quantum transport, and this is the reason why one can speak about the *Violation of Kirchhoff's Laws for a Coherent RC Circuit* [20]. The universality of *<sup>R</sup>*q is rather a consequence of the fact that *<sup>R</sup>*q, differently from the contact resistance *R*c, is related to energy (Joule) dissipation. As electrons propagate coherently within the cavity, they cannot dissipate energy inside it, but only once they reach the lead. We will illustrate in Section 4.5 how this phenomenon is a direct consequence of the possibility to excite particle-hole pairs by driven electrons. At low frequency, dissipation is thus only possible in the presence of a continuum spectrum, accessible in the metallic reservoirs. The expected resistance (44) is recovered only if electrons lose their phase coherence inside the dot [56,162], see also Refs. [165,166], which take Coulomb blockade effects into account. For instance, in the high temperature limit *kBT* Δ, Equation (44) is not recovered. The reason is that, in scattering theory, temperature is fixed by the reservoirs without affecting the coherent/phase-preserving propagation in the mesoscopic capacitor.
