Motivated by the bilayer cuprate superconductors and nickelate superconductor ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$, we investigate the evolution from intralayer to interlayer superconductivity (SC) in finite-width bilayer $t\text{−}J\text{−}{J}_{\perp }$ ladders, where $t$ is the in-plane electron hopping, $J$ is the in-plane spin interaction, and $J_{\perp }$ is the interplane spin interaction. By means of density matrix renormalization group calculations, we map out the quantum phase diagram of the bilayer two-leg ladders by tuning $J_{\perp }$ in a large doping range $\delta =(1/8)–(1/2)$. We find that a large $J_{\perp }$ can always drive interlayer SC by coupling the two layers in both the Luther-Emery liquid (LEL) and Luttinger liquid states. By coupling two LEL states, the in-plane SC evolves to interplane SC either through an intermediate charge-density-wave (CDW) phase or directly, depending on doping ratio. This emergent CDW phase, which exists over a finite doping range, appears to develop from the CDW state of the two-leg ladder at $\delta =1/4$. By coupling two Luttinger liquids, the in-plane Luttinger liquids show a transition to the interplane superconducting phase at large $J_{\perp }$, as reported in previous literature. Interestingly, in the intermediate $J_{\perp }$ regime, we find that, while the in-plane Luttinger liquid features remain stable, the interplane SC can develop an enhanced quasi-long-range order with the power exponent $K_{\mathrm{SC}}^{zz}\sim 1$. Finally, we show that the interlayer SC is also stable by coupling the bilayer three-leg $t\text{−}J$ ladders by a strong $J_{\perp }$ interaction, from both the LEL and Luttinger liquid states.

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• Density matrix renormalization group
• Superconductivity
• High-temperature superconductors
• Strongly correlated systems
• Tensor network methods
• t-J model