You do not need an amplifier, but adding an amp will help you get the most from your new speakers.
The speakers' power rating tells you the maximum recommended power level to use with the speakers. It does not imply a minimum power rating. In other words, while the speakers can handle up to 90 watts RMS, they'll work perfectly well on 18 watts.
The "93dB" is a sensitivity rating. This tells you the speaker's output level with one watt of power. A speaker with a high sensitivity doesn't need as much power to reach a given volume level, compared with a speaker with a lower sensitivity rating. 93dB is actually a relatively high sensitivity rating, which means that these speakers will work fairly well with a limited power source like a head unit.
If you want more power, you might try looking for Alpine's KTP-445 power pack. This is a 45 watt x 4 RMS amplifier that's quite inexpensive, and is designed specifically to use with Alpine head units. It's extremely easy to install, and will more than double the head unit output power.