Self-trapped droplets stabilized by quantum fluctuations have been experimentally realized in dipolar gases and binary Boson mixtures. We propose spinor Bose gases as another candidate for droplet formation in this work. For spin-1 gas, we find that spin fluctuations give a dilute but self-trapped state for two different order parameters where the mean-field picture predicts collapse. A polar droplet phase can be stabilized by spin fluctuations for both antiferromagnetic and ferromagnetic spin-dependent coupling. An antiferromagnetic droplet phase can be stabilized similarly with a negative quadratic Zeeman shift. Furthermore, the beyond mean-field energy of the system depends on the quadratic Zeeman coupling, which provides a mechanism to tune the droplet formation and its density. We discuss the parameters necessary for the experimental realization of such spinor droplets.
Self-bound quantum droplets form when the mean-field tendency of the gas to collapse is stabilized by the effectively repulsive beyond mean-field fluctuations. The beyond mean-field effects depend on Rabi-frequency $ω_R$ and quadratic Zeeman effect $q$ for the Rabi-coupled Bose mixtures and the spinor gases, respectively. The effects of varying $ω_R$ and $q$ on the quantum droplet have recently been examined for unpolarized Rabi-coupled Bose mixture with zero detuning $δ= 0$ and unpolarized spinor gas with $\langle F_z \rangle=0$. In this paper, we theoretically explore the stability of the dr…
Motivated by the recent experiments [H. Kadau et al., Nature (London) 530, 194 (2016); I. Ferrier-Barbut et al., arXiv:1601.03318] and theoretical prediction (F. Wächtler and L. Santos, arXiv:1601.04501), the ground state of a dysprosium Bose-Einstein condensate with strong dipole-dipole interaction is studied using the path-integral Monte Carlo method. It is shown that quantum fluctuation can stabilize the condensate against dipolar collapse.
Liberal Study College, Beijing Institute of Petrochemical Technology（北京石油化工学院文理学院）；Shaanxi Key Laboratory for Theoretical Physics Frontiers, Institute of Modern Physics, Northwest University（西北大学现代物理研究所）；School of Physics and Information Science, Shaanxi University of Science and Technology（陕西科技大学物理与信息科学学院）；Tsientang Institute for Advanced Study（杭州钱塘高等研究院）；Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所凝聚态物理国家实验室）；Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University（特拉维夫大学工程学院电气工程系物理电子学系）；Instituto de Alta Investigación, Universidad de Tarapacá（塔拉帕卡大学高级研究所）
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…
We calculate the collective excitations of a dipolar Bose-Einstein condensate in the regime where it self-binds into droplets stabilized by quantum fluctuations. We show that the filament-shaped droplets act as a quasi-one-dimensional waveguide along which low angular momentum phonons propagate. The evaporation (unbinding) threshold occurring as the atom number $N$ is reduced to the critical value $N_c$ is associated with a monopole-like excitation going soft as $ε_0\sim(N-N_c)^{1/4}$. Considering the system in the presence of a trapping potential, we quantify the crossover from a trap-bound c…
We address the existence and stability of one-dimensional (1D) holes and kinks and two-dimensional (2D) vortex-holes nested in extended binary Bose mixtures, which emerge in the presence of Lee-Huang-Yang (LHY) quantum corrections to the mean-field energy, along with self-bound quantum droplets. We consider both the symmetric system with equal intra-species scattering lengths and atomic masses, modeled by a single (scalar) LHY-corrected Gross-Pitaevskii equation (GPE), and the general asymmetric case with different intra-species scattering lengths, described by two coupled (spinor) GPEs. We fo…
We study the formation of quantum droplets in the mixture of a single-component Bose-Einstein condensate (BEC), and a two-species Fermi superfluid across a wide Feshbach resonance. With repulsive boson-boson and attractive boson-fermion interactions, we show that quantum droplets can be stabilized by attractive fermion-fermion interactions in the Bardeen-Cooper-Schieffer (BCS) side of the resonance, and can also exist in deep BEC regime under weak boson-fermion interactions. We map out the phase diagram for stable droplets with respect to the boson-boson and boson-fermion interactions, and dis…
Quantum droplets have been realized in experiments on binary boson mixtures and dipolar Bose gases. In these systems, the mean-field energy of the Bose-Einstein condensation is attractive, and the repulsive Lee-Huang-Yang energy is crucial for stability. The Bogoliubov theory incorrectly predicts that the phonon mode is dynamically unstable in the long-wavelength limit. In this work, we go beyond the Bogoliubov theory to study how the phonon mode is stabilized in the quantum droplet of a binary boson mixture. Similar to Beliaev's approach to a single-component Bose gas, we compute higher-order…
Ultracold atoms offer valuable opportunities where interparticle interactions can be controlled at will. In particular, by extinguishing the two-body interaction, one can realize unique systems governed by the three-body interaction, which is otherwise hidden behind the two-body interaction. Here we study one-dimensional bosons with a weak three-body attraction and show that they form few-body bound states as well as a many-body droplet stabilized by the quantum mechanical effect. Their binding energies relative to that of three bosons are all universal and the ground-state energy of the dilut…
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 show that attractive spinor Bose-Einstein condensates under the action of spin-orbit coupling (SOC) and Zeeman splitting form self-sustained stable two- and three-dimensional (2D and 3D) states in free space, even when SOC acts in a lower-dimensional form. We find that two-dimensional states are stabilized by one-dimensional (1D) SOC in a broad range of chemical potentials, for atom numbers (or norm of the spinor wavefunction) exceeding a threshold value, which strongly depends on the SOC strength and vanishes at a critical point. The zero-threshold point is a boundary between single-peaked…
Droplet states of ultracold gases which are stabilized by fluctuations have recently been observed for dipolar and two component Bose gases. These systems present a novel form of equilibrium where an instability at the mean field level is arrested by higher order correlations making the droplet states sensitive probes of fluctuations. In a recent paper, we argued that thermal fluctuations can play an important role for droplets even at low temperatures where the non-condensed density is much smaller than the condensate density. We used the Hartree-Fock-Bogoliubov theory together with local den…
We revisit the Bogoliubov theory of quantum droplets proposed by Petrov {[}Phys. Rev. Lett. \textbf{115}, 155302 (2015){]} for an ultracold Bose-Bose mixture, where the mean-field collapse is stabilized by the Lee-Huang-Yang quantum fluctuations. We show that a loophole in Petrov's theory, i.e., the ignorance of the softening complex Bogoliubov spectrum, can be naturally removed by the introduction of bosonic pairing. The pairing leads to weaker mean-field attractions, and also stronger Lee-Huang-Yang term in the case of unequal intraspecies interactions. As a result, the equilibrium density f…
Strongly dipolar Bose gases can form liquid droplets stabilized by quantum fluctuations. In theoretical description of this phenomenon, low energy scattering amplitude is utilized as an effective potential. We show that for magnetic atoms corrections with respect to Born approximation arise, and derive modified pseudopotential using realistic interaction model. We discuss the resulting changes in collective mode frequencies and droplet stability diagram. Our results are relevant for recent experiments with erbium and dysprosium atoms.
Self-bound many-body systems are formed through a balance of attractive and repulsive forces and occur in many physical scenarios. Liquid droplets are an example of a self-bound system, formed by a balance of the mutual attractive and repulsive forces that derive from different components of the inter-particle potential. It has been suggested that self-bound ensembles of ultracold atoms should exist for atom number densities that are 10^8 times lower than in a helium droplet, which is formed from a dense quantum liquid. However, such ensembles have been elusive up to now because they require f…
This brief review summarizes recent theoretical and experimental results which predict and establish the existence of quantum droplets (QDs), i.e., robust two- and three-dimensional (2D and 3D) self-trapped states in Bose-Einstein condensates (BECs), which are stabilized by effective selffirepulsion induced by quantum fluctuations around the mean-field (MF) states [alias the Lee-Huang--Yang (LHY) effect]. The basic models are presented, taking special care of the dimension crossover, 2D -> 3D. Recently reported experimental results, which exhibit stable 3D and quasi-2D QDs in binary BECs, with…
We study two-dimensional (2D) matter-wave solitons in spinor Bose-Einstein condensates (BECs) under the action of the spin-orbit coupling (SOC) and opposite signs of the self- and cross-interactions. Stable 2D two-component solitons of the mixed-mode (MM) type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collap…
This mini-review collects theoretical results predicting the creation of matter-wave solitons by the pseudo-spinor system of Gross-Pitaevskii equations (GPEs) with the self-attractive cubic nonlinearity and linear first-order-derivative terms accounting for the spin-orbit coupling (SOC). In one dimension (1D), the so predicted bright solitons are similar to their well-known counterparts supported by the GPE in the absence of SOC. Completely novel results were recently obtained for 2D and 3D systems: SOC suppresses the collapse instability of the multidimensional GPE, creating fully stable 2D g…
