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I would keep potted tulips in the sun until the buds begin to open. For them to last longer I would then move to the shade. When the flowers are done, put back into sun for the foliage to manufacture food for the bulbs (now split). Most will say to toss them out now. If you most definately want to experiment further, read on.

Bulbs are fertilized when they peep out of the soil and again when first in bloom. Continue to water regularly until the foliage yellows. As folisge ripens water less & less. When it separates from the bulb no more summer water.

Now, to leave the pot baking in the sun or move it to a shadier location? If the pot is plastic, definately out of sun. Clay ?? ummm, either way.

In Sept. tip out the old soil, gather up the splits, put in paper bag and into refrigerator. No apples or ethalene (sp?) gas producing fruits can be stored in frige at same time. End of November plant the splits into fresh soil. Water once to settle soil. No more water til green shoots appear. Do not expect flowers. If they were an old Darwin variety you may have some bloom. In cooler winter climates some of the splits of suitable varieties grow to blooming size in a year, some 2 years, etc.
You have 2 serious obstacles: mild winters & pots.

The other problem is what variety? Some tulips will not repeat no matter how cold the winter. If you are wanting to try again buy Red Impression or Pink Impression Darwins. Apeldoorn has been successful.