**2. Model**

We consider a four-terminal FQH bar in the presence of a QPC (see Figure 1). For a quantum Hall state with filling factor *ν* in the Laughlin sequence *ν* = 1/(<sup>2</sup>*n* + 1) [52], with *n* ∈ N, a single chiral bosonic mode emerges at each edge of the sample [54]. The effective Hamiltonian for right- and left-moving edge states (indicated in the following by *R* and *L* respectively) reads

$$H\_0 = \sum\_{r=R,L} \frac{\upsilon}{4\pi} \int\_{-\infty}^{+\infty} d\mathbf{x} \left[\partial\_\mathbf{x} \Phi\_r(\mathbf{x})\right]^2,\tag{1}$$

with Φ*R*/*L* bosonic operators propagating with velocity *v*, assumed equal for both chiralities, along the edges. Notice that, from now on, we will set ¯*h* = 1 for notational convenience.

The coupling between the electron particle density *ρR*(*x*) = √*ν* 2*π ∂x*Φ*R*(*x*) and the coupling with a generic time dependent voltage gate *V*(*t*) applied to terminal 1 is encoded by the Hamiltonian

$$H\_{\%} = -\varepsilon \int\_{-\infty}^{+\infty} dx \, \Theta(-x - d) V(t) \rho\_{\mathcal{R}}(x). \tag{2}$$

Here, the step function <sup>Θ</sup>(−*<sup>x</sup>* − *d*) models a homogeneous contact which is extended with respect to the Hall sample. In the following, we will focus on a periodic voltage drive of the form

$$V(t) = V\_{dc} + V\_{ac}(t),\tag{3}$$

with

$$\frac{1}{T} \int\_{0}^{T} dt V\_{ac}(t) = 0,\tag{4}$$

T being the period of the drive.

**Figure 1.** Four-terminal setup for an FQH state in the QPC geometry. Contact 1 is driven by a time dependent voltage *V*(*t*) and used as input terminal, contact 4 is grounded, while contacts 2 and 3 are the output terminals where currents and noises are measured. FQH: Fractional Quantum Hall; QPC: Quantum Point Contact.

The time-evolution of the bosonic field Φ*R*, according to the Hamiltonian *H* = *H*0 + *Hg*, is given by [17,19]

$$
\Phi\_R(\mathbf{x}, t) = \phi\_R\left(t - \frac{\mathbf{x}}{v}\right) - e\sqrt{\nu} \int\_{-\infty}^{t - \frac{\mathbf{x}}{v}} dt' V(t'). \tag{5}
$$

Here, *φR* denotes the field which evolves with respect to *H*0 only. These characteristic chiral dynamics are a direct consequence of the linear dispersion of edge modes.

Finally, we allow the tunneling of excitations between the two edges by locally approaching them, creating a QPC at *x* = 0. This process can be effectively described by introducing the tunneling Hamiltonian [55]

$$H\_T = \Lambda \Psi\_R^\dagger(0)\Psi\_L(0) + \text{h.c.}\tag{6}$$

Here, Ψ*R*/*L* (Ψ†*R*/*L*) are the annihilation (creation) operators for quasi-particles with fractional charge *e*∗ = <sup>−</sup>*eν*. The Hamiltonian in Equation (6) describes the dominating tunneling process in the weak-backscattering regime [56]. In this regime, the tunneling Hamiltonian *HT* can be treated as a small perturbation with respect to *H*. As a consequence, the time evolution of quantum operators can be constructed in terms of a perturbative series in the tunneling amplitude Λ.
