**1. Introduction**

The possibility to generate, manipulate and detect single- to few-electron excitations coherently propagating in mesoscopic quantum conductors represents one of the main tasks in condensed matter. Theoretical and experimental studies in this direction culminated with the development of the new field of electron quantum optics (EQO) [1–3]. In this context, a remarkable experimental effort has been devoted to the realization of on-demand sources of electronic wave-packets. With this purpose, two main techniques have been elaborated. The first relies on a driven quantum dot [4,5], which plays the role of a mesoscopic capacitor [6–9], tunnel coupled to a Quantum Hall edge channel. This kind of device allows the periodic injection of an electron and a hole along the edge channels of the Hall bar. An alternative approach involves the generation of purely electronic wave-packets by applying a properly designed train of Lorentzian voltage pulses in time to a quantum conductor [10–13]. The experimental realization of these excitations, dubbed Levitons [14], renewed the interest in the so-called photo-assisted quantum transport, namely the study of the current generated by periodic drives and its fluctuations [15,16].

Even if experimentally challenging, the implementation of EQO in strongly correlated systems, such as Fractional Quantum Hall (FQH) edge channels, could open new and interesting perspectives in the field. In particular, peculiar features like the unexpected robustness of Levitons [17,18] and the crystallization of multiple-electronic wave-packets in the time domain [19,20] have been recently theoretically predicted. The simplest possible set-up to access this new physics requires a periodic train of voltage pulses applied to one of the terminals of a Hall bar in a Quantum Point Contact (QPC) geometry. Here, excitations incoming from the lead are partitioned at the QPC, in an electronic equivalent of the Hanbury–Brown and Twiss optical interferometer [21,22].

Notwithstanding the control of charge transport at the individual excitation level still presenting interesting and fascinating open problems, electric charge is not the only relevant degree of freedom we can look at in the framework of EQO. Indeed, electronic wave-packets also carry energy in a coherent way [23]. This observation is of particular importance in view of the progressive miniaturization of electronic devices, which makes the problem of heat transfer at the nanoscale extremely timely [24], as demonstrated by recent progress in the field of quantum thermodynamics [25]. In this framework, new intriguing challenges are posed by the need of properly transposing concepts such as energy harvesting [26–33], transport [34–40] and exchange [41–43] at the mesoscopic scale.

In this direction, new studies in the EQO domain have also been triggered. For instance, heat current was identified as a useful resource for the full reconstruction of single-electron wave-packets [44]. Intriguingly, also mixed heat-charge correlations [45–47] and heat current noise [48,49] were investigated in the case of single-electron sources with Levitons emerging as robust excitations also for what it concerns heat transport [50]. Despite a direct observation of heat current fluctuations is still lacking, an experimental protocol has been recently proposed in order to extract this quantity from temperature fluctuations [51].

In this paper, we evaluate the photo-assisted mixed and heat noise for FQH states in the Laughlin sequence [52]. Considering the QPC in a weak backscattering regime, we can confine our calculation to the lowest order in the tunneling. The aim will be to investigate the behavior of these quantities by independently tuning the AC and DC components of the voltage drive in the same spirit of what was done for the photo-assisted shot noise generated by electrical current [16,53].

The paper is organized as follows. In Section 2, we illustrate the model used to describe FQH edge channels coupled to a time dependent voltage. The charge current is evaluated at the lowest perturbative order in Section 3, while the corresponding expression for the heat current is derived in Section 4. In Section 5 we consider the charge and heat current fluctuations in terms of mixed and heat noise. Section 6 is devoted to the analysis of the symmetries of the discussed quantities as a function of the dc and ac contribution to the voltages. Finally, in Section 7 we draw our conclusions.
