Further to the good answer provided by James...
The word just expresses, in an emphatic manner, the subjective perception of the near simultaneity of two events. If we say a short time after there is still subjectivity in short, but without just (or only) there is no emphasis on the speaker's perception that very little, if any, time had elapsed between the two events.
For example, I might arrive back at my house, unlock and open the front door, step inside, turn around to close the door, and then turn back around and take only one step into the room when the phone rings. I could say:
The phone rang just as I came in the door.
The phone rang as soon as I came in the door.
The phone rang the second I came in the door.
But another person might take those same actions, but walk all the way into the kitchen and set a kettle to boil on the stove for a cup of tea, when the phone rings. Although three or four minutes might have elapsed, that person might also say
The phone rang just as I came in the door. I didn't even have a chance to sit down.
The phone rang the minute I came in the door. I didn't even have a chance to sit down.