  We have measured motional heating rates of trapped atomic ions, a factor that
can influence multi-ion quantum logic gate fidelities. Two simplified
techniques were developed for this purpose: one relies on Raman sideband
detection implemented with a single laser source, while the second is even
simpler and is based on time-resolved fluorescence detection during Doppler
recooling. We applied these methods to determine heating rates in a
microfrabricated surface-electrode trap made of gold on fused quartz, which
traps ions 40 microns above its surface. Heating rates obtained from the two
techniques were found to be in reasonable agreement. In addition, the trap
gives rise to a heating rate of 300 plus or minus 30 per second for a motional
frequency of 5.25 MHz, substantially below the trend observed in other traps.
