We systematically investigate the properties of three-dimensional dipolar binary Bose mixture at low temperatures. A set of coupled self-consistent equations of motion are derived for the two condensates. In the homogeneous case, useful analytical formulas for the condensate depletion, the anomalous density, the ground-state energy, and the equation of state are obtained. The theory is extended to the inhomogeneous case and the importance of the inhomogeneity is highlighted. Our results open up a new avenue for studying dipolar mixture droplets. Impacts of the dipole-dipole interaction on the stability, density profiles, and the size of the self-bound droplet are deeply discussed. The finite-temperature behavior of such a state is also examined.
In this paper we develop a theory for an infinitely long droplet state of a zero temperature dipolar bosonic gas. The infinite droplet theory yields simpler equations to solve for the droplet state and its collective excitations. We explore the behavior of infinite droplets using numerical and variational solutions, and demonstrate that it can provide a quantitative description of large finite droplets of the type produced in experiments. We also consider the axial speed of sound and the thermodynamic limit of a dipolar droplet.
We describe and benchmark a method to accurately calculate the quantum droplet states that can be produced from a dipolar Bose-Einstein condensate. Our approach also allows us to consider vortex states, where the atoms circulate around the long-axis of the filament shaped droplet. We apply our approach to determine a phase diagram showing where self-bound droplets are stable against evaporation, and to quantify the energetics related to the fission of a vortex droplet into two non-vortex droplets.
The long-range and anisotropic nature of the dipolar interaction provides the so-called supersolid phases in Bose-Einstein condensates (BECs) in an optical lattice. However, in a certain area of dipole interaction parameters, BECs can form into a droplet. In this paper, in order to qualitatively understand the droplet formations, we propose a toy model that allows us to estimate the size and shape of droplets in dipolar Bose-Hubbard system in the optical lattice. We compare results of the toy model with numerical solutions of the mean-field calculation.
Stabilized by quantum fluctuations, dipolar Bose-Einstein condensates can form self-bound liquidlike droplets in the mean-field unstable regime. However in the Bogoliubov theory, some phonon energies are imaginary in the long-wavelength limit, implying dynamical instability of this system. A similar instability appears in the Bogoliubov theory of a binary quantum droplet, and is removed due to higher-order quantum fluctuations as shown recently [1]. In this work, we study the phonon energy of a dipolar quantum droplet in the Beliaev formalism, and find that quantum fluctuations can enhance the…
Recently achieved two-component dipolar Bose-Einstein condensates open exciting possibilities for the study of mixtures of ultra-dilute quantum liquids. While non-dipolar self-bound mixtures are necessarily miscible with an approximately fixed ratio between the two densities, the density ratio for the dipolar case is free. As a result, self-bound dipolar mixtures present qualitatively novel and much richer physics, characterized by three possible ground-state phases: miscible, symmetric immiscible and asymmetric immiscible, which may in principle occur at any population imbalance. Self-bound i…
We investigate the effects of quantum correlations on dipolar quantum droplets. To this end,we derive self-consistent time-dependent Hartree-Fock-Bogoliubov equations that fairly describe the dynamics of the order parameter, the normal, and anomalous quantum correlations of the droplet. We analyze the density profiles, the critical number of particles, the condensate depletion, and the pair correlation function. Our predictions are compared with very recent experimental and Quantum Monte-Carlo simulations results and excellent agreement is found.
The ground state of a free standing, self-bound droplet comprising four hundred dipolar Bose particles with aligned dipole moments, with an additional purely repulsive two-body interaction, is investigated by Quantum Monte Carlo simulations. The focus here is on the evolution of the cluster as the effective range of the repulsive interaction is varied. We identify a ``classical'' regime, in which binding arises exclusively from the dipolar potential energy and the cluster is a quasi-one-dimensional filament, and a ``quantum'' regime of prolate droplets, held together to a significant degree by…
Using a beyond-mean-field model including a Lee-Huang-Yang-type interaction, we demonstrate a supersolid-like spatially-periodic square- and triangular-lattice crystallization of droplets in a polarized dipolar condensate confined by an appropriate three-dimensional (3D) box trap. In this paper we consider a rectangular box (cuboid) trap, a square box (cuboid with two equal sides) trap, a cylindrical box trap and a hexagonal box (hexagonal prism) trap. The droplet lattice is always formed in the $x$-$y$ plane perpendicular to the polarization $z$ direction of dipolar atoms. In contrast to a ha…
We theoretically investigate the droplets formation in a tightly trapped one-dimensional dipolar gas of bosonic atoms. When the strength of the dipolar interaction becomes sufficiently attractive compared to the contact one, we show how a solitonic-like density profile evolves into a liquid-like droplet on increasing the number of particles in the trap. The incipient gas-liquid transition is also signalled by a steep increase of the breathing mode and a change in sign of the chemical potential. Upon a sudden release of the trap, varying the number of trapped atoms and the scattering length, th…
We look into dark solitons in a quasi-1D dipolar Bose gas and in a quantum droplet. We derive the analytical solitonic solution of a Gross-Pitaevskii-like equation accounting for beyond mean-field effects. The results show there is a certain critical value of the dipolar interactions, for which the width of a motionless soliton diverges. Moreover, there is a peculiar solution of the motionless soliton with a non-zero density minimum. We also present the energy spectrum of these solitons with an additional excitation subbranch appearing. Finally, we perform a series of numerical experiments rev…
We study the beyond-mean-field corrections to the energy of a dipolar Bose gas confined to two dimensions by a box potential with dipoles oriented in plane. At a critical strength of the dipolar interaction the system becomes unstable on the mean field level. We find that the ground state of the gas is strongly influenced by the corrections, leading to formation of a self-bound droplet, in analogy to the free space case. Properties of the droplet state can be found by minimizing the extended Gross-Pitaevskii energy functional. In the limit of strong confinement we show analytically that the co…
We study the effect of beyond-mean-field quantum-fluctuation (QF) Lee-Huang-Yang (LHY) and three-body interactions, with quartic and quintic nonlinearities, respectively, on the formation of a stable self-repulsive (positive scattering length $a$) and a self-attractive (negative $a$) self-bound dipolar Bose-Einstein condensate (BEC) droplet in free space under the action of two-body contact and dipolar interactions. Previous studies of dipolar droplets considered either the LHY interaction or the three-body interaction, as either of these interactions could stabilize a dipolar BEC droplet agai…
Gases of doubly-dipolar particles, with both magnetic and electric dipole moments, offer intriguing novel possibilities. We show that the interplay between doubly-dipolar interactions, quantum stabilization, and external confinement results in a rich ground-state physics of supersolids and incoherent droplet arrays in doubly-dipolar condensates. Our study reveals novel possibilities for engineering quantum droplets and droplet supersolids, including supersolid-supersolid transitions and the realization of supersolid arrays of pancake droplets.
We describe a quantum droplet of a Bose-Bose mixture squeezed by an external harmonic forces in one spatial direction. Our approach is based on the self-consistent method formulated in [1]. The true spatial droplet profile in the direction of confinement is accounted for, however local density approximation is assumed in the free directions. We define a numerical approach to find the beyond-mean-field contribution to the chemical potential (Lee-Huang-Yang chemical potential) -- the quantity that determines the droplet's profile. In addition to the numerical approach, we find the Lee-Huang-Yang…
Recent breakthrough experiments on dipolar condensates have reported the creation of supersolids, including two-dimensional arrays of quantum droplets. Droplet arrays are, however, not the only possible non-trivial density arrangement resulting from the interplay of mean-field instability and quantum stabilization. Several other possible density patterns may occur in trapped condensates at higher densities, including the so-called honeycomb supersolid, a phase that exists, as it is also the case of a triangular droplet supersolid, in the thermodynamic limit. We show that compared to droplet su…
Breakthrough experiments have newly explored the fascinating physics of dipolar quantum droplets and supersolids. The recent realization of dipolar mixtures opens further intriguing possibilities. We show that under rather general conditions, the presence of a second component catalyzes droplet nucleation and supersolidity in an otherwise unmodulated condensate. Droplet catalyzation in miscible mixtures, which may occur even for a surprisingly small impurity doping, results from a local roton instability triggered by the doping-dependent modification of the effective dipolar strength. The cata…
Ultracold dipolar droplets have been realized in a series of ground-breaking experiments, where the stability of the droplet state is attributed to beyond-mean-field effects in the form of the celebrated Lee-Huang-Yang (LHY) correction. We scrutinize the dipolar droplet states in a one-dimensional context using a combination of analytical and numerical approaches, and identify experimentally viable parameters for accessing our findings for future experiments. In particular we identify regimes of stability in the restricted geometry, finding multiple roton instabilities as well as regions suppo…
We study coherent dynamics of two interacting Bose-Bose droplets by means of the extended Gross-Pitaevskii equation. The relative motion of the droplets couples to the phases of their components. The dynamics can be understood in terms of the evolution of zero-energy modes recovering symmetries spontaneously broken by the mean-field solution. These are translational symmetry and two U(1) symmetries, associated with the phases of the droplets' two components. A phase-dependent interaction potential and double Josephson-junction equations are introduced to explain the observed variety of differe…
We systematically study the effects of higher-order quantum and thermal fluctuations on the stabilization of self-bound droplets in Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. We calculate the ground-state energy, the droplet equilibrium density, the depletion and anomalous density of the droplets as well as the critical temperature as a function of the relevant parameters. Our findings are compared with previous analytical predictions and diffusion Monte Carlo simulations. We employ our theory together with the local density approximation for quantum and thermal…
We study the equilibrium properties of self-bound droplets in two-dimensional Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. This theory allows one to understand both the many-body and temperature effects beyond the Lee-Huang-Yang description. We calculate higher-order corrections to the excitations, the sound velocity, and the energy of the droplet. Our results for the ground-state energy are compared with the diffusion Monte Carlo data and good agreement is found. The behavior of the depletion and anomalous density of the droplet is also discussed. At finite tempe…
