Quantum Droplets Research Articles

Vortex lattices in strongly confined quantum droplets

Vortex lattices in strongly confined quantum droplets

Two-dimensional symbiotic solitons and quantum droplets in a quasi-one-dimensional optical lattice

Symbiotic solitons (SS) and quantum droplets (QD) are self-trapped localized modes emerging in binary Bose-gas mixtures with intra-component repulsion and inter-component attraction. We have shown that two-dimensional (2D) SS can be stabilized against collapse or decay by means of a quasi-one-dimensional optical lattice (OL). Mobility of SSs along the free direction of the potential allows us to explore interactions and collisions of SSs moving in the same channel and neighboring channels of the quasi-1D OL. For the case of equal atom numbers in both components of the binary Bose–Einstein condensate (BEC) we have developed a variational approach that showed the stability of SS. For parameter settings when the SS stays on the verge of collapse/decay instability we take into account the Lee–Huang–Yang quantum fluctuations (QF) term in the coupled Gross–Pitaevskii equations (GPE). The QF term in the 2D GPE has the form of nonlinearity with a logarithmic factor, which is repulsive for large amplitude waves and attractive for opposite situations. This property of the governing equation supports stable QDs immersed in a gaseous phase of the larger component in particle-imbalanced Bose-gas mixtures. We explore the peculiar properties of QD such as incompressibility and surface tension, which are inherent to liquids. The proposed model of binary BEC loaded in a quasi-1D OL allows us to demonstrate the manifestations of the incompressibility and surface tension of 2D QDs.

Ground-state properties and Bogoliubov modes of a harmonically trapped one-dimensional quantum droplet

Ground-state properties and Bogoliubov modes of a harmonically trapped one-dimensional quantum droplet

Quantum droplets in one-dimensional Bose-Bose mixtures: beyond the Lee-Huang-Yang description

The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.

Modulational instability and droplet formation in Bose-Bose mixtures with Lee-Huang-Yang correction and polaron-like impurity

This letter studies the excitation of nonlinear patterns on the one-dimensional (1D) quantum droplet in the presence of polaron-like impurity under modulational instability (MI) activation. Analytically, it is shown that strong coupling between the impurity and Bose species expands the instability domain, especially in the quantum droplet regime, while pronounced self-repulsive nonlinearity tends to restrict the MI to regions of strong interspecies interactions. In the process, the wavenumber spectrum, capable of supporting MI, also shrinks. Direct numerical simulations confirm the existence of coupled boson-impurity wave patterns, mainly governed by trains of solitary waves of different profiles for the impurity and quasi-droplet waveforms for the boson. Additionally, the two components are mutually trapped, which enhances energy transfer between them, leading to the formation of quasi-droplet structures under a suitable balance between the back action of the impurity on the boson, the Lee-Huang-Yang term, as well as the intrinsic self-repulsive nonlinearity.

Phase Diagram of High-Temperature Electron-Hole Quantum Droplet in Two-Dimensional Semiconductors.

Quantum liquids, systems exhibiting effects of quantum mechanics and quantum statistics at macroscopic levels, represent one of the most exciting research frontiers of modern physical science and engineering. Notable examples include Bose-Einstein condensation (BEC), superconductivity, quantum entanglement, and a quantum liquid. However, quantum liquids are usually only stable at cryogenic temperatures, significantly limiting fundamental studies and device development. Here we demonstrate the formation of stable electron-hole liquid (EHL) with the quantum statistic nature at temperatures as high as 700 K in monolayer MoS2 and elucidate that the high-temperature EHL exists as droplets in sizes of around 100-160 nm. We also develop a thermodynamic model of high-temperature EHL and, based on the model, compile an exciton phase diagram, revealing that the ionized photocarrier drives the gas-liquid transition, which is subsequently validated with experimental results. The high-temperature EHL provides a model system to enable opportunities for studies in the pursuit of other high-temperature quantum liquids. The results can also allow for the development of quantum liquid devices with practical applications in quantum information processing, optoelectronics, and optical interconnections.

Exploring the role of beyond mean-field interaction in the structure and dynamics of one-dimensional quantum droplets

We present simulation results of the ground state structure and dynamics of quantum droplets (QDs) in one-dimensional spin–orbit coupled binary Bose–Einstein condensates. We have considered two cases for this analysis, such as (i) the mean-field term has a vanishingly small contribution utilizing the equal and opposite inter- and intraspecies interactions and (ii) unequal inter- and intraspecies interactions. The QD exhibits remarkably different natures in each case. In the former case, it exhibits a bright sech-like droplet nature, while in the latter case, we find the flattened sech-like shape of the droplet. Further, we analyze the effect of velocity perturbation on the dynamics in both cases. For the first case, we find a systematic change from the solitonic droplet nature to the breathing droplet, which finally has a moving droplet feature upon increasing the velocity. However, the second case shows similar dynamics except having more dynamically stable features than the first. Finally, we present various dynamics that ensued in the QD due to the quenching of the interaction parameters, coupling parameters or allowing the droplet to undergo collisions.

Interaction of one-dimensional quantum droplets with potential wells and barriers

We address static and dynamical properties of one-dimensional (1D) quantum droplets (QDs) under the action of local potentials in the form of narrow wells and barriers. The dynamics of QDs is governed by the 1D Gross–Pitaevskii equation including the mean-field cubic repulsive term and the beyond-mean-field attractive quadratic one. In the case of the well represented by the delta-function potential, three exact stable solutions are found for localized states pinned to the well. The Thomas–Fermi approximation for the well and the adiabatic approximation for the collision of the QD with the barrier are developed too. Collisions of incident QDs with the wells and barriers are analyzed in detail by means of systematic simulations. Outcomes, such as fission of the moving QD into transmitted, reflected, and trapped fragments, are identified in relevant parameter planes. In particular, a counter-intuitive effect of partial or full rebound of the incident QD from the potential well is studied in detail and qualitatively explained.

Excitation and quasi-transition of rogue waves in a one-dimensional quantum droplet

We study the dynamics of rogue wave excitations and quasi-transitions in one-dimensional quantum droplets, whose homogeneous-density flat top provides an applicable background wave for nonlinear excitation. Using a quenching technique and a localized space modulation technique for the nonlinear effect coefficient, the rogue wave is excited on droplets. The excitation modes of rogue waves can be transitioned to dispersive shock waves or Kuznetsov–Ma breathers by regulating their modulation amplitude and width, and their maximal density and width can be regulated by modulation their amplitude. Meanwhile, two Gaussian modulations for droplets also bring the second-order rogue wave’s dynamics, whose maximal amplitude can approach 3.6 times that of the amplitude of droplets. Our results provide guidance for rogue wave excitation and transition on self-bound background waves and deepen our understanding of quantum droplets.

Complex dynamics on the one-dimensional quantum droplets via time piecewise PINNs

The dynamics of one-dimensional quantum droplets and the emerging applications of deep learning in landing technology have become prominent research areas. In this work, we present a novel methodology, termed time piecewise physics-informed neural networks (PINNs), to study the intricate dynamics of one-dimensional quantum droplets by solving the amended Gross–Pitaevskii equation. Our network model exhibits superior training performance in the long time domain compared to conventional PINNs, with each of its subnetwork operating independently and offering high adjustability. By employing time piecewise PINNs with scarce training points, we not only study intrinsic modulations of single droplet and collision between two droplets, but also excite the breather on droplet background. Intriguingly, we obtain an interference pattern from training result of collision between two droplets, which is a significant feature of the interplay of coherent matter waves. The numerical findings demonstrate that in a nonlinear non-integrable system, varying parameters can result in vastly different dynamic behaviors despite sharing the same initial conditions. Our results offer valuable insights and guidance for leveraging deep learning technology to facilitate intrinsic modulations of single droplet, droplets collision, and excitation of breather.

Rotational Dynamics Induced by Low-Energy Binary Collisions of Quantum Droplets

A theoretical analysis of the rotational dynamics induced by off-axis binary collisions of quantum droplets constituted by ultracold atoms is reported. We focus on quantum droplets formed by degenerate dilute Bose gases made from binary mixtures of alkaline atoms under feasible experimental conditions. The stability of the ground state is known to be longer for the chosen heteronuclear gases than for the homonuclear ones. In both cases, we find that the dynamics seem to privilege high similarity of the density of each atomic species. However, the evolution of the phase of the corresponding order parameter differs significantly for heteronuclear admixtures. We evaluate the fidelity as a figure of merit for the overlap between the order parameters of each atomic species. Dynamical evidence of the differences between the phases of the order parameters is predicted to manifest in their corresponding linear and angular momenta. We numerically verify that the total angular and linear momenta are both conserved during the collision. Some direct correlations between the Weber number and the impact parameter with the distribution of the dynamical variables are established.

Effects of higher-order fluctuations on the bulk and surface properties of quantum droplets in a heteronuclear Bose-Bose mixture

Effects of higher-order fluctuations on the bulk and surface properties of quantum droplets in a heteronuclear Bose-Bose mixture

On an intercritical log-modified nonlinear Schrödinger equation in two spatial dimensions

We consider a dispersive equation of Schrödinger type with a nonlinearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with Lee-Huang-Yang correction. Mathematically, it is seen to be mass supercritical and energy subcritical with a sign-indefinite nonlinearity. For the corresponding initial value problem, we prove global in-time existence of strong solutions in the energy space. Furthermore, we prove the existence and uniqueness (up to symmetries) of nonlinear ground states and the orbital stability of the set of energy minimizers. We also show that for the corresponding model in 1D a stronger stability result is available.

Breakup of quantum liquid filaments into droplets

We investigate how the Rayleigh-Plateau instability of a filament made of a $^{41}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ self-bound mixture may lead to an array of identical quantum droplets, with typical breaking times which are shorter than the lifetime of the mixture. If the filament is laterally confined, as it happens in a toroidal trap, and atoms of one species are in excess with respect to the optimal equilibrium ratio, the droplets are immersed into a superfluid background made by the excess species which provides global phase coherence to the system, suggesting that the droplet array in the unbalanced system may display supersolid character. This possibility is investigated by computing the nonclassical translational inertia coefficient. The filament may be a reasonable representation of a self-bound mixture subject to toroidal confinement when the bigger circle radius of the torus is much larger than the filament radius.

Dark quantum droplets and solitary waves in beyond-mean-field Bose-Einstein condensate mixtures

Quantum liquidlike states of matter have been realized in an ongoing series of experiments with ultracold Bose gases. Using a combination of analytical and numerical methods we identify the specific criteria for the existence of dark solitons in beyond-mean-field binary condensates, revealing how these excitations exist for both repulsive and attractive interactions, the latter leading to dark quantum droplets with properties intermediate between a dark soliton and a quantum droplet. The phenomenology of these excitations is explored within the full parameter space of the model, revealing a spatial profile of the excitation that differs significantly from the Zakharov-Shabat soliton, leading to a negative effective mass that is enhanced in the presence of the quantum fluctuations. Finally the dynamics of pairs of the excitations are explored, showing nonintegrable dynamics and dark soliton bound states in the attractive regime.

Solitary waves in a quantum droplet-bearing system

We unravel the existence and stability properties of dark soliton solutions as they extend from the regime of trapped quantum droplets towards the Thomas-Fermi limit in homonuclear symmetric Bose mixtures. Leveraging a phase-plane analysis, we identify the regimes of existence of different types of quantum droplets and subsequently examine the possibility of black and gray solitons and kink-type structures in this system. Moreover, we employ the Landau dynamics approach to extract an analytical estimate of the oscillation frequency of a single dark soliton in the relevant extended Gross--Pitaevskii model. Within this framework, we also find that the single soliton immersed in a droplet is stable, while multisoliton configurations exhibit parametric windows of oscillatory instabilities. Our results pave the way for studying dynamical features of nonlinear multisoliton excitations in a droplet environment in contemporary experimental settings.

Two-dimensional miscible-immiscible supersolid and droplet crystal state in a homonuclear dipolar bosonic mixture

The recent realization of a binary dipolar Bose-Einstein condensate [A. Trautmann et al., Phys. Rev. Lett. 121, 213601 (2018)] presents new exciting aspects for studying quantum droplets and supersolids in a binary mixture. Motivated by this experiment, we study the ground-state phases and dynamics of a Dy-Dy mixture. The dipolar bosonic mixture exhibits qualitatively novel and rich physics. Relying on the three-dimensional numerical simulations in the extended Gross-Pitaevskii framework, we unravel the ground-state phase diagrams and characterize their different possible phases. The emergent phases include single-droplet (SD), multiple-droplets (MD), doubly supersolid (SS), and superfluid (SF) states in both miscible and immiscible phases. Intriguing mixed ground states are observed for an imbalanced binary mixture, including a combination of SS-SF, SS-MD, and SS-SS phases. We also explore the dynamics across the phase boundaries by linear quenches of interspecies and intraspecies scattering lengths. During these dynamical processes, we observe an abrupt change in phase which initially results in some instability in the system and forms some metastable states in the intermediate timescale. However, the state produced in long-time evolution is similar to our predicted ground state. Although we demonstrate the possible results for a Dy-Dy mixture and for a specific parameter range of intraspecies and interspecies scattering lengths, our results are in general valid for other dipolar mixtures and may become an important benchmark for future experimental scenarios.

Collisional dynamics of two-dimensional vortex quantum droplets

A quantum droplet is a self-bound state balanced by the mean-field interaction and Lee–Huang–Yang correction in a Bose–Bose mixture. In this paper, we study the collisional dynamics of two-dimensional quantum droplets with a vortex. By adjusting the initial momentum, the initial phase difference, the topological charge of the quantum droplets, and the total number of particles, we identify three dynamic mechanisms of collisions, namely, splitting, no-splitting, and their crossover according to the states after collision, which are significantly different from the merging, separation, and evaporation of the collisional dynamics of vortex-free droplets. The initial phase difference of the two droplets changes the interference fringes and the manner of splitting of the droplets. We also show that the three-body loss of atoms does not affect the result.

Quasi-one-dimensional harmonically trapped quantum droplets

We theoretically consider effectively one-dimensional quantum droplets in a symmetric Bose-Bose mixture confined in a parabolic trap. We systematically investigate ground and excited families of localized trapped modes which bifurcate from eigenstates of the quantum harmonic oscillator as the number of particles departs from zero. Families of nonlinear modes have nonmonotonous behavior of chemical potential on the number of particles and feature bistability regions. Excited states are unstable close to the linear limit, but become stable when the number of particles is large enough. In the limit of large density, we derive a modified Thomas-Fermi distribution. Smoothly decreasing the trapping strength down to zero, one can dynamically transform the ground-state solution to the solitonlike quantum droplet, while excited trapped states break in several moving quantum droplets.

Two-Dimensional Quantum Droplets in Binary Dipolar Bose-Bose Mixture

We study two-dimensional (2D) isotropic quantum droplets (QDs) in dipolar binary Bose–Einstein condensates (BECs). The QDs are supported by the competition between the 2D form of the Lee-Huang-Yang (LHY) term and the isotropic dipole-dipole interactions (DDIs). Moreover, the DDIs in the 2D plane can be tuned to be either repulsive or attractive. Before that, QDs in dipolar BECs were often explored in three-dimensional (3D) systems, with competition between the attractive DDIs and the repulsive LHY term. Unlike the 3D system, the LHY term of the 2D binary system behaves in a logarithmic form, which can feature both attraction and repulsion. In this case, the QDs can be produced regardless of the interactions (attraction, repulsion, or zero) that the mean-field effect represents. In this paper, we model the aforementioned QDs via the 2D binary dipolar BECs with the competition between isotropic DDIs and the logarithmic LHY term. Their characteristic parameters (the peak density, IP, chemical potential, μ, and effective area, Aeff) using both numerical and theoretical methods are discussed. The centripetal collision and oblique collision between moving QDs are also studied.