As others have noted, experts may feel they have fewer questions to ask. They don't necessarily actually have fewer questions, but they may feel the audience for some of their questions is too small. So for the subset of those questions where they can wait for an answer, you're looking for ways to get them to ask them anyway. Sometimes, from what I've seen, just reminding them that questions aren't only for the current community is enough. I've seen people ask questions that hang around for months or even years and then get a good answer. So long as the asking user has remained involved with the community (so he's still around to care), he benefits from this. The best tools to use in doing this on Stack Exchange are meta and chat.

That's for the questions they don't already know the answers to, but there's another category worth exploring. On a community like Stack Exchange in particular, people can use your site not only to get their questions answered but to anticipate others' questions and share their knowledge. This is where the self-answer comes in. (Here's one example of mine.) Encourage your experts to write good questions that they can answer and then write those answers. They get to share some knowledge, your site gets a good question and at least one answer, and the next person who has that problem can find that answer already present on your site. (And, if extrinsic motivation is important to them, they can get upvotes for both the question and the answer, if they're deserving.)

There is a caveat here: when you already know the answer it's easy to fall into the trap of writing a question that's too terse. The question needs to stand alone as a question; it must not depend on its answer. If writing self-answered questions is new for your community (or for some of your users), pay extra attention to the questions. If your users are invested enough in your community to go down this path at all, they're probably open to making edits in response to comments requesting clarification. But they might need help from other community members to know where they've unintentionally cut corners on the question.

As others have noted, experts may feel they have fewer questions to ask. They don't necessarily actually have fewer questions, but they may feel the audience for some of their questions is too small. So for the subset of those questions where they can wait for an answer, you're looking for ways to get them to ask them anyway. Sometimes, from what I've seen, just reminding them that questions aren't only for the current community is enough. I've seen people ask questions that hang around for months or even years and then get a good answer. So long as the asking user has remained involved with the community (so he's still around to care), he benefits from this. The best tools to use in doing this on Stack Exchange are meta and chat.

That's for the questions they don't already know the answers to, but there's another category worth exploring. On a community like Stack Exchange in particular, people can use your site not only to get their questions answered but to anticipate others' questions and share their knowledge. This is where the self-answer comes in. (Here's one example of mine.) Encourage your experts to write good questions that they can answer and then write those answers. They get to share some knowledge, your site gets a good question and at least one answer, and the next person who has that problem can find that answer already present on your site. (And, if extrinsic motivation is important to them, they can get upvotes for both the question and the answer, if they're deserving.)

There is a caveat here: when you already know the answer it's easy to fall into the trap of writing a question that's too terse. The question needs to stand alone as a question; it must not depend on its answer. If writing self-answered questions is new for your community (or for some of your users), pay extra attention to the questions. If your users are invested enough in your community to go down this path at all, they're probably open to making edits in response to comments requesting clarification. But they might need help from other community members to know where they've unintentionally cut corners on the question.