Artichokes are large and grow fast. It appears in your photo that there are more than one plant in a pot? Are these just side shoots from one plant?
They can grow fine in a 10 gallon pot as an annual. If you live a zone (8-11 USDA) where the temperature never goes below 14 degrees F during the winter you can grow them out in the garden as a perennial. The first year, you won't get buds. For winter cut them off a foot above ground and cover with organic mulch. To be extra careful, place an inverted basket over the top and more mulch...even a little blanket of burlap would work with mulch below and above it.
The first year, grown as a perennial they don't usually produce any buds. The second you'll see buds in late summer and fall. They'll grow to 4' and be 5-6' wide and last up to 5 years.
In colder regions zones 5-7 and sometimes 4, you can get artichokes to make buds the first year by giving them 'the cold treatment'. (Or find Imperial Star made to be grown as an annual in these zones)...
When your plants are around 6 weeks old, keep them between 35 degrees F to 50 degrees F for at least 250 continuous hours or 10 1/2 days. If you can't keep below 50 degrees on some days just continue the cold treatment for 4-6 more weeks. It is easier to put them out in a cold frame, at least 6 weeks before last frost, opening the top during the day.
If the soil temperatures are 60 degrees and all chance of frost is past they should be planted out in the garden...minimum of 2' apart, best is 4' apart.
Your soil should be rich with decomposed organic matter. Use a balanced fertilizer such as 10-10-10. I continually top dress with decomposed organic matter (compost), use fish fertilizer about once a month. I never allow my plants to dry out too much between watering.
As soon as a bud (usually the main terminal bud) is ready to eat, cut it off with a knife at the base. This encourages other buds to form and/or ripen. Keep harvesting as soon as a bud is ready to eat. Artichokes can produce up to 5 years as a perennial.
They require lots of water, soil that drains very well, a slightly more alkaline pH 6.5 - 8.0, regular applications of organic fertilizer, lots of room to grow, protection during the winters and vigilant harvesting for a good crop (4-7 buds is good). I like to allow the last bud to flower before the first frost...stunning purple-blue flower that is great as a cut flower, too.