  Cuprates are d-wave superconductors which exhibit a rich phase diagram: they
are characterized by superconducting fluctuations even above the critical
temperature, and thermal disorder can reduce or suppress the phase coherence.
However, photoexcitation can have the opposite effect: recent experiments have
shown an increasing phase coherence in optimally doped BSCCO with mid-infrared
driving. Time-resolved terahertz spectroscopies are powerful techniques to
excite and probe non-equilibrium states of superconductors, directly addressing
collective modes, such as amplitude (Higgs) oscillations. In this work, we
calculate the full time evolution of the current generated by a cuprate with a
quench-drive spectroscopy setup. Analyzing the response in Fourier space with
respect to both the real time and the quench-drive delay time, we look for the
signature of a transient modulation of higher harmonics as well as the Higgs
mode, in order to characterize the ground state phase. In particular, this
approach can provide a smoking gun for induced or increased phase coherence
when applied to the pseudogap phase. These results can pave the way for future
experimental schemes to characterize and study superconductors alongside
incoherent phases and phase transitions, including induced and transient
superconductivity.
