  A self-consistent theoretical treatment of a triple-well atomtronic
transistor circuit reveals the mechanism of gain, conditions of oscillation,
and properties of the subsequent coherent matterwaves emitted by the circuit. A
Bose-condensed reservoir of atoms in a large source well provides a chemical
potential that drives circuit dynamics. The theory is based on the ansatz that
a condensate arises in the transistor gate well as a displaced ground state,
that is, one that undergoes dipole oscillation in the well. That gate atoms
remain condensed and oscillating is shown to be a consequence of the cooling
induced by the emission of a matterwave into the vacuum. Key circuit parameters
such as the transistor transconductance and output current are derived by
transitioning to a classical equivalent circuit model. Voltage-like and
current-like matterwave circuit wave fields are introduced in analogy with
microwave circuits, as well as an impedance relationship between the two. This
leads to a new notion of a classical coherent matterwave that is the dual of a
coherent electromagnetic wave and which is distinct from a deBroglie matterwave
associated with cold atoms. Subjecting the emitted atom flux to an atomic
potential that will reduce the deBroglie wavelength, for example, will increase
the classical matterwave wavelength. Quantization of the classical matterwave
fields leads to the dual of the photon that is identified not as an atom but as
something else, which is here dubbed a "matteron".

Quantum Physics;Applied Physics;Atomic Physics