Is there a time frame to pay awarded amount to plaintiff in civil legal suit for partial return of security deposit?

I am a lawyer in CT and practice in this area of the law.  The general rule is that a party has 20 days to appeal a judgment.  If more than 20 days go by after the judgment, then you may enforce the judgment.  I suggest placing a judgment lien on the landlord's  property to secure payment of your judgment.  This way, you will have security and guaranteed payment of your judgment.  I suggest that you have a lawyer draft the judgment lien.  The landlord will likely settle with you upon receiving notice of the lien as he does not want to lose his property.