**1. Introduction**

The manipulation of local electrostatic potentials and electron Coulomb interactions has been pivotal to control quantized charges in solid state devices. Coulomb blockade [1–3] has revealed to be a formidable tool to trapping and manipulating single electrons in localized regions behaving as highly tunable artificial impurities, so called quantum dots. Beyond a clear practical interest, which make quantum dots promising candidates to become the building block of a quantum processor [4–6], hybrid [7] quantum dot systems also became a formidable platform to address the dynamics of many-body systems in a controlled fashion, and a comprehensive theory, which could establish the role of Coulomb interactions when these systems are strongly driven out of equilibrium, is still under construction.

Beyond theoretical interest, this question is important for ongoing experiments with mesoscopic devices aimed towards the full control of single electrons out of equilibrium. Figure 1 reports some of these experiments [8–19], in addition to the mesoscopic capacitor [20–26], which will be extensively discussed in this review. These experiments and significant others [27–33] have a common working principle: A fast [34] time-dependent voltage drive *<sup>V</sup>*(*t*), applied either on metallic or gating contacts, triggers emission of well defined electronic excitations. Remarkably, these experiments achieved to generate, manipulate, and detect single electrons on top of a complex many-body state such as the Fermi sea. A comprehensive review of these experiments can be found in Ref. [35].

**Figure 1.** Some recent experiments achieving real-time control of single electrons. (**a**) Leviton generation by a Lorentzian voltage pulse in metallic contacts, generating a noiseless wave-packet carrying the electron charge *e* [8–13]. This wave-packet is partitioned on a Quantum Point Contact (QPC), whose transmission *D* is controlled by the split-gate voltage *V*G. (**b**) Single quantum level electron turnstile [18,19]. Two superconductors, biased by a voltage *V*B, are connected by a single-level quantum dot. Inset—Working principle of the device: A gate voltage controls the orbital energy of the quantum dot, which is filled by the left superconductor and emptied in the right one. (**c**) Long-range single-electron transfer via a radio-frequency pulse between two distant quantum dots QD1 and QD2 [14–17]. The electron "surfs" along the moving potential generated by the radio-frequency source and is transferred along a one-dimensional channel from QD1 to QD2. (**d**) The mesoscopic capacitor [20–26], in which a gate-driven quantum dot emits single electrons through a QPC in a two-dimensional electron gas. This platform will be extensively discussed in this review.

In this context, interactions are usually considered detrimental, as they are responsible for inelastic effects leading to diffusion and dephasing [36]. Interaction screening or, alternatively, the disappearance of such inelastic effects at low driving energies or temperatures [37–42] is thus crucial to identify single-electron long-lived excitations (quasi-particles) close to the Fermi surface. The possibility of identifying such excitations, even in the presence of strong Coulomb interactions, is the core of the Fermi liquid theory of electron gases in solids [43,44], usually identified with the ∝ *T*<sup>2</sup> suppression of resistivities in bulk metals. It is the validity of this theory for conventional metals that actually underpins the success of Landauer–Büttiker elastic scattering theory [45–47] to describe coherent transport in mesoscopic devices.

The aim of this review is to show how a similar approach can also be devised to describe transport in mesoscopic conductors involving the interaction of artificial quantum impurities. In these systems, electron-electron interactions are only significant in the confined and local quantum dot regions, and not in the leads for instance, therefore we use the terminology of a Local Fermi Liquid theory (LFL) in contrast to the conventional Fermi liquid approach for bulk interactions. Originally, the first LFL approach [48] was introduced to derive the low energy thermodynamic and transport properties of Kondo local scatterers in materials doped with magnetic impurities [49]. In this review, we will show how LFLs provide the unifying framework to describe both elastic scattering and strong correlation phenomena in the out-of-equilibrium dynamics of mesoscopic devices. This approach makes also clear how inelastic effects, induced by Coulomb interactions, become visible and unavoidable as soon as

such systems are strongly driven out of equilibrium. We will discuss how extensions of LFLs and related approaches describe such regimes as well.

As a paradigmatic example, we will focus on recent experiments showing the electron transfer with Coulomb interactions [50], (see Figure 2), and, in more detail, on the mesoscopic capacitor [20–26], (see Figure 6). The mesoscopic capacitor does not support the DC transport, and it makes possible the direct investigation and control of the coherent dynamics of charge carriers. The LFL description of such devices entails the seminal results relying on self-consistent elastic scattering approaches by Büttiker and collaborators [51–56], but it also allows one to describe effects induced by strong Coulomb correlations, which remain nevertheless elastic and coherent. The intuition provided by the LFL approach is a powerful lens through which it is possible to explore various out-of-equilibrium phenomena, which are coherent in nature but are governed by Coulomb interactions. As an example, we will show how a bold treatment of Coulomb interaction unveils originally overlooked strong dynamical effects, triggered by interactions, in past experimental measurements showing fractionalization effects in out-of-equilibrium charge emission from a driven mesoscopic capacitor [25].

This review is structured as follows. In Section 2, we give a simple example showing how Coulomb interactions trigger phase-coherent electron state transfer in experiments as those reported in Ref. [50], Figure 2. Section 3 discusses how the effective LFL approach [57–64] provides the unified framework describing such coherent phenomena. In Section 4, we consider the study of the low-energy dynamics of the mesoscopic capacitor, in which the LFL approach has been fruitfully applied [65–69], showing novel quantum coherent effects. Section 5 extends the LFL approach out of equilibrium and describes signatures of interactions in measurements of strongly driven mesoscopic capacitors [25].

### **2. Phase-Coherence in Quantum Devices with Local Interactions**

To illustrate the restoration of phase coherence at low temperatures in the presence of interactions, we consider two counter-propagating edge states entering a metallic quantum dot, or cavity. Such a system was recently realized as a constitutive element of the Mach–Zender interferometer of Ref. [50], reported in Figure 2. In that experiment, the observation of fully preserved Mach–Zehnder oscillations, in a system in which a quantum Hall edge state penetrates a metallic floating island demonstrates an interaction-induced, restored phase coherence [70,71].

The dominant electron-electron interactions in the cavity have the form of a charging energy [1–3]

$$\mathcal{H}\_{\mathbb{C}} = E\_{\mathbb{C}}[N - \mathcal{N}\_{\mathbb{K}}(t)]^2,\tag{1}$$

in which *N* is the number of electrons in the island, *C*g is the geometric capacitance, and Ng = *<sup>C</sup>*g*V*g(*t*)/*<sup>e</sup>* is the dimensionless gate voltage, which corresponds to the number of charges that would set in the cavity if *N* was a classical, non-quantized, quantity. We also define the charging energy *E*c = *<sup>e</sup>*2/2*C*g: The energy cost required the addition of one electron in the isolated cavity. For the present discussion, we neglect the time-dependence of the gate-potential *<sup>V</sup>*g, which will be reintroduced to describe driven settings. In the linear-dispersion approximation, the right/left-moving fermions <sup>Ψ</sup>R,L in Figure 2, moving with Fermi velocity *vF*, are described by the Hamiltonian:

$$\mathcal{H}\_{\text{kin}} = \upsilon\_F \hbar \sum\_{a=\text{R/L}} \int\_{-\infty}^{\infty} dx \, \Psi\_a^\dagger(\mathbf{x}) (-ia\partial\_x) \Psi\_a(\mathbf{x}) \,, \tag{2}$$

with the sign *α* = +/− multiplying the *∂x* operator for right- and left-movers respectively. The floating island occupies the semi-infinite one-dimensional space located at *x* > 0 with the corresponding charge *N* = ∑*α* ∞ 0 *dx* Ψ† *α*(*x*)<sup>Ψ</sup>*α*(*x*). It is important to stress that the model (1)–(2) is general and effective in describing different quantum dot devices. It was originally suggested by Matveev to describe quantum dots connected to leads through a single conduction channel [72] and it equally describes the mesoscopic capacitor, see Sections 4 and 5.

**Figure 2.** Left—Mach–Zehnder interferometer with a floating metallic island (colored in yellow) [50]. The green lines denote chiral quantum Hall edge states, which can enter the floating island passing through a gate-tunable QPC (in blue). An additional QPC separates the floating island from an additional reservoir on its right. Center—The floating island is described by two infinite counter-propagating edges, exchanging electrons coherently thanks to the charging energy *E*c of the island (red arrow). Right—Mach–Zehnder visibility of the device as a function of magnetic field *B*. Oscillation of this quantity as function of *B* signals quantum coherent interference between two paths encircling an Aharonov–Bohm flux. In the situation sketched in box A, the first QPC is closed and the interferometer is disconnected from the island and visibility oscillations are observed, as expected (red line). Remarkably, the oscillations persist (black line) in the situation sketched in box B, where the leftmost QPC is open and one edge channel enters the floating island. The visibility oscillations are only suppressed in the situation sketched in box C, where the rightmost QPC is also open and the island is connected to a further reservoir (blue line).

The model (1)–(2) characterizes an open-dot limit in the sense that it does not contain an explicit backscattering term coupling the *L* and *R* channels. It can be solved exactly by relying on the bosonization formalism [73–76], which, in this specific case, maps interacting fermions onto non-interacting bosons [72,77,78]. Using this mapping, one can show that the charging energy *E*c perfectly converts, far from the contact, right-movers into left-movers. This fact is made apparent by the "reflection" Green function *G*LR [77]:

$$G\_{\rm LR} = \left\langle \mathcal{T}\_{\rm \tau} \Psi\_{\rm L}^{\dagger}(\mathbf{x}, \tau) \Psi\_{\rm R}(\mathbf{x}', 0) \right\rangle \simeq e^{-i2\pi N\_{\rm g}} \frac{T/2\upsilon\_{\rm F}}{\sin\left[\frac{\pi T}{\hbar} \left(\pi + i(\mathbf{x} + \mathbf{x}') - i\frac{\pi \hbar}{\mathbb{E}\_{\rm c} \mathbf{c}^{\dagger}}\right)\right]},\tag{3}$$

which we consider at finite temperature *T*. In Equation (3), T*τ* is the usual time-ordering operator defined as T*τ<sup>A</sup>*(*τ*)*B*(*τ*) = *<sup>θ</sup>*(*τ* − *τ*)*A*(*τ*)*B*(*τ*) ± *<sup>θ</sup>*(*τ* − *<sup>τ</sup>*)*B*(*τ*)*A*(*τ*), in which the sign +/− is chosen depending on the bosonic/fermionic statistics of the operators *A* and *B* [79] and *<sup>θ</sup>*(*τ*) is the Heaviside step function. As first noted by Aleiner and Glazman [77], the form of *G*LR at large (imaginary) time *τ* corresponds to the elastic reflection of the electrons incident on the dot, with a well-defined scattering phase *<sup>π</sup>*N*<sup>g</sup>*. The correlation function (3) would be identical if the interacting dot was replaced with a non-interacting wire of length *vFπh*¯ /*E*c*γ* (with ln *γ* = C 0.5772 being Euler's constant), imprinting a phase *<sup>π</sup>*N*g* when electrons are back-reflected at the end of the wire [80].

The physical picture behind Equation (3) is that an electron entering and thereby charging the island violates energy conservation at low temperature and must escape on a time scale *h*¯ /*Ec* fixed by the uncertainty principle. The release of this incoming electron can happen either elastically, in which case the electron keeps its energy, or inelastically via the excitation of electron-hole pairs. As we discuss in Section 3.1, inelastic processes are suppressed by the phase space factor (*ε*/*Ec*)2, *ε* being the energy of the incoming electron, and they die out at low energy or large distance (time), reestablishing purely elastic scattering despite a nominally strong interaction.

Equation (3) is thus a remarkable example of how interactions trigger coherent effects in mesoscopic devices. It has been derived here for an open dot, a specific limit in which the charge quantization of the island is fully suppressed. However, the restoration of phase coherence at low energy is more general and applies for an arbitrary lead-island transmission, in particular in the tunneling limit where the charge states of the quantum dot are well quantized [1–3]. This quantization is known to induce Coulomb blockade in the conductance of the device, see Figure 3. Nevertheless, a Coulomb blockaded dot acts at low energy as an elastic scatterer imprinting a phase *δ* [81,82] related to its average occupation *N* via the Friedel sum rule, see Section 3.2. For weak transmissions, *N* strongly deviates from the classical value Ng. These features constitute the main characteristics of the local Fermi liquid picture detailed in the forthcoming sections.

**Figure 3.** Coulomb blockade and emergen<sup>t</sup> LFL behavior. When the typical energy of the system (temperature, bias-voltage, . . . ) is smaller than the charging energy *E*c, charge quantization *Q* = *eN* in the dot suppresses the conductance *G* of the system. Degeneracy between different charge occupations lead to conductance peaks, which become larger the stronger the tunnel exchange of electrons with the leads. Conductance peaks and charge quantization disappear in the open-dot limit. For any tunneling strength, the dot behaves as an elastic scatterer described by the LFL theory (8), with potential scattering of strength *W*, inducing a phase-shift *δW* on lead electrons set by the dot occupation *N*.

### **3. What Are Local Fermi Liquids and Why Are They Important to Understand Quantum-Dot Devices?**

In this Section, we introduce the local Fermi liquid theory and discuss its application to quantum transport devices. The general system considered in this paper is a central interacting region, such as a quantum dot, connected to leads described as non-interacting electronic reservoirs. The Hamiltonian takes the general form:

$$
\mathcal{H} = \mathcal{H}\_{\text{res}} + \mathcal{H}\_{\text{res}-\text{d}\text{ct}} + \mathcal{H}\_{\text{d}\text{ct}} + \mathcal{H}\_{\text{c}}.\tag{4}
$$

The first term describes the lead reservoir, which could be either a normal metal [14], a chiral edge state in the quantum Hall regime [22,29], or a superconductor [18]. In the case of a normal metal, it is given by:

Hres = ∑ *k <sup>ε</sup>kc*†*k ck* , (5)

in which *ck* annihilates a fermion in the eigenstate state *k* of energy *εk* in the reservoir. For instance, in Figure 2, the reservoir modes correspond to the *x* < 0 components of the operators ΨR/L. The field <sup>Ψ</sup>res(*x*) = *<sup>θ</sup>*(−*<sup>x</sup>*)<sup>Ψ</sup>R(*x*) + *<sup>θ</sup>*(*x*)<sup>Ψ</sup>L(−*<sup>x</sup>*), with *<sup>θ</sup>*(*x*) the Heaviside step function, unfolds the chiral field onto the interval *x* ∈ [−∞, ∞] and its Fourier transform *ck* = ∞−∞ *dxe*−*ikx*Ψres(*x*) recovers Equation (5) from Equation (2), with *εk* = *hv*¯ *<sup>F</sup>k*.

The single particle physics of the quantum dot is described instead by:

$$\mathcal{H}\_{\text{dot}} = \sum\_{l} (\varepsilon\_{d} + \varepsilon\_{l}) n\_{l} \tag{6}$$

in which *nl* = *d*†*l dl* counts the occupation of the orbital level *l* and *dl* annihilates fermions in that state. The spectrum can be either discrete for a finite size quantum dot or dense for a metallic dot as in the case of Figure 2. We also introduced the orbital energy *εd* as a reference. Hres−dot describes the exchange of electrons between dot and reservoir. It has generally the form of a tunneling Hamiltonian:

$$\mathcal{H}\_{\text{res}-\text{dct}} = t \sum\_{k,l} \left[ c\_k^\dagger d\_l + d\_l^\dagger c\_k \right] \,\prime \tag{7}$$

in which we neglect, for simplicity, any *k* dependence of the tunneling amplitude *t*. The charging energy Hc is given in Equation (1) with the dot occupation operator *N* = ∑*l nl*.

Without any approximation, deriving the out-of-equilibrium dynamics of interacting models such as Equation (4) is a formidable task. The presence of local interactions leads to inelastic scattering events, creating particle-hole pairs when electron scatter on the dot (see Figure 4). From a technical point of view, such processes are difficult to handle and, even if these difficulties are overcome, one has to identify the dominant physical mechanisms governing the charge dynamics. In our discussion, interactions are usually controlled by the charging energy *E*c, which cannot be treated perturbatively in Coulomb blockade regimes. The possibility to rely on Wick's theorem [83], when performing perturbative calculations in the exchange term Hres−dot, is also denied. Thus, one has to look for a more efficient theoretical approach.

**Figure 4.** Difference between elastic (**left**) and inelastic (**right**) events for electrons scattering on a quantum dot. In the elastic case, electrons do not change energy *ε*. The wave function is preserved and the only residual effect of scattering is a phase-shift *δε*. In the inelastic case, many-body interactions trigger the creation of particle-hole pairs. Outgoing electrons are then emitted in a state <sup>Φ</sup>*ε* of energy *ε* different from the initial *ε* and phase coherence is gradually lost.
