Josephson scanning tunneling spectroscopy in superconducting phases coexisting with pair-, charge- and spin-density-waves

We demonstrate that the recent observations of spatial oscillations in the energy position of the superconducting coherence peaks in the cuprate, transition metal dichalcogenide, iron-based and heavy-fermion superconductors are consistent with the possible presence of a pair-, charge- or spin-density-wave phase. We show that for all three cases, the spatial oscillations of the superconducting order parameter, $\mathrm{\Delta }(\mathbf{r})$ can be imaged via the critical Josephson current, $I_c(\mathbf{r})$ measured in Josephson scanning tunneling spectroscopy experiments. Finally, we show that the spatial oscillations of the density waves and of $\mathrm{\Delta }(\mathbf{r})$ exhibit relative phase shifts of $\mathrm{\Delta }\phi =0$ or $\pi$ for a charge-density wave, and $\mathrm{\Delta }\phi =\pi \mathrm{/}2$ for the spin-density wave.