Ultra-dilute self bound droplets in Bose-Einstein condensates have been of great interest since their first prediction in 2015.The self-bound nature of these bosonic gases combined with their extremely low densities make them a prime target of investigation for both experimentalists and theoreticians.In this thesis work dipole-dipole interacting gases are simulated using the Gross-Pitaevskii equation applying a fourth order, split-operator imaginary time method in two dimensions.First, the systems are studied at collapse for dipoles aligned with a magnetic field that tilts the dipoles in the two-dimensional plane.For some angles the dipole interaction becomes partly attractive which can lead to the collapse of the mean field solution.However, quantum fluctuations can stabilize the gas.Introducing the so-called Lee-Huang-Yang (LHY) correction, the system can be stabilized and self-bound droplets may form.Here, we investigate this stabilization for a quasi two-dimensional set-up. Popular science Ultra-cold droplets intrigue physicistsA hot topic within ultra-cold atom physics has been the recent discovery of droplets formed out of Bose-Einstein condensates, a specific form of a super cold and very dilute gas.The condensate is however very different from any gas encountered in everyday life.A normal gas can be thought of as atoms or molecules moving around in all directions.If you cool this gas, all the particles move slower until they either form a liquid or freeze and become a solid.This is not the case for the Bose-Einstein condensate.Instead they spread through space due to quantum mechanics and together they form one entity.This entity, the so called condensate, behaves like no classical gas, as it is mainly governed by quantum mechanics and not by the random chance of a lot of small particles moving about.To add to this strangeness, droplets now seem to form out of these condensates.The existence of these droplets was quite a surprise.Among experimentalists working with creating Bose-Einstein condensates the expected result was a collapse of the system if forces pushing the condensate together won against the forces pushing it apart.However, experimentalists revealed that some other internal force stabilized drops inside the condensate.Not only that, this force seemed to be able to create drops that could exist outside a "trap", which usually is a number of lasers and magnets that keep the condensate together.Some droplets could not be formed without a trap, but instead formed crystal structures, a very intriguing result.This has led to an excited mood within the field of cold atom physics.But what to expect from this anti-collapse force, mainly being described as quantum fluctuations, and the droplets it creates needs further study to find out.In this thesis work we simulate the anti-collapse force and possible droplet systems solving the underlying quantum-mechanical many-particle equations numerically.One system of interest is the very thin disc of condensate, or a quasi two-dimensional condensate.This system is excellent for rotating the condensate in.Why? Imagine having a piece of pizza dough that in this case represents a condensate.If you spin this dough as a skilled pizza maker does, the pizza will become flat due to rotational forces.The same goes for the condensate.As you rotate it it will become flat anyways and therefore having a model where our very cold gas is already flat makes it easier to simulate rotating systems.But before you start rotating your drops, you have to find the drops.And to find the drops you have to study the forces pushing the condensate together and the forces pushing it apart.Therefore we presently aim at studying possible droplet formation in these pizza crust shaped condensates, to make way for the study of rotational excitations.Initial results show some promise that droplets can be found in these quasi two-dimensional systems.
Ultra-dilute self bound droplets in Bose-Einstein condensates have been of great interest since their first prediction in 2015.The self-bound nature of these bosonic gases combined with their extremely low densities make them a prime target of investigation for both experimentalists and theoreticians.In this thesis work dipole-dipole interacting gases are simulated using the Gross-Pitaevskii equation applying a fourth order, split-operator imaginary time method in two dimensions.First, the systems are studied at collapse for dipoles aligned with a magnetic field that tilts the dipoles in the two-dimensional plane.For some angles the dipole interaction becomes partly attractive which can lead to the collapse of the mean field solution.However, quantum fluctuations can stabilize the gas.Introducing the so-called Lee-Huang-Yang (LHY) correction, the system can be stabilized and self-bound droplets may form.Here, we investigate this stabilization for a quasi two-dimensional set-up. Popular science Ultra-cold droplets intrigue physicistsA hot topic within ultra-cold atom physics has been the recent discovery of droplets formed out of Bose-Einstein condensates, a specific form of a super cold and very dilute gas.The condensate is however very different from any gas encountered in everyday life.A normal gas can be thought of as atoms or molecules moving around in all directions.If you cool this gas, all the particles move slower until they either form a liquid or freeze and become a solid.This is not the case for the Bose-Einstein condensate.Instead they spread through space due to quantum mechanics and together they form one entity.This entity, the so called condensate, behaves like no classical gas, as it is mainly governed by quantum mechanics and not by the random chance of a lot of small particles moving about.To add to this strangeness, droplets now seem to form out of these condensates.The existence of these droplets was quite a surprise.Among experimentalists working with creating Bose-Einstein condensates the expected result was a collapse of the system if forces pushing the condensate together won against the forces pushing it apart.However, experimentalists revealed that some other internal force stabilized drops inside the condensate.Not only that, this force seemed to be able to create drops that could exist outside a "trap", which usually is a number of lasers and magnets that keep the condensate together.Some droplets could not be formed without a trap, but instead formed crystal structures, a very intriguing result.This has led to an excited mood within the field of cold atom physics.But what to expect from this anti-collapse force, mainly being described as quantum fluctuations, and the droplets it creates needs further study to find out.In this thesis work we simulate the anti-collapse force and possible droplet systems solving the underlying quantum-mechanical many-particle equations numerically.One system of interest is the very thin disc of condensate, or a quasi two-dimensional condensate.This system is excellent for rotating the condensate in.Why? Imagine having a piece of pizza dough that in this case represents a condensate.If you spin this dough as a skilled pizza maker does, the pizza will become flat due to rotational forces.The same goes for the condensate.As you rotate it it will become flat anyways and therefore having a model where our very cold gas is already flat makes it easier to simulate rotating systems.But before you start rotating your drops, you have to find the drops.And to find the drops you have to study the forces pushing the condensate together and the forces pushing it apart.Therefore we presently aim at studying possible droplet formation in these pizza crust shaped condensates, to make way for the study of rotational excitations.Initial results show some promise that droplets can be found in these quasi two-dimensional systems.
Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.
Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.
Among settings which have recently been elaborated for the creation of stable multidimensional solitons, most advanced, as concerns the theory and experiment alike, is the work with quantum droplets (QDs) in atomic BEC. Important theoretical predictions for fundamental and topologically structured soliton-like (stably self-trapped) QDs have been published in many papers, starting from the pioneering work of Petrov [Phys. Rev. Lett. 115, 155302 (2015)]. Simultaneously, the experimental advancement in the making of QDs in binary BECs, as well as in single-component BECs of atoms carrying magnetic dipole moments, have been really remarkable and unique. Successful experiments followed the first theoretical prediction of the QDs very quickly, and the progress in the experimental work was much faster than in other areas of physics where stable multidimensional solitons were predicted. The basis of these works is a correction to the MF (mean-field) BEC dynamics produced by quantum fluctuations around MF states. The correction was first predicted in 1957 by Lee, Huang, and Yang (LHY). The breakthrough proposal made by Petrov [Phys. Rev. Lett. 115, 155302 (2015)] was to use the effective higher-order (quartic) self-repulsion, produced by the LHY effect in the atomic BEC, as a mechanism arresting the collapse driven by mutual attraction of two self-repulsive components in a binary quantum bosonic gas. The balance between the inter-species attraction and LHY-induced self-repulsion gives rise to stable 3D and quasi-2D (pancake-shaped) localized bound states in the form of the quantum droplets (QDs). In single-component condensates of magnetic atoms, the attraction is provided by the long-range dipole–dipole interaction (Chomaz et al., 2016; Schmitt et al., 2016). The competition between the MF attraction and LHY repulsion maintains a superfluid state whose density (taking very low values) cannot exceed a certain maximum, thus making it effectively incompressible. This is a reason why this quantum macroscopic state of matter is identified as a superfluid. Accordingly, localized states filled by this quantum fluid are called “droplets.” In addition to the 3D QDs, droplets in the effectively 2D setting, which may exist under the action of tight confinement in one direction, imposed by an external potential, were theoretically predicted [Petrov and Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016)] and experimentally created [Cabrera et al., Science 359, 301-304 (2018)] too. This chapter provides a condensed review of basic theoretical results for stable QDs in binary BECs, maintained through the stable balance between the inter-species attraction and intra-species LHY repulsion. Both fundamental modes and ones with embedded vorticity are considered. The presentation is partly based on a review article by Luo et al. [Front. Phys. 16, 32501 (2021)]. In addition to 2D and 3D self-trapped QDs in free space, the present chapter addresses 2D fundamental and vortex modes trapped in a singular attractive potential, ∼−1/r2, which are stabilized by the LHY repulsion. Two subsequent chapters present a summary of experimental findings for QDs in binary BECs and for QDs in single-component BECs of magnetic atoms.
Recent observations of droplets in dipolar and binary Bose-Einstein condensate (BEC) motivates us to study the theory of droplet formation in details. Precisely, we are interested in investigating the possibility of droplet formation in a quasi one dimensional geometry. The recent observation have concluded that the droplets are stabilized by the competition between effective mean-filed and beyond mean-field interaction. Hence, it is possible to map the effective equation of motion to a cubic-quartic nonlinear Schr\odinger equation (CQNLSE). We obtain two analytical solutions of the modified Gross-Pitaevskii equation or CQNLSE and based on their stability we investigate the parameter regime for which droplets can form. The effective potential allows us to conclude about the regions of soliton domination and self bound droplet formations.
We predict a scheme for the creation of isotropic three-dimensional quantum droplets in Rydbeg-dressed Bose gases, which contain both repulsive contact interactions and attractive soft-core interactions. Via manipulating laser detuning, the droplet size and particle number density could be engineered. Quantum fluctuation induced Lee-Huang-Yang (LHY) correction should be considered to stabilize droplets when LHY energy is comparable to the mean-field energy. Self-trapped droplets possess flat-top density distribution for a large particle number, and the collective excitations are calculated to determine the self-evaporation threshold. We further distinguish quantum droplets from solitons according to condensate compressibility. Finally, the dynamic stability of droplets is examined by pairwise-droplet collision. Our system could realize the most dilute quantum droplets, and establish a promising platform for further simulating topological nontrivial droplets. Published by the American Physical Society 2024
This thesis presents a numerical study of the beyond mean field extended Gross-Pitaevskii equation for a two-component self-bound Bose-Einstein condensate, supplemented by analytical work. We extend the quasi-two-dimensional system's mean field particle-particle contact interaction by quantum fluctuations (LHY-correction). We investigate the formation of self-bound droplets without external confinement and confirm their ground-state stability. Furthermore, we examine and discuss the real-time stability and dynamics of vortices in self-bound droplets and find a size dependence for stability and their dynamics of their release from a confined trap.