Objectives

In 2015 the group of researchers from University of Stuttgart led by Tilman Pfau observed, to their surprise, quantum droplets of Dysprosium atoms. These are self-bound very dilute objects which can freely levitate in space in a gradient of magnetic field compensating for the gravity. In contrast to all known ultracold atomic systems, the quantum droplets do not need any external potential to provide their stability. This project is devoted to theoretical studies of these unusual objects.

Quantum droplets are totally new systems of not known properties. They are unique macroscopic objects and their stability is determined by quantum correlations and fluctuations. Energy of quantum fluctuations strongly depends on density of states which, in turn, depends on the space dimensionality. Therefore physical properties of droplets depend on the system geometry. We plan to investigate mainly droplets formed in Bose-Bose mixtures. We want to gain knowledge about their static and dynamic properties such like: collective excitations, vortices and solitons in droplets, coherence and collisions of droplets. We plan to study dynamic of creation of droplets, their thermal properties but also expect to discover new self-bound ultracold systems, in particular we want to investigate a possiblity of droplets formation in Bose-Fermi mixtures.

Methodology

Stability of droplets is related to beyond mean-field effects. In order to account for these effects the extended Gross-Pitaevskii equation was suggested. A term describing energy of Bogoliubov vacuum is incorporated into the mean field description assuming a local density approximation. It allows to study droplets within well established formalism based on the time dependent Gross-Pitaevskii equation. In a case of a two component mixture droplets can be described by a set of two coupled equations. Extended Gross-Pitaevskii equation will be the working horse of our studies.

Expected impact

We expect that our studies of fundamental properties of droplets will have a significant influence on quantum simulations. Quantum droplets should enrich a tool-box of quantum engineers. They provide a unique possibilities of controlling very subtle effects beyond the mean field approximation – quantum fluctuations and correlations. They should also bench-mark all models accounting for quantum fluctuations. Quantum droplets because of their coherence and superfluid character can find applications in interferometry and precision measurements. Because of their excitation spectrum, they might be also a very useful coolants for other quantum systems.

Publications:

**Self-Consistent Description of Bose-Bose Droplets: Harmonically Trapped Quasi-2D Droplets**

Paweł Zin, Maciej Pylak, and Mariusz Gajda*Phys. Rev. A* **106**, 013320 (2022)**Self-Consistent Description of Bose-Bose Droplets: Modified Gapless Hartree-Fock Bogoliubov Method**

Paweł Zin, Maciej Pylak, Zbigniew Idziaszek, Mariusz Gajda*New J. Phys.*** 24** 113038 (2022)**Atoms in a spin dependent optical potential: ground state topology and magnetization**

Piotr Szulim, Marek Trippenbach, Yehuda Band, Mariusz Gajda, Mirosław Brewczyk*New J. Phys.***24** 033041 (2022)**Manifestation of relative phase in dynamics of two interacting Bose-Bose droplets**

Maciej Pylak, Filip Gampel, Marcin Płodzień, and Mariusz Gajda*Phys. Rev. Research* **4**, 013168 (2022)**Zero-energy modes of two-component Bose-Bose droplets**

Paweł Zin, Maciej Pylak, Mariusz Gajda*New J. Phys. ***23** 033022 (2021)**Pauli Crystals-Interplay of Symmetries**

Mariusz Gajda, Jan Mostowski, Maciej Pylak, Tomasz Sowiński and Magdalena Załuska-Kotur*Symmetry* **2020 **12(11) 1886 (2020)**Bistability of Bose-Fermi mixtures **

Tomasz Karpiuk, Mariusz Gajda, Mirosław Brewczyk*New J. Phys.* **22** 103025 (2020)**Revisiting a stability problem of two-component droplets **

Paweł Zin, Maciej Pylak, Mariusz Gajda*Phys. Rev. A* **103**, 013312 (2021)**Self-bound Bose–Fermi liquids in lower dimensions **

Debraj Rakshit, Tomasz Karpiuk, Paweł Zin, Mirosław Brewczyk, Maciej Lewenstein and Mariusz Gajda*New J. Phys.*** 21**, 073027 (2019)**Quantum Bose-Fermi droplets **

Debraj Rakshit, Tomasz Karpiuk, Mirosław Brewczyk and Mariusz Gajda*SciPost Phys.* **6**, 079 (2019)**Modelling quantum aspects of disruption of a white dwarf star by a black hole **

Tomasz Karpiuk, Marek Nikołajuk, Mariusz Gajda, and Mirosław Brewczyk*Scientific Reports ***11*** *2286 (2021)**Quantum Bose-Bose droplets at a dimensional crossover**

Paweł Zin, Maciej Pylak, Tomasz Wasak, Mariusz Gajda and Zbigniew Idziaszek*Phys. Rev. A* **98**, 051603(R) (2018)

The project is supported by the National Science Centre (NCN), grant no. UMO-2017/25/B/ST2/01943.