The incredibly private couple welcomed their first child together in September but, until now, had kept silent about the new arrival.
In an interview with Violet Grey magazine, actress Eva revealed: "We both love the Esmeralda character from the Victor Hugo novel The Hunchback of Notre Dame and just think it's a beautiful name. Her middle is Amada, which was my grandmother's name. It means "beloved" in Spanish."
Eva, 40, told the digital publication that she and Ryan decided early on they wanted Esmeralda to have as much privacy as possible, which is why Eva withdrew from the limelight during her pregnancy.
She added that right now the couple haven't hired a nanny but she is "open to the idea" in the future.
"Esmeralda has two amazing grandmothers and incredible aunties who come over and really help me. But as far as a nanny or a night nurse, for me that part of being a mother at this early stage is the struggle of not being able to sleep and not knowing what I'm doing and really going through it with her and battling out those nights."
The actress also admitted that although she tries to avoid Google for fear of uncovering "frightening" things, she does occasionally visit baby and mother message boards as it makes her realise others are in the same position.
"I find so much solace in that. Just knowing that I'm not alone out there and that other mums have gone through the same thing can at least get me through the night."
Eva and Ryan, 34, have been dating since 2011 when they starred together in movie The Place Beyond The Pines.
A source told People magazine back in September: "Eva's absolutely in heaven being a mum. She's never been happier. She and Ryan are just in awe of this beautiful person they created. It's a really special time for them."
Stars who welcomed babies in 2014:
Copyright: Rex Features WENN/Matthew Taylor/Daniel Deme/Josiah True/Matthre Taylor/Joe
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