Phase-of-the-phase spectroscopy in the few-cycle regime

We present a theoretical study of the phase-of-the-phase spectroscopy in the few-cycle regime and analyze its explicit dependence on the carrier-envelope phase (CEP) and the pulse length. The process of above-threshold detachment of negative ions driven by ultrashort two-color laser pulses is investigated within the strong-field approximation formulated for few-cycle pulses with two carrier frequencies. While the CEP effects are known to be difficult to extract from the photoelectron yield of direct electrons, we demonstrate that the CEP leaves a clear and robust imprint on the phase of the phase, even when only direct electrons are considered. We analyze two-color configurations with parallel and orthogonal polarizations of the driving and perturbation pulses and identify characteristic CEP-dependent structures in both momentum-resolved and energy-averaged phase-of-the-phase spectra. The role of symmetry and its reduction in orthogonally polarized configurations is discussed in detail. Importantly, we show that the CEP dependence of the phase of the phase survives focal averaging and becomes particularly smooth under realistic experimental conditions. In addition, we demonstrate that the phase of the phase exhibits a pronounced sensitivity to the pulse length in the few-cycle regime, while converging for longer pulses, indicating its potential for characterizing ultrashort pulse durations. Our results establish phase-of-the-phase spectroscopy as a sensitive and experimentally viable tool for extracting information about strong-field processes driven by few-cycle laser pulses.