Experiments in atomic, molecular, and optical (AMO) physics rely on lasers at many different wavelengths and with varying requirements on spectral linewidth, power, and intensity stability. Vertical external-cavity surface-emitting lasers (VECSELs), when combined with nonlinear frequency conversion, can potentially replace many of the laser systems currently in use. Here we present and characterize VECSEL systems that can perform all laser-based tasks for quantum information processing experiments with trapped magnesium ions. For photoionization of neutral magnesium, 570.6$\,$nm light is generated with an intracavity frequency-doubled VECSEL containing a lithium triborate (LBO) crystal for second harmonic generation. External frequency doubling produces 285.3$\,$nm light for resonant interaction with the $^{1}S_{0}\leftrightarrow$ $^{1}P_{1}$ transition of neutral Mg. Using an externally frequency-quadrupled VECSEL, we implement Doppler cooling of $^{25}$Mg$^{+}$ on the 279.6$\,$nm $^{2}S_{1/2}\leftrightarrow$ $^{2}P_{3/2}$ cycling transition, repumping on the 280.4$\,$nm $^{2}S_{1/2}\leftrightarrow$ $^{2}P_{1/2}$ transition, coherent state manipulation, and resolved sideband cooling close to the motional ground state. Our systems serve as prototypes for applications in AMO requiring single-frequency, power-scalable laser sources at multiple wavelengths.
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