It very often happens that a new father will feel a little left out when his partner seems to have no time for him, only the new baby. This is unavoidable in the beginning, as the new baby does need plenty of attention. Also, if it is a first baby, the new mother lacks confidence, and needs some time to settle into a routine. Once this happens, then you and your partner can begin to have a social life again, as well as reconnect.
Your aim is to have some time alone with your partner, so arrange for someone to baby sit for a few hours. This can either be a paid babysitter, or a friend or family member. However, if you prefer to stay at home then there is really no need to actually go out on a date. When your baby is asleep, spend some quality time together and focus on each other until your baby wakes for a feed.
You also do not have to only spend time together at night. For instance, travel together on your way to work and have a really good, meaningful conversation. Meet at a restaurant and have lunch together - this way, there is no baby and no household chores to worry about.