Chicago Style requires you to cite AI tools whenever you use the content that they generate in your own work. It does not, however, recommend that you cite your use of AI in a bibliography or reference list unless you provide a publicly-available link to the content you are citing (i.e., sharing a direct link to your conversation with the AI tool). This may change as tools are increasingly providing functionalities which allow users to generate a shareable link to their interaction with the tool. For now, however, Chicago Style treats conversations with AI tools like a personal communication.
To sum things up: You should credit AI tools when you reproduce their words or content within your own work, but unless you are able to include a publicly-available link to your conversation, include that cited information in your text (using parentheses) or in a note--not in a bibliography or reference list. You can read more at the Chicago Manual of Style Online's Q&A section (last updated in 2024).
Chicago Style treats the AI tool--for example, Chat-GPT or Claude--as the "author" of the content, and the companies that developed them (like OpenAI) as the publisher. If you decide to include a footnote or endnote, it might look like this:
Description of what you're citing including the Author, Publisher, Month dd, yyyy, URL.
Example: Text generated by ChatGPT, OpenAI, March 7, 2023, chat.openai.com/chat.
If your prompt hasn't been included in the text, it can be included in the note:
Author, response to "Text of your prompt", Publisher, Month dd, yyyy.
Example: ChatGPT, response to "Explain how to make a homemade sourdough starter," OpenAI, March 7, 2023.