We use exact Quantum Monte Carlo techniques to study the properties of quantum droplets in two-component bosonic mixtures with contact interactions in one spatial dimension. We systematically study the surface tension, the density profile and the breathing mode as a function of the number of particles in the droplet and of the ratio of coupling strengths between intra-species repulsion and inter-species attraction. We find that deviations from the predictions of the generalized Gross-Pitaevskii equation are small in most cases of interest.
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…
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…
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.
We show that the ground state of a dipolar Bose gas in a cylindrically symmetric harmonic trap has a rich phase diagram, including droplet crystal states in which a set of droplets arrange into a lattice pattern that breaks the rotational symmetry. An analytic model for small droplet crystals is developed and used to obtain a zero temperature phase diagram that is numerically validated. We show that in certain regimes a coherent low-density halo surrounds the droplet crystal giving rise to a novel phase with localized and extended features.
We theoretically investigate the finite-temperature structure and collective excitations of a self-bound ultradilute Bose droplet in a flat space realized in a binary Bose mixture with attractive inter-species interactions on the verge of mean-field collapse. As the droplet formation relies critically on the repulsive force provided by Lee-Huang-Yang quantum fluctuations, which can be easily compensated by thermal fluctuations, we find a significant temperature effect in the density distribution and collective excitation spectrum of the Bose droplet. A finite-temperature phase diagram as a fun…
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 …
Quantum droplets have intrigued much attention recently in view of their successful observations in the ultracold homonuclear atoms. In this work, we demonstrate a new mechanism for the formation of quantum droplet in heteronuclear atomic systems, i.e., by applying the synthetic spin-orbit coupling(SOC). Take the Bose-Fermi mixture for example, we show that by imposing a Rashba SOC between the spin states of fermions, the greatly suppressed Fermi pressure can enable the formation of Bose-Fermi droplets even for very weak boson-fermion attractions, which are insufficient to bound a droplet if w…
We define a formalism of a self-consistent description of the ground state of a weakly interacting Bose system, accounting for higher order terms in expansion of energy in the diluteness parameter. The approach is designed to be applied to a Bose-Bose mixture in a regime of weak collapse where quantum fluctuations lead to stabilization of the system and formation of quantum liquid droplets. The approach is based on the Generalized Gross -- Pitaevskii equation accounting for quantum depletion and anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations…
We study the quantum-droplet state in a 3-dimensional (3D) Bose gas in the presence of 1D spin-orbit-coupling and Raman coupling, especially the stripe phase with density modulation, by numerically computing the ground state energy including the mean-field energy and Lee-HuangYang correction. In this droplet state, the stripe can exist in a wider range of Raman coupling, compared with the BEC-gas state. More intriguingly, both spin-orbit-coupling and Raman coupling strengths can be used to tune the droplet density.
Motivated by the recent experiment [H. Kadau \textit{et al.}, arXiv:1508.05007 (2015)], we study roton instability and droplet formation in a Bose-Einstein condensate of $^{164}$Dy atoms with strong magnetic dipole-dipole interaction. We numerically solve the cubic-quintic Gross-Pitaevskii equation with dipole-dipole interaction, and show that the three-body interaction plays a significant role in the formation of droplet patterns. We numerically demonstrate the formation of droplet patterns and crystalline structures, decay of droplets, and hysteresis behavior, which are in good agreement wit…
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 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…
We calculate analytically the quantum and thermal fluctuations corrections of a dilute quasi-two-dimensional Bose-condensed dipolar gas. We show that these fluctuations may change their character from repulsion to attraction in the density-temperature plane owing to the striking momentum dependence of the dipole-dipole interactions. The dipolar instability is halted by such unconventional beyond mean field corrections leading to the formation of a droplet phase. The equilibrium features and coherence properties exhibited by such droplets are deeply discussed. At finite temperature, we find tha…
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…
We theoretically study dynamical formation of a quantum droplet in a two-component Bose-Hubbard system with an external trap potential. Specifically, the superfluid in the central region surrounded by the Mott insulator with double filling forms a quantum droplet, which is self-bound thanks to the discontinuous nature of the quantum phase transition between the two phases. We show how to induce the characteristic behavior of the droplet through the control of the trap potential by using the time-dependent Gutzwiller simulations in a two-dimensional system. The static and dynamical properties o…
Ultradilute quantum droplets are intriguing new state of matter, in which the attractive mean-field force can be balanced by the repulsive force from quantum fluctuations to avoid collapse. Here, we present a microscopic theory of ultradilute quantum droplets in three-, one- and two-dimensional two-component Bose-Bose mixtures, by generalizing the conventional Bogoliubov theory to include the bosonic pairing arising from the interspecies attraction. Our pairing theory is fully equivalent to a variational approach and hence gives an upper bound for the energy of quantum droplets. In three dimen…
We study the statics and dynamics of a dipolar Bose-Einstein condensate (BEC) droplet bound by inter-species contact interaction in a trapped non-dipolar BEC. Our findings are demonstrated in terms of stability plots of a dipolar 164Dy droplet bound in a trapped non-dipolar 87Rb BEC with a variable number of 164Dy atoms and the inter-species scattering length. A trapped non-dipolar BEC of a fixed number of atoms can only bind a dipolar droplet containing atoms less than a critical number for the inter-species scattering length between two critical values. The shape and size (statics) as well a…
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…
An ultradilute quantum droplet is a self-bound liquid-like state recently observed in ultracold Bose-Einstein condensates. In all previous theoretical studies, it is described by a phenomenological low-energy effective theory, termed as the extended Gross\textendash Pitaevskii equation. Here, we microscopically derive the Gross\textendash Pitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive inter-species interaction. We show that the inclusion of the pairing field is essential, in order to have a …
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…
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.
In this work, we present the study of the stationary structures and the breathing mode behavior of a two-dimensional self-bound binary Bose droplet. We employ an analytical approach using a variational ansatz with a super-Gaussian trial order parameter and compare it with the numerical solutions of the extended Gross-Pitaevskii equation. We find that the super-Gaussian is superior to the often used Gaussian ansatz in describing the stationary and dynamical properties of the system. We find that for sufficiently large non-rotating droplets the breathing mode is energetically favourable compared…
Quantum droplets are small clusters of atoms self-bound by the balance of attractive and repulsive forces. Here we report on the observation of a novel type of droplets, solely stabilized by contact interactions in a mixture of two Bose-Einstein condensates. We demonstrate that they are several orders of magnitude more dilute than liquid helium by directly measuring their size and density via in situ imaging. Moreover, by comparison to a single-component condensate, we show that quantum many-body effects stabilize them against collapse. We observe that droplets require a minimum atom number to…
We systematically investigate the ground-state properties of self-bound droplets of quasi-two-dimensional binary Bose gases by using the Gaussian state theory. We find that quantum droplets consists two macroscopic squeezed phases and a macroscopic coherent phase. We map out the phase diagram and determine all phase boundaries via both numerical and nearly analytical methods. In particular, we find three easily accessible signatures for the quantum phases and the stablization mechanism of the self-bound droplets by precisely measuring their radial size. Our studies indicate that binary droplet…
Rabi-coupling-induced three-component quantum droplet in ultracold Bose gases
Beijing National Laboratory for Condensed Matter Physics, 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）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Department of Physics, The Chinese University of Hong Kong（香港中文大学物理系）
We have theoretically investigated the structure of spinning self-bound droplets made of $^{41}$K-$^{87}$Rb Bose mixture by solving the Gross-Pitaevskii equation including beyond-mean-field correction in the Lee-Huang-Yang form. The structure and energetics of vortex formation in the self-bound mixture have been elucidated, showing that the formation of linear vortices in the heavier species is energetically favoured over other configurations. A fake (partially filled) core develops as a consequence in the other species, resulting in a hole which might be imaged in experiments. The interplay b…
Recent experiments have revealed the formation of stable droplets in dipolar Bose-Einstein condensates. This surprising result has been explained by the stabilization given by quantum fluctuations. We study in detail the properties of a BEC in the presence of quantum stabilization. The ground-state phase diagram presents three main regimes: mean-field regime, in which the quantum correction is perturbative, droplet regime, in which quantum stabilization is crucial, and a multi-stable regime. In the absence of a multi-stable region, the condensate undergoes a crossover from the mean-field to th…
We analyze some issues related to the stability and free expansion of a one-dimensional logarithmic Bose-Einstein condensate, particularly its eventual relation to the formation of quantum droplet-type configurations. We prove that the corresponding properties, such as the energy of the associated N-body ground state, differ substantially with respect to its three-dimensional counterpart. Consequently, the free velocity expansion also shows differences with respect to the three-dimensional system when logarithmic interactions are taken into account. The one-dimensional logarithmic condensate t…
Laser induced transitions between internal states of atoms have been playing a fundamental role to manipulate atomic clouds for many decades. In absence of interactions each atom behaves independently and their coherent quantum dynamics is described by the Rabi model. Since the experimental observation of Bose condensation in dilute gases, static and dynamical properties of multicomponent quantum gases have been extensively investigated. Moreover, at very low temperatures quantum fluctuations crucially affect the equation of state of many-body systems. Here we study the effects of quantum fluc…
We theoretically investigate the temperature effect in a Bose-Bose mixture with attractive inter-species interactions, in the regime where a self-bound ultradilute quantum droplet forms due to the subtle balance between the attractive mean-field force and the repulsive force provided by Lee-Huang-Yang quantum fluctuations. We find that in contrast to quantum fluctuations, thermal fluctuations destabilize the droplet state and completely destroy it above a threshold temperature. We show that the threshold temperature is determined by the intra-species interaction energy. For a three-dimensional…
According to the mean-field theory a condensed Bose-Bose mixture collapses when the interspecies attraction becomes stronger than the geometrical average of the intraspecies repulsions, $g_{12}^2>g_{11} g_{22}$. We show that instead of collapsing such a mixture gets into a dilute liquid-like droplet state stabilized by quantum fluctuations thus providing a direct manifestation of beyond mean-field effects. We study various properties of the droplet and find, in particular, that in a wide range of parameters its excitation spectrum lies entirely above the particle emission threshold. The drople…
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 study the properties of self-bound liquid droplets of three-dimensional Bose mixtures in a weak random potential with Gaussian correlation function at both zero and finite temperatures. Using the Bogoliubov theory, we derive useful formulas for the ground-state energy, the equilibrium density, the depletion, and the anomalous density of the droplet. The quantum fluctuation induced by the disorder known as the glassy fraction is also systematically computed. At finite temperature, we calculate the free energy, the thermal equilibrium density, and the critical temperature in terms of the diso…
We study an impurity immersed in the mixture of Bose ultracold gases in the regime where a quantum droplet exists. The quasi-one-dimensional geometry is considered. We find an effective attractive potential that acts by the quantum droplet onto the impurity. The bound states of the impurity in this potential are investigated. These impurity bound states can provide potential probes for the presence of quantum fluctuations effects on the droplet properties. In the case of strong impurity-BEC coupling, we study the properties of the nonlinear local modes on the impurity induced by the quantum fl…
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…
Borromean ring refers to a peculiar structure where three rings are linked together while any two of them are unlinked. Here we propose the realization of its quantum mechanical analog in a many-body system of three-component ultracold bosons. Namely, we identify the {\it Borromean droplet}, where only the ternary bosons can form a self-bound droplet while any binary subsystems cannot. Its formation is facilitated by an additional attractive force induced by the density fluctuation of a third component, which enlarges the mean-field collapse region in comparison to the binary case and renders …
We study a one-dimensional disordered Bose fluid using bosonization, the replica method and a nonperturbative functional renormalization-group approach. We find that the Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale $k$, quantum tunneling between the ground state and low-lying metastable states leads to a rounding of the cusp singularity into a quantum boundary layer (QBL). The width o…
In a joint experimental and theoretical effort, we report on the formation of a macro-droplet state in an ultracold bosonic gas of erbium atoms with strong dipolar interactions. By precise tuning of the s-wave scattering length below the so-called dipolar length, we observe a smooth crossover of the ground state from a dilute Bose-Einstein condensate (BEC) to a dense macro-droplet state of more than $10^4$ atoms. Based on the study of collective excitations and loss features, we quantitative prove that quantum fluctuations stabilize the ultracold gas far beyond the instability threshold impose…
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…
Bose-Bose droplets are self-bound objects emerging from a mixture of two interacting Bose-Einstein condensates when their interactions are appropriately tuned. During droplet formation three continuous symmetries of the system's Hamiltonian are broken: translational symmetry and two U1 symmetries, allowing for arbitrary choice of phases of the mean-field wavefunctions describing the two components. Breaking of these symmetries must be accompanied by appearance of zero-energy excitations in the energy spectrum of the system recovering the broken symmetries. Normal modes corresponding to these e…
Quantum droplets may form out of a gaseous Bose-Einstein condensate, stabilized by quantum fluctuations beyond mean field. We show that multiple singly-quantized vortices may form in these droplets at moderate angular momenta in two dimensions. Droplets carrying these precursors of an Abrikosov lattice remain self-bound for certain timescales after switching off an initial harmonic confinement. Furthermore, we examine how these vortex-carrying droplets can be formed in a more pertubation-resistant setting, by starting from a rotating binary Bose-Einstein condensate and inducing a metastable pe…
