**3. Charge Current**

Charge current operators for right- and left-moving modes can be defined by means of the continuity equations of densities *ρR*/*L*(*<sup>x</sup>*, *t*), namely

$$-\varepsilon \partial\_t \rho\_{R/L}(\mathbf{x}, t) + \partial\_\mathbf{x} I\_{R/L}(\mathbf{x}, t) = 0. \tag{7}$$

Due to the fact that the propagation of the modes long the channel is chiral [19], one finds

$$J\_{\mathbb{R}/L}(\mathbf{x}, \mathbf{t}) = \mp \varepsilon v \varphi\_{\mathbb{R}/L}(\mathbf{x} - v\mathbf{t}),\tag{8}$$

where *ρR*/*L* are the chiral density operators evolving in time according to the total Hamiltonian *H* = *H*0 + *Hg* + *HT*.

Starting from the definition of the chiral current operator, we can define the operators for charge current entering reservoirs 2 and 3 as

$$I\_{2/3}(t) = I\_{\mathbb{R}/L}(\pm d, t),\tag{9}$$

where we recall that the interfaces between edge states and contacts 2 and 3 are placed at *x* = ±*d*, respectively (see Figure 1). The expansion of *J*2/3 in powers of Λ is given by

$$J\_{2/3}(t) = J\_{2/3}^{(0)}(t) + J\_{2/3}^{(1)}(t) + J\_{2/3}^{(2)}(t) + \mathcal{O}(\Lambda^3),\tag{10}$$

with

$$J\_{2/3}^{(0)}(t) = \frac{\varepsilon v \sqrt{\nu}}{2\pi t} \left(\partial\_{\mathbf{x}} \Phi\_{\mathbb{R}/L}(\mathbf{x}, t)\right)\_{\mathbf{x} = \pm d\_{\mathbf{x}}}.\tag{11}$$

$$J\_{2/3}^{(1)}(t) = \pm i \Lambda e \nu \Psi\_R^{\dagger} \left( 0, t - \frac{d}{\upsilon} \right) \Psi\_L \left( 0, t - \frac{d}{\upsilon} \right) + \text{h.c.},\tag{12}$$

$$J\_{2/3}^{(2)}(t) = \pm i \int\_{-\infty}^{t-\frac{d}{v}} dt'' \left[ H\_T(t'')\_\prime + i\Lambda ev \Psi\_R^\dagger \left(0, t - \frac{d}{v}\right) \Psi\_L \left(0, t - \frac{d}{v}\right) + \text{h.c.} \right]. \tag{13}$$

Tunneling contributions entering reservoirs 2 and 3 are connected by the simple relation

$$J\_2^{(1/2)}(t) = -J\_3^{(1/2)}(t). \tag{14}$$

The thermal average of current operators will be performed over the initial equilibrium condition, i.e., in the absence of driving voltages (*Hg*) and tunneling (*HT*). It is worth noting that *J*(1) 2/3(*<sup>x</sup>*, *t*) is zero because it involves a different number of annihilation or creation field operators. Therefore, the average values of charge current operators satisfy

$$
\langle \!/ \!/ \_{2/3}(t) \rangle = \langle \!/ \_{2/3}^{(0)}(t) \rangle + \langle \!/ \_{2/3}^{(2)}(t) \rangle + \mathcal{O}(\Lambda^3),
\tag{15}
$$

with

$$
\langle \langle f\_2^{(0)}(t) \rangle \rangle = \frac{e^2 \nu}{2\pi} V \left( t - \frac{2d}{v} \right),
\tag{16}
$$

$$
\langle J\_3^{(0)}(t) \rangle = 0,\tag{17}
$$

$$
\langle \langle l\_{2/3}^{(2)}(t) \rangle \rangle = \pm i \epsilon \nu \upsilon \int\_{-\infty}^{t-\frac{d}{\overline{\upsilon}}} dt'' \langle \left[ H\_T(t'')\_{\prime} + i \Lambda \Psi\_R^{\dagger}(d - \upsilon t, 0) \Psi\_L(d - \upsilon t, 0) + \text{h.c.} \right] \rangle. \tag{18}
$$

The first term corresponds to the charge current emitted by the reservoir 1, which constitutes the main contribution to the detected currents in reservoir 2. In the absence of tunneling processes (Λ = 0), these zero-order contributions would correspond to a periodic current generated by *<sup>V</sup>*(*t*). For this reason, the integral over one period T gives the total charge C transferred across the edge channel, namely

$$\mathcal{C} = \int\_{-\frac{T}{2}}^{\frac{T}{2}} dt \langle l\_2^{(0)}(t) \rangle = \frac{e^2 \nu}{2\pi} \int\_{-\frac{T}{2}}^{\frac{T}{2}} dt V \left( t - \frac{2d}{\upsilon} \right) = \frac{e^2 \nu}{\omega} V\_{d\varepsilon} = -\varepsilon q\_{\prime} \tag{19}$$

where we have introduced the drive frequency *ω* = 2*π*/T of the voltage in Equation (3) and used the property in Equation (4).

Due to the QPC, some of the excitations emitted by contact 1 are backscattered and *J*(2) 2 (*t*) (*J*(2) 3 (*t*)) represents the transmitted (reflected) current [55,57]. Due to the relation in Equation (14), backscattering currents are equal up to a sign, so that we can define

$$J\_{BS}(t) = \langle J\_3^{(2)}(t) \rangle = -\langle J\_2^{(2)}(t) \rangle. \tag{20}$$

According to Equation (18), this backscattering current can be written as

$$\mathcal{G}\_{BS}(t) = -\varepsilon \left| \Lambda \right|^2 \nu \int\_{-\infty}^{t-\frac{d}{\sigma}} dt'' \left[ G\_{\overline{K}}^{-\zeta} \left( t'', t - \frac{d}{\overline{\upsilon}} \right) G\_{\overline{L}}^{\geq} \left( t'', t - \frac{d}{\overline{\upsilon}} \right) - G\_{\overline{L}}^{<} \left( t'', t - \frac{d}{\overline{\upsilon}} \right) G\_{\overline{K}}^{>} \left( t'', t - \frac{d}{\overline{\upsilon}} \right) + G\_{\overline{L}}^{>} \left( t - \frac{d}{\overline{\upsilon}}, t'' \right) \right]$$

$$-G\_{\overline{K}}^{<} \left( t - \frac{d}{\overline{\upsilon}}, t'' \right) G\_{\overline{L}}^{>} \left( t - \frac{d}{\overline{\upsilon}}, t'' \right) + G\_{\overline{L}}^{<} \left( t - \frac{d}{\overline{\upsilon}}, t'' \right) G\_{\overline{K}}^{>} \left( t - \frac{d}{\overline{\upsilon}}, t'' \right) \Big|\_{} \tag{21}$$

where we introduced the quasi-particle Green's functions

$$\mathcal{G}\_{\mathbb{R}}^{<}\left(t',t\right) = \left<\Psi\_{\mathbb{R}}^{\ \dagger}(0,t')\Psi\_{\mathbb{R}}(0,t)\right> = e^{-i\imath e \int\_{t}^{t'} d\tau V(\tau)} \langle\psi\_{\mathbb{R}}^{\ \dagger}(0,t')\psi\_{\mathbb{R}}(0,t)\rangle,\tag{22}$$

$$G\_L^{<}\left(t',t\right) = \langle \Psi\_L^{\ \dagger}(0,t')\Psi\_L(0,t)\rangle = \langle \psi\_L^{\ \dagger}(0,t')\psi\_L(0,t)\rangle,\tag{23}$$

with *ψR*/*L* the quasi-particle field operators evolving with respect to *H*0 only. Analogous expressions can be derived for *<sup>G</sup>*<sup>&</sup>gt;*R*and *<sup>G</sup>*<sup>&</sup>gt;*L*.

In terms of the bosonized picture [36,58–60], the quasi-particle field operators can be written as coherent states of the bosonic edge modes, namely

$$\psi\_R(\mathbf{x},t) = \frac{\mathcal{F}\_R}{\sqrt{2\pi a}} e^{i\mathbf{k}\_F(\mathbf{x}-\upsilon t)} e^{-i\sqrt{\upsilon}\Phi\_R(\mathbf{x},t)}\,,\tag{24}$$

$$\psi\_L(\mathbf{x},t) = \frac{\mathcal{F}\_L}{\sqrt{2\pi a}} e^{i k\_{\mathbb{F}}(\mathbf{x}+vt)} e^{-i\sqrt{v}\phi\_L(\mathbf{x},t)}\,,\tag{25}$$

with F*R*/*L* Klein factors [59]. Therefore, the expressions for the Green's functions in Equations (22) and (23) become

$$\mathcal{G}\_{\mathbb{R}}^{<}\left(t',t\right) = \left<\Psi\_{\mathbb{R}}^{\dagger}\left(0,t'\right)\Psi\_{\mathbb{R}}\left(0,t\right)\right> = e^{-i\nu\varepsilon\int\_{t}^{t'}d\tau V(\tau)}\frac{e^{ik\cdot\nu\left(t'-t\right)}}{2\pi a}\mathcal{P}\_{\mathbb{V}}\left(t-t'\right),\tag{26}$$

$$G\_L^{<}\left(t',t\right) = \left<\Psi\_L^{\ \dagger}(0,t')\Psi\_L(0,t)\right> = \frac{e^{-ik\_F v(t'-t)}}{2\pi a}\mathcal{P}\_v(t-t'),\tag{27}$$

where we introduced the function

$$\mathcal{P}\_{\nu}(\tau) = \mathcal{e}^{\nu \mathcal{W}(\tau)},\tag{28}$$

with

$$\begin{split} \mathcal{W}(t) &= \left\langle \phi\_{R/L}(0,t)\phi\_{R/L}(0,0) - \phi\_{R/L}^2(0,0) \right\rangle = \\ &= \ln \left[ \frac{\left| \Gamma \left( 1 + \frac{\theta}{\omega\_{\mathbb{C}}} + i\theta t \right) \right|^2}{\left| \Gamma \left( 1 + \frac{\theta}{\omega\_{\mathbb{C}}} \right) \right|^2 \left( 1 + i\omega\_{\mathbb{C}}t \right)} \right] \simeq \ln \left[ \frac{\pi \theta t}{\sinh \left( \pi \theta t \right) \left( 1 + i\omega\_{\mathbb{C}}t \right)} \right]. \tag{29} \end{split} \tag{29}$$

In the above, Equation *θ* is the temperature (*kB* = 1), *ωc* = *v*/*a* is the high energy cut-off, with *a* the finite length cut-off appearing in Equation (25), and <sup>Γ</sup>(*x*) is the Euler's gamma function. The considered approximation is valid as long as *ωc θ*. Let us notice that the expressions are equal for right- and left-movers and, therefore, we have omitted any label *R* or *L*. Moreover, we explicitly used the fact that *W*(*t*) is translationally invariant in time.

By inserting these expressions into Equation (21), one finds

$$J\_{BS}(t) = 2i\nu e \left| \lambda \right|^2 \int\_0^{+\infty} d\tau \sin \left[ \nu e \int\_{t-\tau}^t dt' V(t') \right] \left( \mathcal{P}\_{2\upsilon}(\tau) - \mathcal{P}\_{2\upsilon}(-\tau) \right), \tag{30}$$

where we introduced the rescaled tunneling amplitude *λ* = <sup>Λ</sup>/(<sup>2</sup>*πa*). Notice that the backscattering current satisfies *JBS*(*t*) = *JBS*(*t* + T ) as expected due to the periodicity of the voltage drive *<sup>V</sup>*(*t*). It is thus possible to perform an average over one period of the backscattering current thus finding

$$\overline{f\_{BS}(t)} = 2i\nu e \left| \lambda \right|^2 \int\_0^T \frac{dt}{T} \int d\tau \sin \left[ \nu e \int\_{t-\tau}^t dt' V(t') \right] \mathcal{P}\_{2\nu}(\tau). \tag{31}$$

The presence of a periodic voltage can be conveniently handled by resorting to the photo-assisted coefficients defined by [16,61]

$$p\_l(a) = \int\_{-\mathcal{T}/2}^{\mathcal{T}/2} \frac{dt}{\mathcal{T}} e^{2i\pi a \frac{t}{\mathcal{T}}} e^{-2i\pi a \varrho(t)},\tag{32}$$

with

$$
\varphi(t) = \int\_{-\infty}^{t} \frac{dt'}{\mathcal{T}} \mathcal{V}\_{\rm ac}(t'),
\tag{33}
$$

where *V* ¯ *ac*(*t*) is the AC part of *V*(*t*) with unitary and dimensionless amplitude. Therefore, the backscattering current assumes the final expression

$$\overline{f\_{BS}(t)} = 2i\nu e \left|\lambda\right|^2 \sum\_{l} \left|p\_l(a)\right|^2 \int\_{-\infty}^{+\infty} d\tau \sin\left[\left(q+l\right)\omega\tau\right] \mathcal{P}\_{2\nu}(\tau). \tag{34}$$

By moving to Fourier space and by introducing the function [58,62–65]

$$\tilde{\mathcal{P}}\_{\nu}(E) = \int\_{-\infty}^{+\infty} dt e^{iEt} \mathcal{P}\_{\nu}(t) = \left(\frac{2\pi\theta}{\omega\_{c}}\right)^{\nu - 1} \frac{e^{E/2\theta}}{\Gamma(\nu)\omega\_{c}} \left| \Gamma\left(\frac{\nu}{2} - i\frac{E}{2\pi\theta}\right) \right|^{2},\tag{35}$$

the current in Equation (34) can be recast as

$$\overline{f\_{BS}(t)} = \nu \varepsilon |\lambda|^2 \sum\_{l} |p\_l(a)|^2 \left\{ \vec{\mathcal{P}}\_{2\nu}[(q+l)\omega] - \vec{\mathcal{P}}\_{2\nu}[-(q+l)\omega] \right\}.\tag{36}$$

This final expression explicitly depends on the rescaled amplitudes of both the AC (*α*) and DC (*q*) contribution to the voltage. Even if frequently assumed equal, these two parameters can be tuned independently [16,53].
