  It is known that a semi-classical analysis is not always adequate for
atomtronics devices, but that a fully quantum analysis is often necessary to
make reliable predictions. While small numbers of atoms at a small number of
sites are tractable using the density matrix, a fully quantum analysis is often
not straightforward as the system becomes larger. We show that the fully
quantum positive-P representation is then a viable calculational tool. We
postulate an atomtronic phase-gate consisting of four wells in a Bose-Hubbard
configuration, for which the semi-classical dynamics are controllable using the
phase of the atomic mode in one of the wells. We show that the quantum
predictions of the positive-P representation for the performance of this device
have little relation to those found semi-classically, and that the performance
depends markedly on the actual quantum states of the initially occupied modes.
We find that initial coherent states lead to closest to classical dynamics, but
that initial Fock states give results that are quite different. A fully quantum
analysis also opens the door for deeply quantum atomtronics, in which
properties such as entanglement and EPR (Einstein-Podolsky-Rosen) steering
become valuable technical properties of a device.
