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

Quantum ultra-cold atomtronics