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 2 12 > g11g22. 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 droplet thus automatically evaporates itself to zero temperature, the property potentially interesting by itself and from the viewpoint of sympathetic cooling of other systems.PACS numbers: 03.75. Mn,03.75.Kk The mean-field and the first beyond mean-field contribution, the famous Lee-Huang-Yang (LHY) correction, to the ground state energy of a homogeneous weaklyrepulsive Bose gas read [1]where n is the density and a > 0 and g = 4π 2 a/m are, respectively, the scattering length and coupling constant characterizing the interparticle interaction. The LHY correction originates from the zero-point motion of the Bogoliubov excitations and is thus intrinsically quantum. It is also universal in the sense that it depends only on the two-body scattering length and not on other parameters of the two-body or higher-order interactions. Quite naturally the experimental observation of this fundamental beyond mean-field effect came from the field of ultra-cold gases [2][3][4][5][6][7], where the gas parameter na 3 and, therefore, the relative contribution of the LHY term, can be enhanced by using Feshbach resonances [8]. Note however that the effect is perturbative; for na 3 ∼ 1 higher order terms and processes, in particular, three-body decay, come into play. A different situation is predicted for spinor gases where quantum fluctuations lift the degeneracy in the ground-state manifold [9,10] or lead to quantum mass acquisition [11]. In this Letter we point out that in a Bose-Bose mixture the mean-field term and the LHY term depend on the inter-and intraspecies coupling constants in a different manner. Therefore, one can independently control them and make them comparable to each other without ever leaving the weakly-interacting regime. In particular, an interesting situation, impossible in the singlecomponent case, arises when the mean-field term, ∝ n 2 , is negative and the LHY one, ∝ n 5/2 , is positive. Because of its steeper density scaling the quantum LHY repulsion neutralizes the mean-field attraction and stabilizes the system against collapse. The mixture can then exist as a droplet in equilibrium with vacuum without any external trapping [12]. This phenomenon naturally suggests a proof-of-principle experiment for observing the LHY quantum correction. The droplet can be prepared from currently available homo-and heteronuclear atomic mixtures by tuning the inter-and intraspecies scattering lengths into the unstable (from the mean-field viewpoin...

We discuss the regimes of quantum degeneracy in a trapped 1D gas and obtain the diagram of states. Three regimes have been identified: the Bose-Einstein condensation (BEC) regimes of a true condensate and quasicondensate, and the regime of a trapped Tonks gas (gas of impenetrable bosons). The presence of a sharp crossover to the BEC regime requires extremely small interaction between particles. We discuss how to distinguish between true and quasicondensates in phase coherence experiments.

We discuss the behavior of weakly bound bosonic dimers formed in a two-component cold Fermi gas at a large positive scattering length a for the interspecies interaction. We find the exact solution for the dimer-dimer elastic scattering and obtain a strong decrease of their collisional relaxation and decay with increasing a. The large ratio of the elastic to inelastic rate is promising for achieving Bose-Einstein condensation of the dimers and cooling the condensed gas to very low temperatures.

We calculate the energy of one-and two-dimensional weakly interacting Bose-Bose mixtures analytically in the Bogoliubov approximation and by using the diffusion Monte Carlo technique. We show that in the case of attractive inter-and repulsive intraspecies interactions the energy per particle has a minimum at a finite density corresponding to a liquid state. We derive the Gross-Pitaevskii equation to describe droplets of such liquids and solve it analytically in the one-dimensional case. DOI: 10.1103/PhysRevLett.117.100401 According to van der Waals' theory the fundamental property of a liquid to form self-bound states with free surface is due to the shape of the interaction potential which typically has a repulsive core and a more extended attractive part. Usual liquids are dense and almost incompressible since particles prefer to be at the potential minima. A qualitatively different type of liquid, dilute one, has very recently been observed in a Bose-condensed Dy gas characterized by anisotropic dipolar interactions [1,2] and a similar phenomenon has been predicted to occur in three-dimensional Bose-Bose mixtures with isotropic contact interactions [3]. In both cases the system, collapsing from the mean-field viewpoint, is stabilized by quantum many-body effects; each particle feels the attractive meanfield interaction proportional to the density n compensated by the positive Lee-Huang-Yang correction ∝ n 3=2 [1][2][3][4][5]. Such liquids and their finite-size droplets remain dilute and weakly interacting allowing for a well-controlled perturbative description. They also have quite peculiar features: their very existence is a direct manifestation of beyond mean-field effects, they require no trapping and their bulk density and shape are tunable by changing interactions, in the absence of external trapping they can reach zero temperature by evaporation, etc.In this Letter, motivated by the enhanced role of beyondmean-field effects in low dimensions [6], we consider twoand one-dimensional Bose-Bose mixtures and show that with decreasing the dimensionality the liquid phase not only persists, but becomes more ubiquitous and remarkable. We find that in the two-dimensional case the energy per particle is proportional to n½lnðn=n 0 Þ − 1 (n 0 is the equilibrium density), the liquid state exists as long as the interspecies interaction is weakly attractive and the intraspecies ones are weakly repulsive. This contrasts the threedimensional case where a critical interspecies attraction is needed to liquefy the mixture. Interestingly, we find that a three-dimensional mixture in the gas phase can become liquid if confined to the quasi-two-dimensional geometry. In the one-dimensional case the liquid phase originates from the competition of a repulsive mean-field term ∝ n and attractive beyond mean-field correction ∝ −n 1=2 . Counterintuitively, this means that a one-dimensional mixture, stable from the mean-field viewpoint, is actually unstable towards the formation of a liquid droplet. We analytically describe its shape...

We consider a three-boson system with resonant binary interactions and show that for sufficiently narrow resonances three-body observables depend only on the resonance width and the scattering length. The effect of narrow resonances is qualitatively different from that of wide resonances revealing novel physics of three-body collisions. We calculate the rate of three-body recombination to a weakly bound level and the atom-dimer scattering length and discuss implications for experiments on Bose-Einstein condensates and atom-molecule mixtures near Feshbach resonances.

Recent developments in the physics of ultracold gases provide wide possibilities for reducing the dimensionality of space for magnetically or optically trapped atoms. The goal of these lectures is to show that regimes of quantum degeneracy in two-dimensional (2D) and one-dimensional (1D) trapped gases are drastically different from those in three dimensions and to stimulate an interest in low-dimensional systems. Attention is focused on the new physics appearing in currently studied low-dimensional trapped gases and related to finite-size and finite-temperature effects.

We study the lifetime of a Bose gas at and around unitarity using a Feshbach resonance in lithium 7. At unitarity, we measure the temperature dependence of the three-body decay coefficient L(3). Our data follow a L(3)=λ(3)/T(2) law with λ(3)=2.5(3)(stat)(6)(syst)×10(-20) (μK)(2) cm(6) s(-1) and are in good agreement with our analytical result based on zero-range theory. Varying the scattering length a at fixed temperature, we investigate the crossover between the finite-temperature unitary region and the previously studied regime where |a| is smaller than the thermal wavelength. We find that L(3) is continuous across the resonance, and over the whole a<0 range our data quantitatively agree with our calculation.

We consider bosonic dipolar molecules in an optical lattice prepared in a mixture of different rotational states. The 1/r 3 interaction between molecules for this system is produced by exchanging a quantum of angular momentum between two molecules. We show that the Mott states of such systems have a large variety of quantum phases characterized by dipolar orderings including a state with ordering wave vector that can be changed by tilting the lattice. As the Mott insulating phase is melted, we also describe several exotic superfluid phases that will occur.