Quicktip: Registered Letter by La Poste
A lot of issues arise when you decide to leave an apartment. Lots of nice landlords can turn into assholes because they know you're a foreigner and you don't know French law. Worse if you let your landlord know you are leaving France.
First things first don't mention you're leaving France/Paris/wherever. Second you MUST perform a L'état des lieux de sortie (the apartment must be completely empty). I'm not even sure why this is a question. We have these in America as well. You should be going through your apartment before you move in and after. Also always take pictures. The number of people who don't do this step is mind boggling, especially when they're mandatory at home (US).
So you've done all of those steps and your landlord is still ignoring you? They have two months to return the deposit (assuming you will get the full return because they have not given you an itemized bill of repairs. No, lightbulbs should never be deducted). After two months they're in breach.
I've heard of people having great relationships with their landlords and then their landlords ghost them. Shock said no one who has actually lived in Paris. Your next step should be to send a registered letter by post. We talked about this in a previous post about untangling the near impossible relationship with Free Mobile. If you no longer live in France, La Poste offers this handy dandy little service (this should tell you just how often this happens). You may access it here: https://www.laposte.fr/envoi-courrier-en-ligne/creer-lettre/lrel. I love that they have a category labeled "various terminations".
Et voila. No more worrying about that security deposit. If they ignore those letters? Oof, you'll just have to get a plane ticket back to France to go to court 🤷♀️