This theoretical project concerns ultracold gases in which atoms interact via both non-local and short-range interactions. In the situation when these two types of interactions are almost canceling each other, one can expect new quantum phases such as recently observed quantum droplets or this year's result - supersolid state. In task 1 of the project is to find the quantum phase diagram in abstract models, in which the ground states for various types of non-local interactions will be examined. Then we intend to investigate which areas of the phase diagram are available in experiments with dipolar atoms and, atoms in resonant cavities, which have not yet been studied in this context. In the task 2 we will deal with finding elementary and collective excitations. The last part of the project is to investigate the decay of the novel phases as a result of heating and losses of particles. One of the goals of this third task is to find characteristic temperatures at which quantum phases disappear in favour of a typical thermal gas. All problems will be studied by two complementary methods. We are going to combine the many-body quantum approach with methods emerging from the theory of the mean-field, such as the generalized Gross-Pitaevski equation with Lee-Huang-Yang's corrections and the recently proposed model emerging from the local density approximation and the Lieb-Liniger model [1]. The results of the parallel study will allow for benchmarks, but they will also be used to understand the relations between the many-body phenomena and the semi-classical picture of a single-particle moving in the mean-field of other atoms. The many-body theory is necessary to examine the correlations between particles, in particular phase coherence and spatial correlations between atoms. These measurable quantities will detect the presence of a supersolid. In addition to standard calculation methods such as DMRG, we will use a method developed by us that combines the Bogoliubov approximation with a full quantum model [2]. during this five-years project, we will try to learn the new methods, based on machine learning. According to the recent publications, the neural-network-based methods are proving to be a serious competitor to the calculation methods used so far. On the other hand, the advantage of the approximate single-particle methods is the possibility to perform calculations for large systems containing hundreds of thousands of atoms. Using the approximate methods it is possible to study collective excitations, vortices and solitons. The next step will be to use these equations to study a quantum gas at finite temperature using the classical field method. It will give insight into statistical properties - the size of quantum droplets or supersolid or possible new phases which we will find as a function of temperature. Observing new unexpected states of matter, quantum droplets, revolutionizes the research on ultracold gases. It suddenly turned out that the system which seemed to be well known hides fundamentally different phases than the Bose-Einstein condensate. Moreover, these new phases are stable thanks to quantum fluctuations resulting from many-body physics. This puts scientists in a situation analogous to when the first Bose-Einstein condensate was observed. At this point it is necessary to examine these new phases, to search for them in other physical systems, to find a practical application for them. [1] W. Golletz, W. Górecki, R. Ołdziejewski, and K. Pawłowski Dark solitons revealed by particle losses in the 1D Bose gas, arXiv:1905.04604 (2019) [2] R. Ołdziejewski, W. Górecki, K. Pawłowski, and K. Rzążewski. Strongly correlated quantum droplets in quasi-1D dipolar Bose gas, arXiv:1908.00108 (2019)
