Yes, the door is automatically locked each time it’s closed
Arcane lock states:
You touch a closed door, window, gate, chest, or other entryway, and it becomes locked for the duration.
The spell doesn’t state that it ends when the door is opened, so it continues to lock the door. In fact, the door never stops being locked, even while you’re manipulating it; instead, you are simply able to open it normally (as if it weren’t locked).
If the door didn’t automatically relock, then listing a password duration wouldn’t make sense (as speaking the password would be equivalent to permanently dispelling the lock):
You can also set a password that, when spoken within 5 feet of the object, suppresses this spell for 1 minute.
The spell gives no way for the caster to manually relock the door, so it’s clear that the spell is handling that automatically.
Don’t forget about the object interaction
Note that the door will probably need to start off open. The wizard only gets one object interaction for free, which can include:
open or close a door
Depending on how pedantic your group is, the wizard might not be able to both open and close the door for free in the same turn.1
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