The operation of a quantum computer is inherently nonlocal due to the
entanglement of separated quantum gates. This paper is concerned inste
ad with the use of nonlocal effects to construct the quantum gates the
mselves. A nonlocal interaction in which pairs of atoms exchange two p
hotons is expected to produce a large nonlinear phase shift. An intere
sting feature of this effect is that energy is not conserved at the lo
cation of an individual atom but it is conserved overall. The prelimin
ary results from a series of experiments in a sodium vapor cell are in
apparent agreement with the predictions of the theory. These nonlinea
r phase shifts can be used to construct quantum logic gates, such as a
Controlled-NOT.