Periodic Reporting for period 1 - FLAC (Fluctuations in Atomtronic Circuits) Reporting period: 2020-12-15 to 2022-12-14 Summary of the context and overall objectives of the project Ultracold quantum gases trapped in magnetic and optical traps are an ideal highly-controllable system for precision measurements and quantum-technological applications. In recent years this has led to the emergence of the field of `atomtronics' loosely associated with the dynamics of ultracold atomic gases in closed `circuit' geometries, aimed at understanding fundamental dynamics in such closed geometries, and how to use them for potential quantum-technological applications. The aim of this project was to use state-of-the-art numerical techniques and models to understand dynamics in a range of atomtronics systems.Atomtronics-based quantum technologies are gradually emerging as an alternative avenue of precision measurements and sensors, which could e.g. provide distinct devices measuring accelaration (including gravity), or rotation. Such devices, may operate (be more sensitive) in regimes where current devices are not idealy-suited, so they could potentially provide more flexibility and better accuracy than existing/established devices, or alternative measurement schemes, in the decades to come.The first project was associated with understanding in detail, optimizing and generalizing the atomic analogue of a superconducting quantum interference device, a device used as a quantum sensor (e.g. a magnetometer) The intention was to analyze experiments on such device, and try to optimize their regimes of operation, in close contact to relevant experimental groups. Following completion of that, the intention was to characterize open issues in the cooling and generation of atomtronic transistors, focussing on issues ranging from fundamental understanding of the formation of coherence (or Bose-Einstein condensation) in such systems, to how this can be controlled in actual experimentally-relevant devices. Finally, the intention was to combine such features for the understanding of `transistor-like' devices involving coupled multiple closed ring geometries containing ultracold (coherent) quantum gases.The aim was to also perform such tasks within secondments to Crete and Paris: the first one kept being delayed due to the COVID situation and labs being closed to external visitors; the second one was meant for later on in the project -- but the Fellow moved on to another post before reaching such time when secondments could be facilitated. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The first task in this ambitious project was to provide a more in-depth understanding of an atomtronic quantum interference device (AQUID), and to this aim we undertook to analyse experiments done at Los Alamos in the US (Malcolm Boshier group).In their latest (2020) work, this group demonstrated experimentally the observation of quantum interference of currents in an atomtronic quantum interference device studying the flow of neutral atomic current through two `junctions' embedded within a closed ring (doughnut-shaped) geometry, and paved the way for using such devices for quantum metrology, or rotational sensing. This experiment had studied such dynamics and characterised it in terms of in-house numerical simulations. The experiment was three-dimensional, but the simulations had only been performed in two dimensions. Although very good agreement was observed in general, there were some experimental observations that were still not fully understood, and we set out to study those.While trying out different dynamical protocols to see the motion, we studied different rotation protocols and the role of adiabaticity in initiating the sensor. It turns out that the more three-dimensional the system is, the more sensitive it is to the particular dynamical initiation scheme. Experimental results were well reproduced by us again in two dimensions, but only qualitatively so in three-dimensional systems (differing slightly in the critical atom number dependence on external rotation as a function of transverse confinement -- which merits further integrated experimental-theoretical study): this is even though we found no clear evidence of strong excitations in the third direction. Our analysis indicated the potential existence of an intermediate/transitional regime (at "early" dynamical evolution times of up to few 100 ms) between the different dynamical regimes, indicating an early-time sensitivity of the atomic dynamics across the junctions which depended sensitively on dimensionality (how close to two dimensions the system actually was, with more three-dimensional systems exhibiting a much richer, qualitatively different, atomic population difference dynamics). While this is not a limiting factor to such future sensors, they do indicate the need both for careful benchmarking and for carefully-tailored geometries and atom number combinations, as such features seemed to be largely absent in (effectively) two-dimensional geometries.We also generalised our model to include stochastic fluctuations (via the so-called stochastic Gross-Pitaevskii equation) and found that the inclusion of noise in the dynamical equation could partly explain observed discrepancies between experimental findings (dependence of critical atom number on external rotation) and simple mean-field predictions. This looked like a highly-promising area of research -- unfortunately the work was terminated ahead of schedule due to the Fellow moving to another academic position abroad prior to completion of the manuscript we were working on. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) This was the first time that such atomtronic quantum interference devices had been numerically/theoretically studied either in a full three-dimensional setting, or with the addition of noise mimicking the randomness of atomic collisions. and quantum and thermal fluctuations, typically present (even if not dominant) in all experimental devices. After correctly initializing the system dynamics, we identified distinct atomic population oscillation modes, consistent with an intermediate dynamical (transient) regime. Such features, which still require further investigation, were present even in the pure mean field limit. Addition of stochastic noise seemed indeed to improve numerical predictions in three dimensions in relation to the experiment, with the randomisation originating from such fluctuations seemingly enough to decrease the critical atom number with increasing rotation (compared to the case where external rotation is absent). This is a potentially significant result, which may lead to a stringent requirement of the purity of the quantum gas in future atomtronics quantum interference devices, but more work still needs to be done to fully characterize such potential limitations.The early Fellowship termination due to the fellow accepting another job offer abroad did not allow such matters to be investigated to publication level. Atomic profile in a ring-trap ultracold quantum interference device.