  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.
