In my web page there is a link which should be opened in private browsing mode always. So how can I do it using HTML/javascript/jquery/php?
That's up to the user. You can't force a user to open the browser in private browsing mode.
HTML and Javascript are used on a wide variety of programs including browsers. Even among the browsers, only a few browsers have private/incognito mode. So there cannot be a native HTML/Javascript feature to target the private browsing mode.