Spin dynamics in the pressure-induced two-leg ladder cuprate superconductor Sr${}_{14-x}$Ca${}_x$Cu${}_{24}$O${}_{41}$

Within the two-leg $t$-J ladder, the spin dynamics of the pressure-induced two-leg ladder cuprate superconductor Sr${}_{14-x}$Ca${}_x$Cu${}_{24}$O${}_{41}$ is studied based on the kinetic energy driven superconducting mechanism. It is shown that in the pressure-induced superconducting state, the incommensurate spin correlation appears in the underpressure regime, while the commensurate spin fluctuation emerges in the optimal pressure and overpressure regimes. In particular, the spin-lattice relaxation time is dominated by a temperature linear dependence term at low temperature followed by a peak developed below the superconducting transition temperature, in qualitative agreement with the experimental observation on Sr${}_{14-x}$Ca${}_x$Cu${}_{24}$O${}_{41}$.