Inspired by the recent experimental advances in cold atom quantum simulators, we explore the experimentally implemented bosonic $t–t^{\prime }–J$ model on the square lattice using large-scale density matrix renormalization group simulations. By tuning the doping level $\delta$ and hopping ratio $t^{\prime }/t$, we uncover six distinct quantum phases, several of which go far beyond the conventional paradigm of phase-coherent superfluidity (SF) expected for bosonic systems. In particular, in the presence of antiferromagnetic (AFM) order, doped holes are tightly bound into pairs, giving rise to a pair density wave phase at low doping and small $|t^{\prime }/t|$, which is suppressed on the $t^{\prime }<0$ side, resulting in a disordered pair density wave state that lacks coherence of either individual bosons or pairs. Upon further doping, bosons can regain phase coherence and form an SF* state, characterized by condensation at emergent incommensurate momenta concurrent with an incommensurate magnetic order. On the $t^{\prime }>0$ side, the sign-induced kinetic frustration inherently disfavors local AFM correlations, leading to a phase separation in which doped holes cluster into ferromagnetic (FM) domains spatially separated by undoped AFM regions. Upon further doping, this inhomogeneous state evolves into a uniform $\mathrm{SF}+xy$-FM phase. Finally, we propose a concrete experimental scheme to realize both signs of $t^{\prime }/t$ in Rydberg tweezer arrays, with an explicit mapping between model parameters and experimentally accessible regimes. Our results reveal competing and intertwined orders in doped antiferromagnets, which are relevant to central issues in high-$T_c$ superconductivity, reflecting the frustrated interplay between doped holes and spin background.

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• Pair density wave
• Quantum simulation
• Superfluidity
• Mott insulators
• Rydberg atoms & molecules
• Density matrix renormalization group
• Tensor network methods
• t-J model