Jesuits have always had a special relationship to the pope. It started when St. Ignatius was not able to follow his dream of preaching the Gospel to Muslims in the Holy Land. When he arrived in Jerusalem without a cent in his pocket, the Franciscans shook their heads and told him to go home. Thank God for the Franciscans! They were in charge of the Holy Land and obviously had more sense than he did.
That left Ignatius with Plan B: putting himself and his companions at the service of the pope to be sent wherever he desired. This is the origins of the "fourth vow" taken by professed Jesuits along with their vows of poverty, chastity and obedience. The text of the vow reads: "I further promise a special obedience to the Sovereign Pontiff in regard to the missions according to the same apostolic letters and Constitutions."
So what happens to a Jesuit who becomes pope? Does he look in the mirror and tell himself what he should do each morning? Is he still a Jesuit?
I really doubt that he talks to himself, but there is no question that he is still a Jesuit. While canon law does not say anything specific about a religious who becomes a pope, it is clear that a religious who becomes a bishop is still a member of his religious community but in a different way. The pertinent canons applied to Bergoglio when he first became a bishop, and they apply to him now as the bishop of Rome.
Canon 705 is clear: "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." In other words, any Jesuit bishop, including the bishop of Rome, is still a Jesuit but he does not have to follow orders of any Jesuit superior, even the Jesuit superior general. A Jesuit who does not have to follow the orders of a Jesuit superior is certainly a strange bird, but he is still a Jesuit.
Canon 706 provides that any "goods" (property, money, etc.) acquired by a professed religious who is a bishop become the property of the diocese or the Holy See, not his religious community. And when he retires, he can live with his religious community or separately. If he lives separately, the diocese has an obligation to support him in his retirement.
Although Jesuits take a special vow not to seek higher office, there have been scores of Jesuit bishops. Ignatius did not want Jesuits to be bishops because he did not want to lose his best men to the episcopate and he did not want the Society corrupted by careerism. Popes have appointed Jesuits as bishops anyway, especially in missionary countries. John Paul went further and appointed them to major sees, like Milan (Cardinal Carlo Maria Martini) and Buenos Aires.
Jesuit bishops have been rare in the United States even though the very first American bishop, John Carroll, was a Jesuit. Today there are three Jesuit bishops in the U.S., two retired and one active in Youngstown, Ohio.
So, is it Pope Francis, S.J.? Well, the pope can do whatever he wants, but traditionally, popes do not put any letters after their names. No Ph.D., S.T.D., J.C.D., etc. They don't even have a last name!
More important than the letters after his name, Pope Francis brings with him the Jesuit spirituality that unites a person with Christ in his mission of preaching the Gospel and building the kingdom of God, a kingdom of love, justice and peace. In the first week of the Spiritual Exercises of St. Ignatius, the retreatant experiences the mercy of God, a theme Pope Francis has repeatedly stressed during his first week as bishop of Rome. After experiencing God’s mercy and love, Jesuit spirituality asks one to be open to the Spirit, which can always surprise us, and Francis has certainly surprised people. There is also a practical side to Jesuit spirituality—if one thing does not work, try something else. This will also help him as he faces the daunting tasks before him.