Spectral properties of interacting helical channels driven by Lorentzian pulses

Precise shaping of coherent electron sources allows the controlled creation of wavepackets into a one dimensional (1D) quantum conductor. Periodic train of Lorentzian pulses have been shown to induce minimal excitations without creating additional electron-hole pairs in a single non-interacting 1D electron channel. The presence of electron-electron (e-e) interactions dramatically affects the non-equilibrium dynamics of a 1D system. Here, we consider the intrinsic spectral properties of a helical liquid, with a pair of counterpropagating interacting channels, in presence of time-dependent Lorentzian voltage pulses. We show that peculiar asymmetries in the behavior of the spectral function are induced by interactions, depending on the sign of the injected charges. Moreover, we discuss the robustness of the concept of minimal excitations in presence of interactions, where the link with excess noise is no more straightforward. Finally, we propose a scattering tunneling microscope setup to spectroscopically access and probe the non-equilibrium behavior induced by the voltage drive and e-e interactions. This allows a diagnosis of fractional charges in a correlated quantum spin Hall liquid in presence of time-dependent drives.

Time-dependent voltage pulses [21,49] can be exploited to inject single or multiple electrons in a quantum conductor [50][51][52][53][54][55][56][57][58][59][60][61][62]. However, the underlying presence of a Fermi sea may result in the excitation of electron-hole pairs, which will produce noisy and unwanted signals [21,49]. Levitov and coworkers have predicted that by properly engineering the shape of the voltage pulse, it is possible to inject "clean" electron wavepackets, whose real-time profile is a Lorentzian pulse, without creating any unwanted electron-hole pairs in a non-interacting single-channel quantum conductor [63][64][65]. Indeed, it has been measured that these voltage pulses, called Levitons, produce zero excess noise, equivalent to the absence of extra electron-hole pairs, in integer quantum Hall devices [21,49].
In this paper, we will consider a pair of counterpropagating helical channels, subject to a time-dependent drive, where a richer phenomenology arises compared to more conventional single channel and spinless conductors. This scenario naturally raises the question on: how electron-electron interactions affect the dynamics of injected wavepackets and Levitons in helical channels? Are the concepts of clean signal and minimal excitations still valid in presence of repulsive interactions and counterpropagating channels? In order to answer these questions we will analyze the intrinsic spectral properties of an interacting helical liquid in the presence of a timedependent drive. We will focus on the injection of integer Levitons, which in a noninteracting system are the cleanest signals, to better enlighten the effect induced by interactions. We will demonstrate that, in this non-equilibrium situation, peculiar charge asymmetries are visible in the spectral function, but those can be masked in a conventional excess noise experiment. Indeed, these two quantities are related but, as we will discuss, they are totally equivalent solely in the absence of interactions.
Spectral properties and charge fractionalization phenomena of 1D systems can be inspected via scanning tunneling spectroscopy. In particular, fractionalization of charge in Luttinger liquid wires has been highlighted, showing that for an infinite nanotube, fractional charges can be identified through the measurement of both the autocorrelation noise and the cross correlation noise, measured at the extremities of the nanotube [66][67][68][69][70][71]. Experimental results of tunneling spectroscopy of topological insulators [72][73][74] have been recently reported. In this work, we will consider a setup involving a polarized tip, whose advantage consists in the possibility to separately probe all dynamical spectral properties of counterpropagating channels.
The paper is organized as follows: after a pedagogical overview of basic concepts in Section 2, we introduce model and general settings in Section 3. Section 4 is dedicated to analyze the out-of-equilibrium spectral functions in the presence of a time-dependent drive. Finally, in Section 5 we will propose a possible setup to probe non-equilibrium spectral properties.

Levitons as minimal excitations
For sake of clarity, it is useful to briefly review and introduce some concepts and definitions regarding time-dependent voltage pulses. Despite the fact that the interest in Lorentzian-shaped pulses is quite recent and has been triggered by the experimental implementations in EQO [21], the peculiar properties of this kind of drive were theoretically investigated several decades ago [63,64]. In particular, these authors provided the answer to the question whether it is possible to generate in a non-interacting 1D quantum conductor a minimal excitation by applying a specific voltage drive V (t).
Here, minimality means that the generation of electron-or of hole-like excitations by the drive produces no additional neutral quasiparticle, i.e. electron-hole pairs. In other words, the number of extra holes or electrons generated by the drive must vanish. Levitov's result [63,64] states that a minimal Lorentzian pulse carrying an integer particle number q = (−e/h) dtV (t) (or a superposition of such pulses with integer charges of the same sign) produces minimal excitations. In particular, we will focus on a periodic train (from here on we set = 1) with T the period, w the width of each pulse and q an integer number representing the charge per period (in units of −e) of the drive.
Here, we consider a 1D non-interacting system with linear dispersion = v F k, v F being the Fermi velocity. Let us focus in this Section on the case of a minimal electronic excitation by choosing q > 0. We also consider the zero temperature case, in order to get rid of possible thermally-excited electron-hole pairs. The requirement that no extra holes are generated can be written as N h = k<κ F Ĉ kĈ † k = 0, where k F is the Fermi momentum andĈ k is the annihilation operator for an electron with momentum k. Equivalently, we could say that the variation in the electron occupation number (or electron momentum distribution) ∆n(k) = Ĉ † kĈ k − ĉ † kĉ k must be positive for every k. Here,ĉ k is the annihilation operator in the equilibrium situation, when no drive is present. Thus, by variation (indicated with the notation ∆) we mean that we consider just the effects induced by the drive, by properly subtracting the equilibrium contribution [15,17,51,[55][56][57]59]. Furthermore, since the system is non-interacting, an equivalent information is provided by the variation of the local spectral function, ∆A < (ω). The lesser local spectral function is defined as where the corresponding Green function is withΨ(x, t) denoting the time evolution of the fermionic field in the Heisenberg picture. The variation of the spectral function is ∆A < (ω; x) the equilibrium spectral function, defined as in (2) by replacing the Green function with the equilibrium one G < 0 (t 1 , t 2 ; x) = i ψ † (x, t 2 )ψ(x, t 1 ) . Here,ψ(x, t) is the time evolution of the fermionic field at equilibrium, with no applied drive.
In a 1D free system, the dispersion relation directly connects ∆n(k) and ∆A < (ω; x), which therefore provide equivalent information, so that a minimal excitation is associated with a positive-definite excess lesser spectral function ∆A < (ω; x). In addition, from Equation (2), it is possible to show the following sum rule: In summary, while the integral of ∆A < (ω; x) is always the same, only a positive-definite ∆A < (ω; x) is a signature of a minimal excitation and regions where this spectral function is negative are related to unwanted particle-hole pairs. drive barrier current fluctuations Figure 1. Excitations generated in a 1D non-interacting ballistic conductor by a periodic drive V (t) are partitioned by a tunnel barrier (for instance a quantum point contact). Current fluctuations due to this tunneling process are measured after the barrier and can be directly linked to the concept of "minimal excitation".
From an experimental point of view, signatures of this minimality can be found by considering the zero-frequency noise S produced when the generated excitations are partitioned by a tunnel barrier (typically a QPC [21,36]), as sketched in Figure 1. It is defined as where δÎ(t) =Î(t) − Î (t) denotes fluctuations with respect to the average current. Indeed, it was shown [49,65] that the number of extra holes excited by the drive is proportional to the excess noise at zero temperature Therefore, the excess noise vanishes if the drive V (t) generates minimal electron excitations. This feature was experimentally observed [21]. As far as interacting systems are concerned, the situation becomes more complicated. First of all, ∆n(k) and ∆A < (ω; x) are no more related to each other; still, the sum rule in Equation (4) has to be satisfied. Therefore, it is natural to refer to an excitation as minimal if its spectral function has a definite sign, as it was the case for a non-interacting system. As we will discuss, this notion of minimality is in general unrelated to a vanishing excess noise, which is instead due to a different property of the spectral function. To fully characterize the behavior of Levitons in an interacting system we therefore will focus on its intrinsic spectral properties out of equilibrium.

Model and general setting
We consider a pair of interacting helical channels, capacitively coupled to an external voltage source. The Hamiltonian isĤ =Ĥ HLL +Ĥ g . The Hamiltonian of the helical channels readsĤ HLL =Ĥ 0 +Ĥ int , wherê describes a pair of free counterpropagating channels, with right-(Ψ R ) and left-(Ψ L ) moving electrons having spin up and down, respectively, due to the so-called spinmomentum locking [13,44]. The interacting partĤ int accounts for short range Coulomb interactions [75]: with g 2 and g 4 the inter-and intra-channel coupling constants, respectively, and n r = :Ψ † rΨ r : the particle density operators. It is useful to introduce bosonic operatorŝ Φ r (x) via the bosonization identity [76,77] withF r a Klein factor ensuring the proper fermionic anticommutation rules and a a short length cutoff. The HamiltonianĤ HLL becomeŝ where u = (2π) −1 (2πv F + g 4 ) 2 + g 2 2 represents the renormalized velocity andΦ η are new chiral bosonic fields, related toΦ r via the relation Here, is the Luttinger parameter describing repulsive interactions: K = 1 corresponds to the non-interacting case, whereΦ ± =Φ R/L . The presence helical channels are capacitively coupled to an external gatê where U (x, t) = F (x)V (t) encodes the spatial and temporal profile of the external voltage. Hereafter we consider an extended gate, by choosing F (x) = Θ(−x) (with Θ(x) the Heaviside step function), so that the drive V (t) is applied in the region (−∞, 0). In the following we will consider a periodic train of pulses with period T , as in Equation (1), specifying only later the precise form of the pulse V (0) (t). Recall that the charge per period of the pulse (in units of −e) is The equations of motion of fieldsΦ η , obtained from the full HamiltonianĤ are whose solution can be written aŝ withφ ± (x∓ut, 0) the chiral evolution of bosonic fields without external drive. The time evolution of fermion operatorsΨ r (x, t) is thus obtained by using (11) and (9). Finally, a generic expectation value of an operatorÔ(t), is obtained as Ô (t) = Tr[ Ô (t)]. Here, is the time-independent equilibrium density matrix at t = −∞ when no voltage is applied and thus originating only from the HamiltonianĤ HLL .

Excess particle density
In order to understand how excitations are generated as a result of the drive, a first quantity to look at is the average particle density n(x, t) = n R (x, t) +n L (x, t) . In particular, we are interested in the deviations from the equilibrium situation withn 0 denoting the particle density operator in the absence of the drive. It is useful to introduce the chiral particle density operatorsn η (x, t) = −η K/(2π)∂ xΦη (x, t), since the total particle density can be written asn( This shows that the effect of the drive is to induce excitations propagating both to the right (η = +) and to the left (η = −). Notice that this expression is independent of the temperature and that the chiral right-(left-) moving excitation contributes only at x > 0 (x < 0), see Figure 2. Moreover, from the relation we see that excitations for each chirality η are composed of contributions coming from both channels r = R, L. By combining (18) and (19), the spatial profile of the excitation on channel r moving in the η direction is as schematically depicted in Figure 2. It is worth to note that all these contributions can be distinguished, since R and L channels have opposite spin projections.
In the non-interacting case K = 1, only ∆n R,+ and ∆n L,− are present (red pulses), consistently with the fact that for free fermions R and L channels are right-and left-moving, respectively, and do not mix. On the contrary, in the interacting case also the (R, −) and (L, +) are involved (blue pulses), due to charge fractionalization phenomena [78][79][80][81]. The charge ‡ per period Q r,η carried on each channel is obtained by integrating over one period the corresponding contribution to the current flowing away from the point x = 0. Thus, by fixing a detection point d > 0, we find where Equation (14) has been used and coefficients A ηϑr are defined in (11). The charge Q r,η carried by each excitation is an interaction-dependent fraction of the charge q injected by the drive V (t), which is the experimentally tunable parameter. Notice that ±q would be the charge carried in the non-interacting system (K = 1) by the excitation on channel (R, +)/(L, −).

General properties
In this section we discuss the intrinsic spectral properties [75,76,[82][83][84]] of the interacting system in presence of a periodic drive. To this end, we will focus on the variation of the spectral function, with respect to its equilibrium value (when V (t) = 0). It is therefore useful to introduce the variation of the lesser/greater local Green function: Here,ψ r (x, t) denotes the time evolution of the fermion operator for r-electrons in the absence of the drive. The exponential factor is a Wilson line, ensuring gauge invariance of the correlators [56,62]. Being out of equilibrium, the Green functions depend both on the time difference τ = t 1 − t 2 and on the average time t = (t 1 + t 2 )/2. Therefore, we define the local (excess) spectral functions as the Fourier transform with respect to τ and we further average over the period of the drive T : By resorting to standard bosonization techniques [76,77], Green functions can be written as where the contribution of the excitation on the channel (r, η) has the structure Due to the function Θ(ηx), the term related to the excitation with η = + (−) contributes only at positive (negative) values of x. Taking advantage of this fact, we can write ∆A ≷ r (ω; x) = η=± Θ(ηx)∆A ≷ r,η (ω), defining the greater/lesser spectral function ∆A ≷ r,η (ω) associated with the excitation (r, η) as Here, (27) represents the equilibrium Green function at zero temperature and is independent of the channel index r. The term P < r,η (τ ) = (iηuτ − aϑ r )(iηuτ ) −1 = P > r,η (−τ ) stems from the point splitting procedure and ensures that the diagonal limit τ → 0 correctly reproduces the excess particle density ∆n r (x, t) = lim τ →0 ∆G < r (t + τ /2, t − τ /2; x), as already discussed in [46]. This factor is only relevant at small values of τ and thus affects the corresponding spectral function ∆A ≷ r,η (ω) at high energies. The effects of the time-dependent external drive are encoded in the phase factor where we recognize the fractionalization factors √ KA ηϑr already encountered when discussing the excess particle density, see also (11). In the presence of a periodic drive, the phase factor can be conveniently decomposed in a Fourier series [49,85] Here, the photoassisted amplitudes p represent the probability amplitude for an electron to emit ( < 0) or absorb ( > 0) | | photons of energy Ω as a consequence of the ac drive. Their particular expression depends on the functional form of V (t). By using (29) it is easy to show that so that ∆A < r,η (ω) reads with Ω = ( + Q r,η )Ω. Notice that the photoassisted coefficients depend on the charge Q r,η of the different excitations. Now, some general properties can be discussed. First of all, the following sum rules are fulfilled: This indicates that the integral over energies of the lesser excess spectral function on channel r, η gives the charge per period carried by the excitation on that channel, in the same way as the integral over time of the excess charge density [see Equation (21)]. This important sum rule allows us to introduce the notion of minimality of an excitation, by requiring that the corresponding spectral function has everywhere the same sign as the one dictated by its sum rule. Physically, every excess spectral function represents a perturbation with respect to equilibrium and this perturbation is globally bigger if the function has in some regions an opposite sign with respect to what the sum rules requires, showing the presence of additional positive/negative charge. Moreover, looking at Equations (27) and (28) some symmetry relations between greater and lesser components of spectral functions can be deduced. In particular, we have Here, for sake of clarity, we have explicitly included the dependence on the parameter q [see Equation (14)]. Going from positive to negative q simply means flipping the sign of V (t). Thanks to Equation (33) we can simply focus on ∆A < R,± (ω, q) and obtain from these all other contributions by properly changing the sign of q and ω. Finally, in the non-interacting case K = 1, some additional symmetry relations are satisfied: As a consequence, the total excess spectral function, defined as ∆A r,η (ω, q) = ∆A < r,η (ω, q) + ∆A > r,η (ω, q) = ∆A < r,η (ω, q) + ∆A < r,η (−ω, −q) , vanishes when K = 1 independently of the drive. In the presence of interactions, instead, ∆A r,η (ω) = 0. Therefore, a measure sensitive to ∆A r,η (ω) would be able to clearly distinguish between an interacting and a non-interacting system, see Section 5.

Lorentzian pulses
As summarized in Section 2, periodic trains of Lorentzian pulses play a special role in the context of EQO [21,36] (being the best choice for the generation of minimal excitations for non-interacting systems) and have triggered an intense research activity [49,54,55,59,[85][86][87]. Here, we will thus focus on this particular shape of wavepackets, in order to elucidate the effects caused by the presence of repulsive interactions. The real-time shape of a single Lorentzian pulse is given by In order to calculate the spectral functions we need to specify the photoassisted coefficients appearing in (31) and then compute the integral. They read [49,55] p (q) = q +∞ s=0 (−1) s Γ(q + + s) e −2πw(2s+ )/T Γ(q + 1 − s)Γ(1 + s)Γ(1 + + s) .
We now restrict our attention to the case of Lorentzian pulses with integer values of Q r,η for the corresponding channel (r, η). As a consequence of the particular form of the photoassisted coefficients, the spectral function shows some remarkable properties for these integer values. The behavior of the lesser spectral function ∆A < R,+ for rightmoving wavepackets is shown in Figure 3 for different values of the interaction strength K. Note that this contribution is present also in the non-interacting case (K = 1), where Q R,+ = q. Different panels correspond to integer but opposite value of the injected charge Q R,+ = ±1. In absence of interactions (K = 1), as one can argue from (34), the spectral function for Q R,+ = −1 can be obtained by a rotation by π around the origin of the one with Q R,+ = +1, resulting in a vanishing total spectral function ∆A R,+ = ∆A < R,+ + ∆A > R,+ = 0. On the other hand, a remarkable asymmetry appears in presence of interactions (K < 1), where the variation of the lesser spectral function for positive or negative charge are now independent. Another feature evident in Figure 3 is that ∆A < R,+ (ω) ∝ Θ(−ω) when Q R,+ = −1, independently of the interaction. This behavior is uniquely due to the specific shape of integer Lorentzian pulses and, in particular, to the following peculiar property of their photoassisted coefficients: In the non-interacting case K = 1, the additional symmetry relation (34) is responsible for the appearance of a Θ(+ω) also at the positive charge value Q R,+ = +1. This feature is spoiled by interactions if K < 1, where ∆A < R,+ (ω) is finite for both ω ≷ 0. Importantly, in the non-interacting case only the channel (R, +), and the related (L, −), has a finite spectral weigth, while for K < 1 other two channels are also present. The presence of these additional contributions in the non-equilibrium spectral function, and its variation, are thus a unique fingerprint of interactions. The variation ∆A < R,− (ω) is shown for various interaction strengths K < 1 in Figure 4, where, again, the two panels refer to opposite injected integer charges Q R,− = ±1. The plots show that ∆A < R,− (ω) is nonvanishing for both ω ≷ 0 when its charge is negative, while a Θ(−ω) appears for a positive charge. This shows once more that the presence of interactions results in an asymmetric behavior of the non-equilibrium spectral function in response to positive or negative excitations.
Another aspect present in the variation of the spectral functions ( Figure 3 and Figure 4) concerns their sign. We have already pointed out in (32) that the integral of ∆A < r,η (ω) yields the charge Q r,η . By inspecting the plots, we see that for the channel (R, +) in Figure 3 the sign of the A < R,+ is everywhere the same of the one of its integral. It is actually possible to prove that the spectral function is always positive-definite for integer Q R,+ (see Appendix for details). This shows that for the channel (R, +) Lorentzian pulses with associated integer charges remain minimal even in presence of interactions. On the contrary, for the channel (R, −) the sign of the spectral function is not definite and at low ω it is actually the opposite of the one required by the sum rule (see Appendix). For this reason, the function ∆A < R,− is not minimal also in the case of associated integer charges.
Having described the peculiarities of Lorentzian pulses with associated integer charges Q r,η , a comment on a generic situation of non-integer charge is in order. In this case qualitative differences appear and have to be considered, since it is in general not possible to have all charges Q r,η simultaneously integer, unless for very specific values of the interaction strength. As an example, in Figure 5 we plot the function ∆A < R,+ (ω) for Q R,+ = ±0.6, directly obtained from Equation (31). The main difference to be appreciated with respect to the integer case in Figure 3 is the absence of the Θ(−ω) and that the sign of the spectral function is not defined. Here, we show an example of non-integer charge, with Q R,+ = ±0.6. We set T = 50w and a = 0.01uw.

Possible experimental signatures
In this Section we show how the intrinsic properties of the spectral functions can be probed by relying on a spectroscopic tool, i.e. a spin-polarized tip, kept at a given (but tunable) bias V tip with respect to the helical channels. As a result of this coupling, a tunnel current flows between the tip and the system. The spin polarization of the tip allows us to access all possible channels of the helical liquids, exploiting the spinmomentum locking [41,71]. Recently, this technique has been successfully used to probe the surface states of three-dimensional topological insulators [72][73][74]. We will argue that information about the spectral functions can be obtained from the current flowing into the tip and its fluctuations.
We consider a spin-polarized tip, placed at a fixed position d > 0, as sketched in Figure 6. The tip is coupled to the system via the tunneling Hamiltonian whereχ σ is the annihilation operator of electrons with spin projection σ on the tip. The spin-up polarization of the tip is described by α ↑ = 1 and α ↓ = 0, the spin-down one by α ↑ = 0 and α ↓ = 1. We only include spin-preserving tunneling, therefore, in the Hamiltonian (39),χ ↑ andχ ↓ are only coupled toΨ R andΨ L , respectively [71]. The tunnel current flowing between the system and the tip, when the latter is polarized with spin ↑ or ↓, can be written aŝ The noise associated with its fluctuations is S tip,r = 2 where δÎ tip,r (t) =Î tip,r (t) − Î tip,r (t) .
polarized tip Figure 6. Sketch of the proposed setup. The interacting helical channels are driven by the periodic time dependent voltage V (t), applied in the region x < 0. A spin-polarized tip is placed at d > 0 and allows the spin-preserving tunneling of electron, selecting their spin according to its polarization. The tip is modeled as a non-interacting system and is biased with a voltage V tip with respect to the chemical potential of the helical channels. This setup measures the tunnel current between the system and the tip.
Both quantities I tip = dt T Î tip,r (t) and S tip,r are evaluated to lowest order in the coupling constant λ, finding Here, A ≷ r (ω, d) = A ≷ 0 (ω) + ∆A ≷ r (ω, d), with the latter term defined in (23) and the equilibrium one is nothing but the Fourier transform of (27) [88]: Since d > 0, spectral functions are then only related to the excitation on the channel (r, +) [see Equation (24)]. For notational convenience, we will not include the index + in this Section, since there is no ambiguity. Concerning the tip, its spectral function is defined as where the tip Green functions, G < tip (τ ) = i χ † σ (0)χ σ (τ ) and G > tip (τ ) = −i χ σ (0)χ † σ (−τ ) , are independent of the spin σ. Since the tip is non-interacting, their expressions are obtained from (44), with A − = 0 and u = v F . Then, A ≷ tip (ω) = (2πv F ) −1 Θ(±ω). Therefore, the deviations ∆I tip,r in the current (42) due to the effect of the drive V (t) only can be expressed as Similarly, the deviations ∆S tip,r in the noise (43) read It is worth noticing that ∆I tip,r = 0 even at zero static bias (V tip = 0), because the helical channels are driven out of equilibrium by V (t), which has a non-zero dc component. Moreover, while in general the deviations ∆I tip,r and ∆S tip,r are different from I tip,r and S tip,r obtained in (42) and (43), the difference disappears at V tip = 0. We can now introduce the following excess noises by combining the last two equations: These quantities represent the deviations of the noise from its Poissonian limiting value.
It is now possible to extract information about the intrinsic spectral properties of the helical channels. First of all, the variation of the total spectral distribution ∆A r (ω) = ∆A < r (ω) + ∆A > r (ω) can be obtained from the excess differential conductance, namely Given Equation (35), we expect that in a non-interacting system ∆A r (ω) = 0, regardless the shape of the drive. This does not hold anymore as soon as interactions are present. Indeed, by considering the case of a Lorentzian drive, we show in Figure 7 the variation of the excess differential conductance for different values of the interaction strength, in the case where the tip is polarized with σ =↑. Thanks to the sharply different behavior between interacting and non-interacting case, it is possible from a measure of the current ∆I tip,r to probe whether the system is interacting or not. in units of e 2 |λ| 2 w vFuT , as a function of eV tip , in units of w −1 . This plots are obtained for a Lorentzian drive with Q R,+ = 1 and T = 50w and directly give, up to a sign, the total excess spectral function ∆A R (ω), as established by Equation (49). Notice that in the non-interacting case, the result is zero, due to the symmetry (34) of the spectral functions.
Additional information can be obtained by taking the derivative of the two excess noises introduced in (48): This relation makes it possible, by varying V tip , to reconstruct both the greater and lesser spectral functions and access all the features presented in the previous Section. Notice also that spectral functions of both channels R and L can be investigated by simply changing the polarization of the tip. Further comments about the excess noise in (48) can be argued. At V tip = 0, the two quantities ∆S (±) exc,R (0) vanish when the excitation on the channel (R, +) is an integer charge with Lorentzian shape. In particular ∆S (+) exc,R (0) = 0 when Q R,+ is a positive integer (electron-like excitation, with q > 0), while ∆S (−) exc,R (0) = 0 when Q R,+ is a negative integer (hole-like excitation, with q < 0). This is due to the fact that Figure 3). Through the same reasoning we see that ∆S (±) exc,L (0) = 0 when Q L,+ is a negative [q > 0, see (21)] or positive (q < 0) integer, respectively. Let us focus on ∆S (+) exc,r (0) and analyze the conditions for it to vanish. When r = R, we need Q R,+ = n, with n ∈ N + . In terms of the initial injected value q, this means Likewise, when r = L, we need Q L,+ = −m, with m ∈ N + , namely (a) (b) Here, the zeros are located according to (52). Note that no signal is present at K = 1. In both panels we set w = 0.1T and u(aΩ) −1 = 10.
In Figure 8 we present the behavior of excess noise ∆S (+) exc,R/L (0) as a function of the experimentally tunable parameter q, showing that the zeros are indeed located at the points given by (51) and (52). By varying the interactions, the zeros in panel (a) move to higher values of q, as required by Equation (51), while the opposite is true in panel (b), according to Equation (52). In the latter case we do not have any signal at K = 1, since in this case the spectral function on the L channel vanishes. This discussion demonstrates that a measure of the excess noise could be used to extract the value of the interaction strength. Indeed, by looking for instance at the value of q at which the n-th zero in Figure 8(a) occurs, the Luttinger parameter can be determined by solving Equation (51) for K. As it is clear from the above discussion, a vanishing excess noise is only due to the presence of proper Θ functions in the spectral functions. This property is uniquely determined by the Lorentzian drive and is therefore robust with respect to the presence of interactions. Indeed a vanishing excess noise can be achieved at any interaction strength, provided that the conditions in (51) or (52) are met. We recall that a vanishing excess noise in the case of Lorentzian pulses producing excitations with integer charge has been already reported in a QPC geometry for noninteracting systems [21,49,65] as well as in the integer [50,60] and fractional quantum Hall effect [55]. Indeed, one can recognize that expressions for the excess noise of a QPC are equivalent to the ones in (42) and (43). We emphasize however that interactions in counterpropagating helical channels result in a richer phenomenology in the excess noise, where the positions of the zeros depend on K.
As discussed in Section 2 and Section 4, in a non-interacting system a vanishing excess noise directly implies that ∆A < R (ω) has a definite sign. This is not anymore true if K = 1. For instance, when a Lorentzian pulse with Q L,+ a negative integer is generated on the channel (L, +), both ∆A ≷ L,+ (ω) do not have a definite sign, but still ∆S (+) exc,L (0) = 0, as we see in Figure 8. We conclude that, apart from the case K = 1, a vanishing excess noise is not necessarily related to a minimal spectral function (in the sense of absence of additional positive/negative charge). This fact was already noticed in a different context [89,90]. One of the stricking results of the latter analysis is that contrary to common belief that minimal wave packets need to bear an integer charge, in strongly correlated systems such as a quantum spin Hall Luttinger liquid system, the excess noise vanishes for non integer charges.

Conclusions
We have analized the non-equilibrium spectral properties of interacting helical channels subjected to a time-dependent drive. In order to better elucidate the effects induced by e-e interactions, we have focused on the case of periodic train of integer Lorentzian pulse which, in the non-interacting case produce minimal excitations. We have shown that peculiar asymmetries, related to the sign of the injected charge, appear as a function of the interaction strength. Moreover, the concept of minimal excitations has to be properly considered and is no more directly related to the vanishing of the excess noise as in the case of free fermions. These findings can be tested by looking at the tunneling current and its fluctuations with a polarized tip, which allows for a spectroscopic investigation of the intrinsic spectral properties of all counterpropagating channels. The spin-polarized tip experiment suggested in this work shows that in a correlated electron systems such that the quantum spin Hall Luttinger liquid, the transmitted charges per period associated with the voltage pulse which minimize the excess noise deviate from integer values. This is a direct consequence of the fact that the Luttinger parameter K is smaller than one. Therefore, our analysis can be considered as a novel diagnosis for detecting fractional charges in quantum spin Hall Luttinger liquids in presence of time-dependent drives.
Here, the tilde indicates that we adapted the definition of the spectral function to a single-pulse drive, by replacing the integral T −1 T /2 −T /2 dt with +∞ −∞ dt. By evaluating the previous integral we arrive at the following result: This results shows that the spectral function ∆A < R,+ is always positive and nonvanishing for both ω ≷ 0, precisely as observed in the main text.
When Q R,+ = m < 0, by following the same steps as before, we find: This shows that, in this case, the spectral function is always negative and vanishes for ω > 0. Concerning the spectral function ∆A < R,− (ω), we show in Figure A1 that its sign is not definite and that a change of sign occurs in the high-energy tail at ω < 0, ensuring that the sum rule (32) is satisfied.