Since I've been getting this question a lot over the last few days, here goes. (And, for good measure, another photo of two Jesuit brothers.)
A Jesuit is a Jesuit forever, from the day he enters the novitiate until his death, unless he formally leaves the Society of Jesus or, in religious parlance, is "dismissed." (The three dates listed on a Jesuit's tombstone are: Natus, the date of his birth; Ingressus, the date he entered the novitiate; and Obiit, the date of his death.) Most Jesuits, when they have finished their Jesuit training, or "formation," are invited to not only to pronounce the three vows of poverty, chastity and obedience (which they first pronounced at the end of their novitiate), and the special "fourth vow" to the Sovereign Pontiff "with regards to missions" (that is, an openness and willingess to be sent anywhere in the world, or on any mission by the pope), but also to make several "promises." Among these is a promise is not to "strive or ambition" for any high office or "dignity" in the Society of Jesus or the church. This promise was intended by St. Ignatius to prevent Jesuits from the kind of clerical climbing that he found so distasteful in his time. So Jesuits are supposed to avoid all such offices.
But sometimes the Vatican will ask a Jesuit to become a bishop, or an archbishop, often in places where the church has fewer local vocations, or when the Jesuit is considered by the Vatican as an outstanding candidate for the episcopacy. When that happens the Jesuit will ask the permission of the Superior General, and it is almost always granted. (I believe that this is a courtesy; technically, the Vatican can do what it likes and ask whom it likes.) The Jesuit of course can turn down this invitation, as ordination to the episcopacy is a sacrament and cannot be coerced. If the Jesuit accepts the invitation (and he almost always does out of a desire to help the universal church) the Jesuit is then formally "dispensed" from his vows of obedience and poverty. Obedience because he obviously is not taking orders from the Superior General any longer. Poverty because under canon law a bishop must own property. (There is a promise in the final vows that the Jesuit will be open to taking advice from the General, if he offers it.)
But the man is still considered a Jesuit by tradition--if not canonically. I'll leave it to the canonists to figure that one out: I've gotten multiple responses in response to that question. However, Canon 705 states: "A religious raised to the episcopate remains a member of his institute but is subject only to the Roman Pontiff by virtue of the vow of obedience and is not bound by obligations which he himself prudently judges cannot be reconciled with his condition." And usually they themselves consider themselves as member of their orders, even after their ordination to the episcopacy. Often "Jesuit bishops," for example, will stay in the Jesuit residence in a city they are visiting, and they almost always retire in the Jesuit infirmary with their brothers, and are buried in the Jesuit cemetery.
The best "proof" of all this may be the official communique from the Jesuit Curia on the election of Pope Francis, in which the Superior General of the Society of Jesus, Father Adolfo Nicolas referred to him first as "Cardinal Jorge Mario Bergoglio, S.J.," using the traditional abbreviation for a Jesuit (S.J.) and then called him "our brother." Cardinal Bergoglio himself used the "S.J." when signing letters addressed to the Archdiocese of Buenos Aires. Also, Jesuit bishops and archbishops and cardinals are always listed on the first page of the local Jesuit catalog. Avery Cardinal Dulles, S.J., for example, was, after his creation as a cardinal by Pope John Paul II, listed on Page 1 of the New York Province catalog until his death. (He is buried in the Jesuit cemetery at Auriesville, N.Y.). When the Pope met Father General last week, Francis said that Father Nicolas should treat him like "any other Jesuit."
So yes, traditionally, the Pope is still considered a Jesuit.
And to answer two other questions that have come up frequently: Yes, the Superior General of the Society of Jesus is obedient to Pope Francis, not the other way around. In a few days, Father General will meet with the Pope Francis to "formally" to offer his own obedience, as Superiors General have done with every Pope. And no, I seriously doubt that Cardinal Bergoglio asked the permission of the Superior General to accept his election as pope; besides, he was locked away in the conclave.
Updated: This post has been updated to answer questions posed in the comments below, especially with regard to Canon 705.