The first stability tests applied sudden, discontinuous changes of to undamped two- and four-ion crystals. In both cases, the crystalline structure was destroyed well before the Mathieu instability. For the two-ion crystal, the alternating voltage was instantaneously changed from V () to V (); after equilibration, was raised to V (), then to V (). As shown in Tab. , decreased as was raised, while the amplitude of radial oscillations due to micromotion increased. When the crystal melted at , there was a gain in kinetic energy which, averaged over s, amounted to a heating rate of . The melted two-ion configuration is in file 2sudden.trp.
Table shows the trend of decreasing as q was raised discontinuously for a planar four-ion crystal. Like the two-ion crystal, the four-ion crystal melted far before reached the Mathieu instability: after s of simulation at with m, the crystal's kinetic energy and increased substantially.
These results demonstrate that crystals in region A are not stable when sudden changes in Mathieu parameters are applied. Qualitatively, this ``instability'' can be understood by noting that sudden changes in the Mathieu parameters disrupt the phase and amplitude of the crystal's micromotion oscillations and that quantitative features of the crystal (i.e., ) are Mathieu parameter-dependent. When a Mathieu parameter undergoes a sudden change, the crystalline configuration is no longer a steady-state (analogous to an eigenstate) solution for the given system. Rather, it may be interpreted as a superposition of solutions, some of which may correspond to heating configurations and enable melting via diffusive energy gain from the oscillating potential.
Table: Structural response of 2-ion crystal to instantaneous changes in magnitude of applied oscillating voltage, . Note the increase in micromotion amplitude with increasing magnitude of the oscillating potential, . source file: 2CrA.trp.
Table: Change in coupling parameter () of a 4-ion crystal in response to instantaneous changes in magnitude of applied oscillating voltage, . source file: 4CrA.trp.