I am not sure about asparagus in a raised bed ! The roots do not go very deep in fact they tend to grow horizontally a few inches down and I would worry that they would be contained by the sides of the raised bed. I have had a large asparagus bed for the last 10 years and I can cut at the peak of the season 150 plus spears in a week. The problem that I have is that I have to grow short rooted vegetables such as beetroot , lettuce or spring onions adjacent to the asparagus bed as the roots of the asparagus can extend for one to two metres away from the bed so potatoes, carrots, parsnips etc are a no go. But back to the question I cut down my asparagus ferns around the end of November ( here in the UK ) and cover the bed with well rotted farmyard manure but then around the end of March early April I remove all the manure that has not been taken in by the worms and wait for the 1st pickings which depending on the temperature is around mid April. If the temperature drops it sulks and will not produce spears until it warms up. I hope that bit of information helps but it works for me.
