*4.4. The Open-Dot Limit*

We address now the role of the charging energy *E*c and come back to our initial example of the open dot limit, considered in Section 2. The possibility to rely on an exact bosonized solution for the model (1)–(2), made possible the derivation of the admittance A(*ω*) in linear-response theory for a fully transparent point contact (*r* = 0) and a finite-sized cavity [65]:

$$\mathcal{A}(\omega) = -i\omega \mathbb{C}\_{\mathbb{S}} \left( 1 - \frac{i\omega \tau\_{\mathbb{C}}}{1 - e^{i\omega \tau\_{\mathbb{I}}}} \right)^{-1}. \tag{45}$$

This expression is important as it makes possible to study the interplay between two different time-scales, namely the time of flight *τ*f of electrons inside the cavity, already present in the previous discussion, and *τ*c = *hC*g/*e*<sup>2</sup> the time scale corresponding to the charging energy *E*c. We mention that, interestingly, the admittance (45) was also found to describe the coherent transmission of electrons through interacting Mach–Zehnder interferometers [167,168].

What is quite remarkable about the admittance (45) is that, to linear order in *ω*, the two time scales *τ*f and *τ*c still combine into the universal charge relaxation resistance *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> and a series of a geometrical and quantum capacitance *C*q = *e*2*τ*f/*<sup>h</sup>* [51–53] (see also Equation (42)):

$$\frac{1}{\mathcal{C}\_0} = \left[\frac{1}{\mathcal{C}\_\mathcal{g}} + \frac{h}{e^2 \tau\_\mathcal{f}}\right].\tag{46}$$

The low-frequency behavior of Equation (45) illustrates how interacting systems behave as if interactions were absent at low energies. What is then also implicit in Equation (45) is that, to observe separate effects on the charge dynamics, induced by free propagation (*τ*f) or interactions (*τ*c), one has to consider proper out-of-equilibrium/high-frequency regimes. These regimes will be addressed in Section 5.

Nevertheless, interactions still matter even in low-frequency regimes. Consider the infinite-size (metallic) limit for the cavity, *τ*f → ∞. In this limit, also describing the experiment in Figure 2, one implicitly assumes that the driving frequency *ω* is larger than the internal level spacing of the dot Δ,

$$
\hbar\omega \gg \Delta.\tag{47}
$$

The discrete spectrum of the dot can thus be treated as a continuum, which allows for energy dissipation also inside the cavity, see Figure 9. In particular, averaging the admittance (45) over a finite bandwidth *δω*, such that *ω δω* Δ, one exactly recovers the admittance of a classical RC circuit of capacitance *C*g and charge-relaxation resistance *Rq* = *h*/*e*<sup>2</sup> [65]:

$$\mathcal{A}(\omega) = \frac{-i\omega \mathcal{C}\_{\mathcal{S}}}{1 - i\omega \mathcal{C}\_{\mathcal{S}} \frac{\hbar}{c^2}}.\tag{48}$$

The mesoscopic crossover *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> → *h*/*e*<sup>2</sup> is an exquisite coherent effect triggered by interactions. This phenomenon has fundamentally the same origin of the elastic electron transfer exemplified by the correlation function (3), considered at the very beginning of this review.

Remarkably, the universality of the charge-relaxation resistance holds in the presence of backscattering at the dot entrance, without affecting the mesoscopic crossover *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> → *h*/*e*<sup>2</sup> [65]. Nevertheless, the possibility to interpret the differential capacitance as a series of two separate geometric and quantum term as in Equation (46), is lost. If we locate the entrance of the dot at *x* = 0, backscattering corrections to the model (1)–(2) read:

$$\mathcal{H}\_r = -\hbar r \upsilon\_F \left[ \Psi\_\mathcal{R}(0)^\dagger \Psi\_\mathcal{L}(0) + \Psi\_\mathcal{L}(0)^\dagger \Psi\_\mathcal{R}(0) \right],\tag{49}$$

and compromise a non-interacting formulation of the problem, even in its bosonized form [65,72,148].

It becomes then important to understand why and to which extent quantities such as the charge-relaxation resistance show universal coherent behavior even in the presence of interactions. The extension of the LFL theory in the quasistatic approximation provides the unified framework to understand the generality of such phenomena.

**Figure 9.** Mesoscopic crossover in the charge relaxation resistance. Top—In a small dot, the level spacing Δ is larger than the driving energy *h*¯ *ω* and energy levels in the dot are not excited. The universal resistance *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> of the equivalent *RC* circuit is furnished exclusively by the lead electron reservoir. Bottom—Excitation of energy levels inside the dot are permitted in the large dot limit, which acts as a further dissipative reservoir in series to the lead.

### *4.5. The Tunneling Limit and the Quasi-Static Approximation*

In Section 3, we showed that a large class of models of the form (4) are effectively described, in the low-energy limit, by a LFL theory (8), in which the potential scattering coupling constant *W*

depends on the orbital energy of the dot *εd*. The expansion of the charging energy Hamiltonian (33) made apparent that this energy is renormalized by the gate potential *εd* → *εd* − *eVg*(*t*). For an AC bias voltage, we consider then a periodic function of time oscillating at the frequency *ω*:

$$
\varepsilon\_d(t) = \varepsilon\_d^0 + \varepsilon\_\omega, \cos\left(\omega t\right). \tag{50}
$$

The *quasi-static approximation* consists in substituting Equation (50) directly in Equation (8). This condition assumes that the low energy Hamiltonian (8), derived for the equilibrium problem, follows, without any delay, the orbital oscillations expected from the parent, high-energy, model. The quasi-static approximation is then a statement about a behavior close to adiabaticity.

We consider then the linear response regime and expand the coupling *<sup>W</sup>*(*<sup>ε</sup>d*) in *εω*. Focusing on the single channel case, the extension to multiple channels being straightforward, Equation (8) becomes:

$$\mathcal{H} = \sum\_{k} \varepsilon\_{k} \mathbf{c}\_{k}^{\dagger} \mathbf{c}\_{k} + \left[ \mathcal{W}(\boldsymbol{\varepsilon}\_{d}^{0}) + \mathcal{W}'(\boldsymbol{\varepsilon}\_{d}^{0}) \boldsymbol{\varepsilon}\_{\omega} \cos \left( \omega \mathbf{t} \right) \right] \sum\_{kk'} \mathbf{c}\_{k}^{\dagger} \mathbf{c}\_{k}'.\tag{51}$$

We diagonalize the time independent part of this Hamiltonian [104]:

$$\mathcal{H} = \sum\_{kk'} \varepsilon\_k a\_k^\dagger a\_k + \frac{\mathcal{W}'(\varepsilon\_d^0)}{1 + \left[\pi \upsilon\_0 \mathcal{W}(\varepsilon\_d^0)\right]^2} \varepsilon\_\omega \cos\left(\omega t\right) \sum\_{kk'} a\_k^\dagger a\_{k'},\tag{52}$$

where the operators *a* and *a*† describe the new quasi-particles diagonalizing the time independent part of the Hamiltonian (51). The Friedel sum rule (11) establishes that:

$$\chi\_{\rm c} = \frac{\nu\_0 \mathcal{W}'(\varepsilon\_d^0)}{1 + \left[\pi \nu\_0 \mathcal{W}(\varepsilon\_d^0)\right]^2},\tag{53}$$

and the Hamiltonian (52) can be cast in the more compact and transparent form:

$$\mathcal{H} = \sum\_{kk'} \varepsilon\_k a\_k^\dagger a\_k + \frac{\chi\_{\rm c}}{\nu\_0} \varepsilon\_{\omega \prime} \cos \left( \omega \prime t \right) \sum\_{kk'} a\_k^\dagger a\_{k'} \,. \tag{54}$$

This Hamiltonian shows the mechanism responsible for energy dissipation at low energy for the rich variety of strongly interacting systems satisfying the Friedel sum rule and LFL behavior at low energy. The time dependent term pumps energy in the system, which is then dissipated by the creation of particle-hole pairs. Crucially, this term is controlled by the static charge susceptibility *χ*c of the quantum dot. The non-interacting Hamiltonian (54) explains why non-interacting results also hold for the universal charge relaxation resistance in the presence of interactions on the quantum dot.

We now illustrate how the Hamiltonian (54) implies the validity of the KS relation and thus universality of the charge-relaxation resistance *<sup>R</sup>*q. The proof was originally devised for spin-fluctuations [169] and we extend it here to the case of charge fluctuations. For drives of the form (50), the power dissipated by the system is proportional to the imaginary part of the dynamic charge susceptibility, see Appendix C,

$$\mathcal{P} = \frac{1}{2} \varepsilon\_{\omega}^{2} \omega \text{Im} \chi\_{\mathbb{C}}(\omega) \,. \tag{55}$$

A direct calculation of Im*χ*c is a difficult task and this is where the low-energy model (54) becomes useful. Similarly as for Equation (55), the LFL theory (54) predicts the dissipated power:

$$\mathcal{P} = \frac{1}{2} \varepsilon\_{\omega}^{2} \omega \text{Im} \chi\_{A}(\omega) \,, \tag{56}$$

where the linear response function *<sup>χ</sup>A*(*<sup>t</sup>* − *t*) = *ih*¯ *θ*(*t* − *<sup>t</sup>*)[*A*(*t*), *<sup>A</sup>*(*t*)]0 is a correlator at different times of the potential scattering operator:

$$A = \frac{\chi\_c}{\nu\_0} \sum\_{kk'} a\_k^\dagger a\_{k'} \, , \tag{57}$$

responsible for the creation of particle-hole pairs. The Fourier transform of the response function reads:

$$\chi\_A(\omega) = -\frac{1}{\hbar} \frac{\chi\_c^2}{\nu\_0^2} \sum\_{pp'} f(\varepsilon\_P) [1 - f(\varepsilon\_{p'})] \left[ \frac{1}{\omega + \frac{\varepsilon\_{p'} - \varepsilon\_{p'}}{\hbar} + i0^+} - \frac{1}{\omega + \frac{\varepsilon\_{p'} - \varepsilon\_p}{\hbar} + i0^+} \right],\tag{58}$$

in which *f*(*ε p*) = 1/(*eβε p* + 1) is the Fermi distribution. We consider the electron lifetime as infinite, i.e., much longer than the typical time scales *τ*c and *τ*f. Taking the imaginary part and the continuum limit for the spectrum in the wide-band approximation, one finds, at zero temperature,

$$\mathrm{Im}\chi\_A(\omega) = \pi\hbar\omega\chi\_\varepsilon^2. \tag{59}$$

The two dissipated powers (55) and (59) have to be identical, implying the Korringa–Shiba relation (32), enforcing then a universal value for the charge relaxation resistance *<sup>R</sup>*q = *h*/2*e*2.

### *4.6. The LFL Theory of Large Quantum Dots: The Mesoscopic Crossover R* q = *h*/2*e*<sup>2</sup> → *h*/*e*<sup>2</sup>

The above demonstration has to be slightly adapted to show the mesoscopic crossover *<sup>R</sup>*q = *h*/2*e*<sup>2</sup> → *h*/*e*2. This crossover takes place for the CBM (12), in the infinite-size limit of the dot. As implicit in the effective description (21) of the CBM, the dot and the lead constitute two separate Fermi liquids. Sections 2 and 3 illustrated how the energy cost *E*c prevents the low-energy transfer of electrons between the dot and lead [77]. The electrons of both these gases are then only backscattered at the lead/dot boundary with opposite amplitudes. In the quasi-static approximation, all the steps carried in the previous discussion apply for the Hamiltonian (21). In this case, the time variation of the orbital energy *εd* also drives particle-hole excitations in the dot. The operator responsible for energy dissipation becomes:

$$A = \frac{\chi\_c}{\nu\_0} \left( \sum\_{kk'} c\_k^\dagger c\_{k'} - \sum\_{ll'} d\_{l'}^\dagger d\_{l'} \right),\tag{60}$$

in which the operators *c*†*k* and *d*†*l* create lead and dot electrons of energy *<sup>ε</sup>k*,*<sup>l</sup>* respectively. This formulation of the operator *A* adds a further contribution to Equation (58), analogous to the contribution of particle-hole pairs excited in the lead, namely:

$$-\frac{1}{\hbar} \frac{\chi\_{\mathbf{c}}^{2}}{\nu\_{0}^{2}} \sum\_{ll'} f(\varepsilon\_{l}) [1 - f(\varepsilon\_{l'})] \left[ \frac{1}{\omega + \frac{\varepsilon\_{l'} - \varepsilon\_{l'}}{\hbar} + i0^{+}} - \frac{1}{\omega + \frac{\varepsilon\_{l'} - \varepsilon\_{l}}{\hbar} + i0^{+}} \right] . \tag{61}$$

The limits *ω* → 0 and Δ → 0 do not commute in the above expression. This fact has a clear physical interpretation: If the frequency is sent to zero before the level spacing, energy cannot be dissipated in the cavity and no additional contribution to Im*χ*c(*ω*) is found. If the opposite limit is taken, the condition (47) is met and the Korringa–Shiba relation is then modified by a factor two:

$$\mathrm{Im}\chi\_{\mathbb{C}}(\omega) = 2\pi\hbar\omega\chi\_{\mathbb{C}}^2,\tag{62}$$

which doubles the universal value of the single-channel charge relaxation resistance *<sup>R</sup>*q = *h*/*e*2. The relation (62) was originally shown by explicit perturbation theory in the tunneling amplitude, close and away from charge degeneracy points [65]. As summarized in Figure 9, driving at a frequency higher than the dot level spacing induces the creation of particle/hole pairs inside the dot as well, enhancing energy dissipation with respect to the small dot limit *h*¯ *ω* < Δ. As energy can be *coherently* dissipated in two fermionic baths (dot and lead), the dot acts effectively as a further (Joule) resistor in series with the lead, leading to a doubled and still universal charge relaxation resistance.

### *4.7. The Multi-Channel Case and Universal Effects Triggered by Kondo Correlations*

The above discussion also extends to the *M* channels case, leading to a generalized KS relation:

$$\mathrm{Im}\chi\_{\mathrm{c}}(\omega)|\_{\omega \to 0} = \hbar \pi \omega \sum\_{\sigma} \chi\_{\sigma}^{2},\tag{63}$$

which corresponds to a non-universal expression for the charge relaxation resistance [68,69]

$$R\_{\rm q} = \frac{h}{2e^2} \frac{\sum\_{\sigma} \chi\_{\sigma}^2}{\left(\sum\_{\sigma} \chi\_{\sigma}\right)^2} \,. \tag{64}$$

This expression is analogous to the one obtained by Nigg and Büttiker [54]. In their derivation leading to Equation (A48), the densities of states, or dwell-times *τσ*, of the *σ* channel in the dot, replace the susceptibilities *χσ*. The single channel case is remarkable in that the numerator simplifies with the denominator in Equation (64), leading to the universal value *<sup>h</sup>*/(<sup>2</sup>*e*<sup>2</sup>), which is thus physically robust. Otherwise, in the fine-tuned case that all the channel susceptibilities are equal, one finds *<sup>R</sup>*q = *h*/2*e*2*M*.

Spinful systems in the presence of a magnetic field are the simplest ones to study how the charge-relaxation resistance is affected by breaking the symmetry between different conduction channels. Indeed, lifting the orbital level degeneracy by a magnetic field breaks the channel symmetry and the charge relaxation resistance is no longer universal, as it was originally realized in studies of the AIM (13) relying on the Hartree–Fock approximation [54].

Nevertheless, the self-consistent approach misses important and sizable effects triggered by strong Kondo correlations. These were originally observed relying on the Numerical Renormalization Group (NRG) [147]. The numerical results, reported in Figure 10, showed that, for Zeeman splittings of the order of the Kondo temperature *T*K, the charge relaxation resistance can reach up to 100 times the universal value of *<sup>R</sup>*q = *<sup>h</sup>*/(<sup>4</sup>*e*<sup>2</sup>), which would be expected in the two-fold spin degenerate case.

**Figure 10.** Left—Dependence of *<sup>R</sup>*q on the Zeeman splitting Δ*Z* in the Kondo regime from Ref. [147]. These results have been obtained by Numerical Renormalization Group (NRG) calculations with Γ = 0.02 and *E*c = 0.2 (both quantities are measured in units of the contact bandwidth *D* and the definition of the hybridization energy Γ is provided in Appendix A.3). They show that, for Zeeman energies of the order of the Kondo temperature, a giant non-universal peak appears in the charge relaxation resistance. Right—Comparison of *<sup>R</sup>*q as a function of the magnetic field between NRG calculations (dots) (extracted from Ref. [147]) and our Bethe ansatz results (solid lines) for different *<sup>ε</sup>d*/*U* and *U*/Γ = 20, showing excellent agreemen<sup>t</sup> [67,68].

*Entropy* **2020**, *22*, 847

The LFL approach allows the analytical quantification and physical interpretation of such giant dissipative phenomenon [67,68]. For two spin channels, the total charge on the dot is the sum of the two spin occupation *N* = *N*↑ + *N*↓. Equation (64) can then be recast in the useful form:

$$R\_{\rm q} = \frac{h}{4e^2} \left( 1 + \frac{\chi\_m^2}{\chi\_c^2} \right) \tag{65}$$

in which we introduce the usual charge susceptibility: *χ*c = −*∂N*/*∂ε<sup>d</sup>* and the *charge-magneto* susceptibility

$$
\chi\_m = -2\frac{\partial \left< m \right>}{\partial \varepsilon\_d}.\tag{66}
$$

This quantity is twice the derivative of the dot magnetization *m* = (*<sup>N</sup>*↑ − *<sup>N</sup>*↓)/2, with respect to the orbital energy *εd*. The charge-magneto susceptibility is an atypical object to study quantum dot systems, where the *magnetic* susceptibility *χH* = −*∂ m* /*∂H* is rather considered to study the sensitivity of the local moment of the quantum dot to variations of the magnetic field *H*. Equation (65) shows that the susceptibility of the *magnetization* of the dot, and not its charge, is responsible for the departure from the universal quantization *<sup>h</sup>*/(<sup>4</sup>*e*<sup>2</sup>) of the charge relaxation resistance. Equation (65) also separates explicitly charge and spin degrees of freedom of the electrons in the quantum dot. They can display very different behaviors in correlated systems, as illustrated in Figure 11 in the Kondo regime, defined for one charge blocked on the dot and Zeeman energies below the Kondo temperature (26).

In particular, Kondo correlations strongly affect the dot magnetization, but not its occupation. The points where *χm* differs from *χ*c correspond to non-universal charge relaxation resistances. In Figure 10, the values derived with the LFL approach (65) are compared to those obtained with NRG [147], showing excellent agreement. Additionally, the LFL approach also allows to derive an exact analytical description of this peak, showing a genuinely giant dissipation regime: Simultaneous breaking of the SU(2) (*H* = 0) and particle-hole symmetry (*<sup>ε</sup>d* = −*U*/2) trigger a peak in *<sup>R</sup>*q which scales as the 4th(!) power of *U*/Γ and has its maximum for Zeeman splittings of the order of the Kondo temperature. This effect is caused by the fact that breaking the Kondo singlet by a magnetic field activates spin-flip processes, which dissipate energy through creation of particle-hole pairs [147].

We conclude by discussing the deviations of the differential capacitance *C*0 from the local density of states of the cavity, which is clearly apparent in Kondo regimes. The spin/charge separation arising in the AIM allows to observe important physical effects on the differential capacitance of strongly interacting systems. Charge and spin on the dot are carried by different excitations: Holons and spinons. We report in Figure 12 the density of states of these excitations in the particle-hole symmetric case *εd* = −*U*/2. In the absence of interactions (*U*/Γ = 0), they have the same shape, but they start to strongly differ as the interaction parameter *U*/Γ is increased. They develop well pronounced peaks, but at different energies, signaling the appearance of separated charge and spin states. In the case of holons, the excited charge state appears close to *ε* = *U*/2, the energy required to change the dot occupation at particle-hole symmetry. In Ref. [170], it is shown that the density of states of the holons equals the static charge susceptibility *χ<sup>c</sup>*, coinciding then with the differential capacitance *C*0. At particle-hole symmetry, this quantity scales to zero as <sup>8</sup>Γ/*π<sup>U</sup>*2, see Equation (29). Instead, the spinon density of states develops a sharp peak at zero energy, known as the Abrikosov–Suhl resonance [118], signaling the emergence of the strongly correlated Kondo singlet. The differential capacitance *C*0 is completely insensitive to this resonance, which dominates the *total* density of states on the dot. Such an effect was distinctly observed in carbon nanotube devices coupled to high-quality-factor microwave cavities [145]. These systems efficiently probe the admittance (30) also in quantum dots with more than two internal degrees of freedom [150,152], such as extensions of the AIM to SU(4) regimes, relevant for quantum dots realized with carbon nanotubes [68,171–177].

**Figure 11.** From Ref. [68]. Top—Charge occupation and magnetization of the dot for *U*/Γ = 20 as function of the orbital energy *<sup>ε</sup>d* and magnetic field *H*. The insets show the same quantities on a logarithmic scale. The light green lines in the linear plots correspond to *H* = 2*εd*, and separate regions with different charge occupations, while the dark green lines in the insets correspond to *T*K, Equation (26), and separate regions with different magnetization. The charge is not sensitive to the formation of the Kondo singlet for Zeeman energies below the Kondo temperature (green line), while the magnetization becomes zero. Bottom—Corresponding charge susceptibility and charge-magneto susceptibility. The susceptibilities are in units of 1/Γ. In the insets the same quantities are plotted on a logarithmic scale and the zone of appearance of the giant peak of the charge relaxation resistance can be appreciated. It is the region, following *T*K, in which *χ*c is close to zero, while *χm* acquires important values because of the formation of the Kondo singlet.

**Figure 12.** From Ref. [170]. (**Left**)—Density of states of local holons on the dot N*h*(*ε*). *ε* is the excitation energy. A Coulomb peak emerges increasing the interaction parameter *U*/Γ and vanishes at zero energy as <sup>8</sup>Γ/*πU*2. This quantity coincides with *χ<sup>c</sup>*, plotted in Figure 11. (**Right**)—Density of states of local spinons N*s*(*ε*). It behaves as the holonic one for *U*/Γ = 0 and develops the Abrikosov–Suhl resonance at zero energy, the signature of the formation of the strongly correlated Kondo singlet state.

The above discussion completes the review of the application of the LFL theory to study the low-energy dynamics of quantum impurity driven systems. Further applications could be envisioned to describe various correlation effects on different aspects of weakly driven interacting quantum-dot systems, as long as they can be described by an effective theory of the form (8). An important case involves the driving of the coupling term Hres−dot → Hres−dot(*t*) in the Hamiltonian (4), which has been implemented experimentally, with important metrologic applications [27–32]. Another interesting perspective concerns the application of the LFL theory to energy transfer [178–180], or coupling quantum-dot systems to mechanical degrees of freedom [181–187], which are described by similar models as quantum-dot devices embedded in circuit-QED devices [150,153–158].

Deviations from universal and coherent behaviors are expected in non-LFL regimes, arising when the reservoirs are Luttinger Liquids [188,189] or in over-screened Kondo impurities, in which the internal degrees of freedom of the bath surpass those of the impurity [85,190].

We have thus illustrated how a coherent and effectively non-interacting LFL theory accounts for strong correlation effects in the dynamics of quantum dot devices. It has to be clarified how interaction are supposed to affect proper out-of-equilibrium regimes. As a direct example, consider again the admittance (45). Its expansion to low-frequencies completely reproduces the self-consistent predictions of Refs. [51–53], but it 'hides' the qualitative difference between the two time scales *τ*c and *τ*f, associated to interactions and free-coherent propagation respectively. Higher-frequency driving will inevitably unveil this important difference, as we are going to demonstrate by giving a new twist to past experimental data in the next conclusive section.

### **5. What about Out-Of-Equilibrium Regimes? A New Twist on Experiments**

We conclude this review by showing how interaction inevitably dominate proper out-ofequilibrium or fastly-driven regimes. We will focus, also in this case, on the mesoscopic capacitor. In particular, we will show that past experimental measurements, showing fractionalization effects in out-of-equilibrium charge emission from a driven mesoscopic capacitor [25], also manifest previously overlooked signatures of non-trivial many-body dynamics induced by interactions in the cavity.

As a preliminary remark, notice that the circuit analogy (31) does not apply for a non-linear response to a gate voltage change or to fast (high-frequency) drives. An important example is a large step-like change in the gate voltage *<sup>V</sup>*g(*t*) = *<sup>V</sup>*g*<sup>θ</sup>*(*t*), *θ*(*t*) being the Heaviside step function, which is relevant to achieve triggered emission of quantized charge [24]. Such a non-linear high-frequency response has been considered extensively for non-interacting cavities [22,24,191–195], where the current response to a gate voltage step at time *t* = 0 was found to be of the form of simple exponential relaxation [22,191,193,195]:

$$I(t) \propto e^{-t/\tau \mathbb{R}} \theta(t). \tag{67}$$

For a cavity in the quantum Hall regime the relaxation time *τ*R = *τ*f/(<sup>1</sup> − |*r*| 2), where *τ*f is the time of flight around the edge state of the cavity, see Figures 6 and 7, and *r* the reflection amplitude of the point contact.

There have been relatively few studies of the out-of-equilibrium behavior of the mesoscopic capacitor in the presence of interactions. The charging energy leads to an additional time scale *τ*c = 2*πhC*¯ g/*e*<sup>2</sup> for charge relaxation. The limit 1 − |*r*| 2 1 of a cavity weakly coupled to the lead, such that it can effectively be described by a single level, was addressed in Refs. [196–201].

The full characterization of the out-of-equilibrium dynamics behavior of the mesoscopic capacitor, with a close-to-transparent point contact, was carried out in Ref. [148], extending the analysis of Ref. [65] to a non-linear response in the gate voltage *<sup>V</sup>*g. A main result, spectacular in its simplicity, is that for a fully transparent contact (*r* = 0) the linear-response admittance (45) also describes the non-linear response, i.e., the correction terms in Equation (30) vanish for an ideal point contact connecting cavity and lead [136,202–204]. The Fourier transform of the admittance (45) describes the real-time evolution of the charge *Q*(*t*) after a step change in the gate voltage.

Figure 13 illustrates that initially, for times up to *τ*f, *Q*(*t*) relaxes exponentially with time *τ*c, whereas at time *t* = *τ*f the capacitor abruptly enters a regime of exponentially damped oscillations, the period and the exponential decay, controlled by a complex function of *τ*f and *τ*c, which does not correspond to any time scale extracted from low-frequency circuit analogies. This behavior is not captured by Equation (67), derived in the non-interacting limit. These oscillations correspond to the emission of initially sharp charge density pulses, which are damped and become increasingly wider after every charge oscillation. Such complex dynamics is exquisitely coherent, but totally governed by interactions.

**Figure 13.** From Ref. [148]. Time-evolution of the current/charge density following a sudden gate voltage shift at time *t* = 0 for large interaction strength, *τ*f = 20*τ*c. Top—Charge response Δ*Q*(*t*) as a function of time *t*. Bottom—Series of snapshots of the current/charge density *j*(*<sup>x</sup>*, *t*) at different times. In the inset of panel (**a**), the real-space representation (reproducing the one adopted in Figure 7 with the dot site - = *L*) of the mesoscopic capacitor with the profiles of the emitted charge pulses is given. The times at which the snapshots are taken are indicated by vertical dashed lines in the top panel. Notice that the scale changes along the vertical axis in the different panels. At time *t* = 0, two charge pulses of width ∼ *vFτ*c and opposite sign emerge from the point contact (**<sup>a</sup>**,**b**), one pulse entering the cavity and one pulse entering the chiral edge of the bulk two-dimensional electron gas. Both pulses have a net charge approaching *<sup>C</sup>*gΔ*V*g. The pulse that is emitted into the cavity returns to the point contact at time *t* = *τ*f. As that pulse leaves the cavity, a second pulse-antipulse pair is generated (**c**), partially canceling the original charge pulse that leaves the cavity at *t* = *τ*f. The resulting pulse exiting the cavity is the sum of the dashed profiles. The repetition of this mechanism leads to the widening and lowering of successive pulses (**d**,**<sup>e</sup>**) (notice the change of scale between snapshots). Finally, the asymptotic configuration is attained with a charge *<sup>C</sup>*Δ*V*g uniformly distributed along the cavity edge (**f**).

Additionally, it is also interesting to consider the effect of a small reflection amplitude *r* in the point contact. In this case, the charge *Qr* acquires nonlinear terms in the gate voltage *<sup>V</sup>*g,

$$Q\_{\mathcal{I}}(t) = Q(t) - \frac{e\tilde{r}}{\pi \mathcal{C}} \int dt' \mathcal{A}(t - t') \sin[2\pi \mathcal{Q}(t')/\varepsilon],\tag{68}$$

in which A(*t*) and *Q*(*t*) are the Fourier transform of the admittance and charge for the case of a point contact with perfect transparency, *r* = 0, see Equations (30) and (45). The parameter *r*˜ involves both the (weak) backscattering amplitude *r* and temperature *T*, details can be found in Ref. [148].

### *Experimental Signatures of the Effects of Interaction in Quantum Cavities Driven out of Equilibrium*

The prediction that, in the open dot limit, interactions trigger the emission of a series of subsequent charge density pulses led to the possible explanation of additional effects that relate to a, so far not satisfactorily explained, part of the Hong-Ou-Mandel current noise measurements at the LPA [25,205]. The experimental setup is the solid state realization of the Hong-Ou-Mandel experiment, see the left panel in Figure 14: When two electrons collided at the same time on the QPC from different sources (states 1 and 2), they could not occupy the same state because of Pauli's exclusion principle and their probability to end up in different leads (states 3 and 4) was increased. As a consequence, the current noise was suppressed [132], see the right panels in Figure 14. More generally, <sup>Δ</sup>*q*(*τ*) measures the cross-correlation (or overlap) in time of the two incoming currents at the level of the QPC. If the two incoming currents are identical in each input, one should ge<sup>t</sup> <sup>Δ</sup>*q*(*τ* = 0) = 0 and the rest of the curve will reflect on the time trace of the current. However, because of small asymmetries in the two electronic paths and the two electron sources, the noise suppression is not perfect [205]. In Ref. [25], the current noise Δ*q* as a function of the time delay *τ* with which electrons arrived at the QPC from different sources was measured in more detail for the outer and inner edge of the filling factor *ν* = 2 (central and right panel in Figure 14). The current in the inner edge channel was induced by inter-edge Coulomb interactions and could be computed with a plasmon-scattering formalism [25,206]. In addition to what this plasmon scattering model predicted, unexpected oscillations as a function of *τ* were observed. These could be satisfactorily explained by our prediction [148] of further charge emission triggered by interactions in the electron sources [205]. From independent calibration measurements, the total RC time constant of the source could be measured to set the constrain (*τ*<sup>−</sup><sup>1</sup> f + *τ*<sup>−</sup><sup>1</sup> *c* )−1/2 = *τRC* = 21 ps. Combining Equation (30) and (45) with the plasmon scattering formalism one could compute the current noise in the ouput of the Hong-Ou-Mandel interferometer <sup>Δ</sup>*q*(*τ*) with only one fitting parameter: The ratio *<sup>τ</sup>*f/*<sup>τ</sup>*c. The minimization procedure gave the most-likely result: *τ*f = 136 ps. The comparison is shown on the right panels of Figure 14, where the model for *τ*f = 2*τRC* [*<sup>τ</sup>*c → <sup>∞</sup>, i.e., no interaction within the dot] described the fractionalization process due to interedge Coulomb interactions but not the interactions within the dot itself. This provides a reasonable qualitative and quantitative agreemen<sup>t</sup> with the experimental data reported in Ref. [25]. On the bottom-left panel in Figure 14 we compared, for *τ*f = 136 ps, the charge exiting the dot *Q*(*t*) with the applied square pulse sequence on the top-gate *V*(*t*) which had a finite rise time of 30 ps. In particular, it could explain the appearance of extra rebounds in Δ*q* for time delays *τ* between 70 and 450 ps which was not possible with a non-interacting dot (*<sup>τ</sup>*f = 0). This was directly due to the additional effects coming from the interactions within the dot itself and could not be explained by the fractionalization mechanism. Indeed, relying exclusively on the model describing fractionalization, we could not reproduce the pronounced additional rebound for |*τ*| = 200 ps for *τ*f < 100 ps. This highlights the relevance of Coulomb interactions in the open dot dynamics.

**Figure 14.** Top-Left—Hong-Ou-Mandel experiment from Ref. [25]: Two single electron sources, as that shown in Figure 6, inject single electron towards the same QPC, which works as a beamsplitter. Bottom-left—Simulation based on Ref. [148] of the charge exiting the open dot when applying a square pulse sequence *V*(*t*) with a rise time of 30 ps, one clearly sees the additional pulses coming from interactions. Top-Right—Normalized Hong-Ou-Mandel current noise Δ*q* of the outer edge as a function of the time delay *τ* with which charge arrives on the QPC from different sources. Noise is suppressed for *τ* = 0 because of anti-bunching effects, but additional oscillations were observed for *τ* = 0, which the theory in Ref. [148] contributed to explain. The points are the experimental data while the solid and dashed lines are theoretical curves with different fittings for the time of flight *τ*f for electrons in the cavity. Bottom-Right—Same as in the central panel but for the inner edge.
