We report on experiments that probe the stability of a two-dimensional jammed granular system formed by imposing a quasistatic simple shear strain $gamma_{rm I}$ on an initially stress free packing. We subject the shear jammed system to quasistatic cyclic shear with strain amplitude $deltagamma$. We observe two distinct outcomes after thousands of shear cycles. For small $gamma_{rm I}$ or large $deltagamma$, the system reaches a stress-free, yielding state exhibiting diffusive strobed particle displacements with a diffusion coefficient proportional to $deltagamma$. For large $gamma_{rm I}$ and small $deltagamma$, the system evolves to a stable state in which both particle positions and contact forces are unchanged after each cycle and the response to small strain reversals is highly elastic. Compared to the original shear jammed state, a stable state reached after many cycles has a smaller stress anisotropy, a much higher shear stiffness, and less tendency to dilate when sheared. Remarkably, we find that stable states show a power-law relation between shear modulus and pressure with an exponent $betaapprox 0.5$, independent of $deltagamma$. Based on our measurements, we construct a phase diagram in the $(gamma_{rm I},deltagamma)$ plane showing where our shear-jammed granular materials either stabilize or yield in the long-time limit.
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