Quantum collapse in three and two dimensions (3D and 2D) is induced by attractive potential ~ -1/r^2. It was demonstrated that the mean-field (MF) cubic self-repulsion in the 3D bosonic gas suppresses the collapse and creates the missing ground state (GS). However, the cubic nonlinearity is not strong enough to suppress the 2D collapse. We demonstrate that the Lee-Hung-Yang (LHY) quartic term, induced by quantum fluctuations around the MF state, is sufficient for the stabilization of the 2D gas against the collapse. By means of numerical solution of the Gross-Pitaevskii equation including the LHY term, as well as with the help of analytical methods, such as expansions of the wave function at small and large distances from the center and the Thomas-Fermi approximation, we construct stable GS, with a singular density, ~ 1/r^{4/3}, but convergent integral norm. Counter-intuitively, the stable GS exists even if the external potential is repulsive, with the strength falling below a certain critical value. An explanation to this finding is given. Along with the GS, singular vortex states are produced too, and their stability boundary is found analytically. Unstable vortices spontaneously transform into the stable GS, expelling the vorticity to periphery.
Supersonic Flow Past an Obstacle in a Quasi-Two-Dimensional Lee-Huang-Yang Quantum Fluid
Universidade de São Paulo（圣保罗大学）；Russian Academy of Sciences（俄罗斯科学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Higher School of Economics（高等经济大学）
We study the dipolar Gross-Piteavskii functional with the Lee-Huang-Yang (LHY) correction term without trapping potential and in the regime where the dipole-dipole interaction dominates the repulsive short-range interaction. We show that, above a critical mass, the functional admits minimizers and we prove their regularity and exponential decay. We also estimate the critical mass in terms of the parameters of the system.
We consider a homogeneous heteronuclear Bose mixture with contact interactions at the mean-field collapse, i.e. with interspecies attraction equal to the mean geometrical intraspecies repulsion. We show that the Lee-Huang-Yang (LHY) energy functional is accurately approximated by an expression that has the same functional form as in the homonuclear case. The approximated energy functional is characterized by two exponents, which can be treated as fitting parameters. We demonstrate that the values of these parameters which preserve the invariance under permutation of the two atomic species are …
The beyond-mean-field Lee-Huang-Yang (LHY) correction is ubiquitous in dilute ultracold quantum gases. However, its effects are often elusive due to the typically much larger influence of the mean-field energy. In this work, we study an ultracold mixture of $^{23}$Na and $^{87}$Rb with tunable attractive interspecies interactions. The LHY effects manifest in the formation of self-bound quantum liquid droplets and the expansion dynamics of the gas-phase sample. A liquid-to-gas phase diagram is obtained by measuring the critical atom numbers below which the self-bound behavior disappears. In sta…
The perturbative Lee-Huang-Yang (LHY) interaction proportional to $n^{3/2}$, where $n$ is the density, creates an infinitely repulsive potential at the center of a Bose-Einstein condensate (BEC) with net attraction, which stops the collapse to form a self-bound state in a dipolar BEC and in a binary BEC. However, recent microscopic calculations of the non-perturbative beyond-mean-field (BMF) interaction indicate that the LHY interaction %with the $n^{3/2}$ term is valid only for very small values of gas parameter $x$. We show that a realistic non-perturbative BMF interaction can stop collapse …
We consider a dilute and ultracold bosonic gas of weakly-interacting atoms. Within the framework of quantum field theory we derive a zero-temperature modified Gross-Pitaevskii equation with beyond-mean-field corrections due to quantum depletion and anomalous density. This result is obtained from the stationary equation of the Bose-Einstein order parameter coupled to the Bogoliubov-de Gennes equations of the out-of-condensate field operator. We show that, in the presence of a generic external trapping potential, the key steps to get the modified Gross-Pitaevskii equation are the semiclassical a…
We investigate the properties of self-bound ultradilute Bose-Bose mixtures, beyond the Lee-Huang-Yang description. Our approach is based on the determination of the beyond mean-field corrections to the phonon modes of the mixture in a self-consistent way and calculation of the associated equation of state. The newly obtained ground state energies show excellent agreement with recent quantum Monte Carlo calculations, providing a simple and accurate description of the self-bound mixtures with contact type interaction. We further show numerical results for the equilibrium properties of the finite…
We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates (BECs) in the presence of a quadratic Zeeman effect, and derive the Lee-Huang-Yang (LHY) corrections to the ground-state energy, sound velocity, and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECs that can be realized experimentally. We also examine the stability of each phase against quantum fluctuations and the quadratic Zeeman effect. Furthermore, we discuss a relationship between the number of symmetry generators that are spontaneously broken and that of Nambu-Goldstone (NG) modes. It is …
Lee-Huang-Yang (LHY) fluids are an exotic quantum matter dominated purely by quantum fluctuations. Recently, the three-dimensional LHY fluids were observed in ultracold atoms experiments, while their low-dimensional counterparts have not been well known. Herein, based on the Gross-Pitaevskii equation of one-dimensional LHY quantum fluids in two-component Bose-Einstein condensates, we reveal analytically and numerically the formation, properties, and dynamics of matter-wave structures therein. Considering a harmonic trap, approximate analytical results are obtained based on variational approxim…
Rogue waves in extended Gross-Pitaevskii Models with a Lee-Huang-Yang correction
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Missouri University of Science and Technology（密苏里科技大学）；Northumbria University（诺森比亚大学）；National and Kapodistrian University of Athens（雅典国立卡波迪斯特里亚斯大学）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）
We observe monopole oscillations in a mixture of Bose-Einstein condensates, where the usually dominant mean-field interactions are canceled. In this case, the system is governed by the next-order Lee-Huang-Yang (LHY) correction to the ground state energy, which describes the effect of quantum fluctuations. Experimentally such a LHY fluid is realized by controlling the atom numbers and interaction strengths in a $^{39}$K spin mixture confined in a spherical trap potential. We measure the monopole oscillation frequency as a function of the LHY interaction strength as proposed recently by Jørgens…
We present the numerical results of the structure and dynamics of the self-bound ground state arising solely because of the presence of beyond mean field quantum fluctuation in spin-orbit coupled binary Bose-Einstein condensates in one dimension. Depending upon spin-obit (SO) and Rabi couplings, we observe that the ground state exhibits either quantum-bright (plane) or quantum-stripe soliton nature. We find an analytical soliton solution for non-zero SO coupling that matches quite well with the numerical results. Further, we investigate the dynamical stability of these solitons by adopting thr…
Understanding the effects of interactions in complex quantum systems beyond the mean-field paradigm constitutes a fundamental problem in physics. Here, we show how the atom numbers and interactions in a Bose-Bose mixture can be tuned to cancel mean-field interactions completely. The resulting system is entirely governed by quantum fluctuations -- specifically the Lee-Huang-Yang correlations. We derive an effective one-component Gross-Pitaevskii equation for this system, which is shown to be very accurate by comparison with a full two-component description. This allows us to show how the Lee-Hu…
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 address the existence and dynamics of one-dimensional harmonically confined quantum droplets, appearing in two-component mixtures by deploying a nonperturbative approach. We find that, in symmetric homonuclear settings, beyond Lee-Huang-Yang correlations result in flat-top droplet configurations for either decreasing intercomponent attraction or larger atom number. Asymmetric mixtures feature spatial mixing among the involved components with the more strongly interacting or heavier one exhibiting flat-top structures. Applying quenches on the harmonic trap we trigger the lowest-lying collect…
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…
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 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.
Collapse in dipolar Bose-Einstein condensates may be arrested by quantum fluctuations. Due to the anisotropy of the dipole-dipole interactions, the dipole-driven collapse induced by soft excitations is compensated by the repulsive Lee-Huang-Yang contribution resulting from quantum fluctuations of hard excitations, in a similar mechanism as that recently proposed for Bose-Bose mixtures. The arrested collapse results in self-bound filament-like droplets, providing an explanation to recent dysprosium experiments. Arrested instability and droplet formation are novel general features directly linke…
The properties of a supersolid state (SS) in quasi-one-dimensional dipolar Bose-Einstein condensate is studied, considering two possible mechanisms of realization - due to repulsive three-body atomic interactions and quantum fluctuations in the framework of the Lee-Huang-Yang (LHY) theory. The proposed theoretical model, based on minimization of the energy functional, allows evaluating the amplitude of the SS for an arbitrary set of parameters in the governing Gross-Pitaevskii equation (GPE). To explore the dynamics of the SS first, we numerically construct its ground state in different settin…
In the first experimental realization of dilute Bose-Bose liquid drops using two hyperfine states of $^{39}$K some discrepancies between theory and experiment were observed. The standard analysis of the data using the Lee-Huang-Yang beyond mean-field theory predicted critical numbers which were significantly off the experimental measurements. Also, the radial size of the drops in the experiment proved to be larger than expected from this theory. Using a new functional, which is based on quantum Monte Carlo results of the bulk phase incorporating finite-range effects, we can explain the origin …
We analyze the role played by quantum fluctuations on a Raman Spin-Orbit Coupled system in the stripe phase. We show that beyond mean-field effects stabilize the collapse predicted by mean-field theory and induce the emergence of two phases: a gas and a liquid, which also show spatial periodicity along a privileged direction. We show that the energetically favored phase is determined by the Raman coupling and the spin-dependent scattering lengths. We obtain the ground-state solution of the finite system by solving the extended Gross-Pitaevskii equation and find self-bound, droplet-like solutio…
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…
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…
We measure the excitation spectrum of a stable dipolar Bose--Einstein condensate over a wide momentum-range via Bragg spectroscopy. We precisely control the relative strength, $ε_{\rm dd}$, of the dipolar to the contact interactions and observe that the spectrum increasingly deviates from the linear phononic behavior for increasing $ε_{\rm dd}$. Reaching the dipolar dominated regime $ε_{\rm dd}>1$, we observe the emergence of a roton minimum in the spectrum and its softening towards instability. We characterize how the excitation energy and the strength of the density-density correlations at t…
We predict that Lee-Huang-Yang effect makes it possible to create stable quantum droplets (QDs) in binary Bose-Einstein condensates with a hetero-symmetric or hetero-multipole structure, i.e., different vorticities or multipolarities in their components. The QDs feature flat-top shapes when either chemical potential μ_1,2 of the droplet approaches an edge of a triangular existence domain in the (μ_1,μ_2) plane. QDs with different vorticities of their components are stable against azimuthal perturbations, provided that the norm of one component is large. We also present multipole states, in whi…
We introduce a system combining the quadratic self-attractive or composite quadratic-cubic nonlinearity, acting in the combination with the fractional diffraction, which is characterized by its Lévy index $α$. The model applies to a gas of quantum particles moving by Lévy flights, with the quadratic term representing the Lee-Huang-Yang correction to the mean-field interactions. A family of fundamental solitons is constructed in a numerical form, while the dependence of its norm on the chemical potential characteristic is obtained in an exact analytical form. The family of \textit{quasi-Townes …
We study the quantum fluctuations in the condensates of a mixture of bosonic atoms and molecules with interspecies p-wave interaction. Our analysis shows that the quantum phase of coexisting atomic and molecular condensates is unstable at the mean-field level. Unlike the mixture of s-wave interaction, the Lee-Huang-Yang correction of p-wave interaction is unexpectedly found here to exhibit an opposite sign with respect to its mean-field term above a critical particle density. This quantum correction to the mean-field energy provides a remarkable mechanism to self-stabilize the phase. The order…
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…
We study a liquid quantum droplets in a mixture of two-component Bose-Einstein condensates under a variable confinement introduced along one or two spatial dimensions. Despite the atom-atom scattering has a three-dimensional character, discreetness of the available modes in the reduced dimension(s) strongly influences the zero-point energy -- the Lee-Huang-Yang term. In a weakly interaction limit, it is the leading correction to the mean-field energy at the crossover from three to two dimensions, or from three to one dimension. We analyze the properties of the droplets at the dimensional cross…
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…
We demonstrate the possibility of creating a self-bound stable three-dimensional matter-wave spherical boson-fermion quantum ball in the presence of an attractive boson-fermion interaction and a small repulsive three-boson interaction. The three-boson interaction could be attractive or repulsive whereas the fermions are taken to be in a fully-paired super-fluid state in the Bardeen- Cooper-Schreifer ( quasi-noninteracting weak-coupling) limit. We also include the Lee-Huang-Yang (LHY) correction to a repulsive bosonic interaction term. The repulsive three-boson interaction and the LHY correctio…
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…
Attractive Multidimensional Solitons in Trapping Potentials
Physical-Technical Institute of Uzbekistan Academy of Sciences（乌兹别克斯坦科学院物理技术研究所）；Department of Physics ”E.R. Caianiello”, University of Salerno（萨莱诺大学 E.R. Caianiello 物理系）
We consider a binary bosonic condensate with weak mean-field (MF) residual repulsion, loaded in an array of nearly one-dimensional traps coupled by transverse hopping. With the MF force balanced by the effectively one-dimensional attraction, induced in each trap by the Lee-Hung-Yang correction (produced by quantum fluctuations around the MF state), stable onsite-centered and intersite-centered semi-discrete quantum droplets (QDs) emerge in the array, as fundamental ones and self-trapped vortices, with winding numbers, at least, up to 5, in both tightly-bound and quasi-continuum forms. The appl…
We study the formation of a stable self-trapped spherical quantum ball in a binary Bose-Einstein condensate (BEC) with two-body inter-species attraction and intra-species repulsion employing the beyond-mean-field Lee-Huang-Yang and the three-body interactions. We find that either of these interactions or a combination of these can stabilize the binary BEC quantum ball with very similar stationary results, and for a complete description of the problem both the terms should be considered. These interactions lead to higher-order nonlinearities, e.g. quartic and quintic, respectively, in a nonline…
It is well known that attractive potential which is inversely proportional to the squared distance from the origin gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical analysis, chiefly published in several original papers, which demonstrates suppression of the collapse caused by this potential, and the creation of the otherwise missing ground state in a 3D gas of bosonic dipoles pulled by the same potential to the central charge, with repulsive contact interactions between them, represented…
