I'm not sure that's possible, since basic authentication at least requires an authentication challenge to be sent to the browser, which AFAIK always results in the login pop-up. the relevant RFC is here. A short explanation is here

If you implement your own authentication scheme you can do whatever you want, usually by binding a login (from a standard HTML form) to a session. See for example CGI::Session.

Update/note: some browsers allow you to use URLs like http://username:password@www.example.com to automatically log the user in. You may be able to do something with that.

Update2: if you use digest authentication you may be able to invalidate a current login (thus logging a user out again). You can't do that at all using Basic Authentication, since the credential sent to the server is just a base64 encoded user:password string and never changes unless the username and/or password changes, and the browser will remember those until you restart it.