The Constitution does not refer to the phrase "separation of powers", nor does it use the word "branch" to describe the three main parts of the federal government. However, those concepts are implicit in that document's language, and the independence of the judiciary is particularly important in maintaining the rule of law.
Associate Justice John Paul Stevens, the longest-serving current member of the U.S. Supreme Court, feels so strongly about that, he has spoken out against a recent trend that he perceives to be a symbolic attack on judicial independence.
Justices of the Supreme Court take two oaths of office (and not because they get it wrong the first time, as they sometimes do when administering the oath to a president). Apparently those two are the constitutional oath and the judicial oath, although I've never understood the difference.
In recent years, new justices have taken their first oath at the White House, and the second at the Supreme Court building. Stevens advocates the procedure that was used when he joined the Court in 1975, which was to take both oaths in the court's building. The president who appointed Stevens, Gerald Ford, went to the court to witness the oath, rather than expecting the new justice to come to the White House for the ceremony.
I agree with Stevens that such symbolism is important. However, I think he overestimates the degree of independence that the justices have had from presidents.
Members of the Supreme Court have accepted temporary job assignments from presidents. Chief Justice John Jay represented President Washington in the negotiations for a treaty with Great Britain in 1794. Associate Justice Robert Jackson acted as a prosecutor at the post-World War II war crimes trial at Nuremberg. And Chief Justice Earl Warren chaired the presidential commission that produced the official report on the assassination of President Kennedy.
After President Lyndon Johnson appointed Abe Fortas as an associate justice in 1965, Fortas reportedly continued to serve as an unofficial adviser to Johnson.
Is Stevens attempting to maintain a degree of purity in the separation of the branches that never really existed?