Quantum droplets are formed in quantum many-body systems when the competition of quantum corrections with the mean-field interaction yields a stable self-bound quantum liquid. We predict the emergence of a quantum droplet when a Bose-Einstein condensate is placed in a transverse-pumped optical cavity. The strong coupling between the atoms and a cavity mode induces long-range interactions in the atoms, and a roton mode for negative cavity detuning emerges. Using Bogoliubov theory, we show that the roton mode competes with the repulsive atomic s -wave scattering. Due to the favorable scaling of the quantum fluctuations with respect to the volume, a self-bound stable quantum liquid emerges.
Quantum droplets are formed in quantum many-body systems when the competition of quantum corrections with the mean-field interaction yields a stable self-bound quantum liquid. We predict the emergence of a quantum droplet when a Bose-Einstein condensate is placed in a transverse-pumped optical cavity. The strong coupling between the atoms and a cavity mode induces long-range interactions in the atoms, and a roton mode for negative cavity detuning emerges. Using Bogoliubov theory, we show that the roton mode competes with the repulsive atomic s -wave scattering. Due to the favorable scaling of the quantum fluctuations with respect to the volume, a self-bound stable quantum liquid emerges.
We investigate a dilute Bose gas with both a short-range contact and an effective long-range interaction between the atoms. The latter is induced by the strong coupling to a cavity light mode and is spatially characterized by a periodic signature and a tunable envelope rooted in the pumping of the cavity. We formulate a Bogoliubov theory based on a homogeneous mean-field description and quantum fluctuations around it. The competition between the repulsive contact interaction and the long-range interaction allows the formation of self-bound quantum droplets. This generic approach is applied to two cavity setups, one without and one with a momentum-conserving effective long-range interaction between the atoms in the form of a driven dispersive cavity mode and a multimode cavity, respectively. For both cases we show analytically how the size and the central density of the cavity-induced quantum droplets depend on the contact interaction strength and on the shape of the spatial envelope of the long-range interaction.
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 self-repulsion 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 the inter-component attraction slightly exceeding the MF self-repulsion in each component, and in single-component condensates of atoms carrying permanent magnetic moments, are presented in some detail. The summary of theoretical results is focused, chiefly, on 3D and quasi-2D QDs with embedded vorticity, as the possibility to stabilize such states is a remarkable prediction. Stable vortex states are presented both for QDs in free space, and for singular but physically relevant 2D modes pulled to the center by the inverse-square potential, with the quantum collapse suppressed by the LHY effect.
Quantum droplets are small clusters of atoms self-bound by the balance of attractive and repulsive forces. Here, we report on the observation 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. We show that the droplets are stablized against collapse by quantum fluctuations and that they require a minimum atom number to be stable. Below that number, quantum pressure drives a liquid-to-gas transition that we map out as a function of interaction strength. These ultradilute isotropic liquids remain weakly interacting and constitute an ideal platform to benchmark quantum many-body theories.
Multimode optical cavities can be used to implement interatomic interactions which are highly tunable in strength and range. For bosonic atoms trapped in an optical lattice we show that, for any finite range of the cavity-mediated interaction, quantum self-bound droplets dominate the ground state phase diagram. Their size and in turn density is not externally fixed but rather emerges from the competition between local repulsion and finite-range cavity-mediated attraction. We identify two different regimes of the phase diagram. In the strongly glued regime, the interaction range exceeds the droplet size and the physics resembles the one of the standard Bose-Hubbard model in a (self-consistent) external potential, where in the phase diagram two incompressible droplet phases with different filling are separated by one with a superfluid core. In the opposite weakly glued regime, we find instead direct first order transitions between the two incompressible phases, as well as pronounced metastability. The cavity field leaking out of the mirrors can be measured to distinguish between the various types of droplets.
Among settings which have recently been elaborated for the creation of stable multidimensional solitons, most advanced, as concerns the theory and experiment alike, is the work with quantum droplets (QDs) in atomic BEC. Important theoretical predictions for fundamental and topologically structured soliton-like (stably self-trapped) QDs have been published in many papers, starting from the pioneering work of Petrov [Phys. Rev. Lett. 115, 155302 (2015)]. Simultaneously, the experimental advancement in the making of QDs in binary BECs, as well as in single-component BECs of atoms carrying magnetic dipole moments, have been really remarkable and unique. Successful experiments followed the first theoretical prediction of the QDs very quickly, and the progress in the experimental work was much faster than in other areas of physics where stable multidimensional solitons were predicted. The basis of these works is a correction to the MF (mean-field) BEC dynamics produced by quantum fluctuations around MF states. The correction was first predicted in 1957 by Lee, Huang, and Yang (LHY). The breakthrough proposal made by Petrov [Phys. Rev. Lett. 115, 155302 (2015)] was to use the effective higher-order (quartic) self-repulsion, produced by the LHY effect in the atomic BEC, as a mechanism arresting the collapse driven by mutual attraction of two self-repulsive components in a binary quantum bosonic gas. The balance between the inter-species attraction and LHY-induced self-repulsion gives rise to stable 3D and quasi-2D (pancake-shaped) localized bound states in the form of the quantum droplets (QDs). In single-component condensates of magnetic atoms, the attraction is provided by the long-range dipole–dipole interaction (Chomaz et al., 2016; Schmitt et al., 2016). The competition between the MF attraction and LHY repulsion maintains a superfluid state whose density (taking very low values) cannot exceed a certain maximum, thus making it effectively incompressible. This is a reason why this quantum macroscopic state of matter is identified as a superfluid. Accordingly, localized states filled by this quantum fluid are called “droplets.” In addition to the 3D QDs, droplets in the effectively 2D setting, which may exist under the action of tight confinement in one direction, imposed by an external potential, were theoretically predicted [Petrov and Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016)] and experimentally created [Cabrera et al., Science 359, 301-304 (2018)] too. This chapter provides a condensed review of basic theoretical results for stable QDs in binary BECs, maintained through the stable balance between the inter-species attraction and intra-species LHY repulsion. Both fundamental modes and ones with embedded vorticity are considered. The presentation is partly based on a review article by Luo et al. [Front. Phys. 16, 32501 (2021)]. In addition to 2D and 3D self-trapped QDs in free space, the present chapter addresses 2D fundamental and vortex modes trapped in a singular attractive potential, ∼−1/r2, which are stabilized by the LHY repulsion. Two subsequent chapters present a summary of experimental findings for QDs in binary BECs and for QDs in single-component BECs of magnetic atoms.
We study a transversely pumped atomic Bose-Einstein condensate coupled to a single-mode optical cavity, where effective atom-atom interactions are mediated by pump and cavity photons. A number of experiments and theoretical works have shown the formation of a superradiant state in this setup, where interference of pump and cavity light leads to an optical lattice in which atoms self-consistently organize. This self-organization has been extensively studied using the approximate Dicke model (truncating to two momentum states), as well as through numerical Gross-Pitaevskii simulations in one and two dimensions. Here, we perform a full mean-field analysis of the system, including all relevant atomic momentum states and the cavity field. We map out the steady-state phase diagram vs pump strength and cavity detuning, and provide an in-depth understanding of the instabilities that are linked to the emergence of spatiotemporal patterns. We find and describe parameter regimes where the mean field predicts bistability, regimes where the dynamics form chaotic trajectories, instabilities caused by resonances between normal mode excitations, and states with atomic dynamics but vanishing cavity field.