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arXiv:2603.17745

Bosonic quantum mixtures with competing interactions: quantum liquid droplets and supersolids

Sarah Hirthe, Leticia Tarruell

cond-mat.quant-gasphysics.atom-phquant-phSubmitted Mar 18, 2026• Announced Mar 19, 2026• 0 citations

Abstract

These lecture notes contain an introduction to quantum simulation of bosonic systems in the continuum, focusing on weakly interacting Bose-Bose mixtures with competing mean-field interactions. When the values of such interactions are fine-tuned to almost completely cancel the mean-field energy, quantum fluctuations become apparent and dominate the behavior of the system, stabilizing an ultradilute quantum liquid phase. An analogous situation appears in single-component dipolar quantum gases. We review the mechanism that gives rise to this exotic quantum liquid, which can form droplets that are self-bound in the absence of any external confinement, and discuss their properties and dynamics in both the mixture and the dipolar cases. In dipolar gases, arrays of dipolar droplets stabilized by quantum fluctuations can establish global phase coherence and form supersolids. In bosonic mixtures, supersolidity can emerge already at the mean-field level through spin-orbit coupling. We discuss the properties of such spin-orbit-coupled supersolids, comparing them to their dipolar counterparts. Specifically, we focus on their periodic density modulation, phase coherence, and peculiar excitation spectrum, which hosts both superfluid and crystal excitations. Finally, we conclude by discussing open research directions in the areas of quantum liquid droplets and spin-orbit-coupled supersolids, in particular at the interface of the two research topics.

Comments: This manuscript has been submitted to appear in the Proceedings of the Course 214 "Quantum Computers and Simulators with Atoms" of the International School of Physics "Enrico Fermi" (Varenna, July 2024). 50 pages, 24 figures

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These lecture notes contain an introduction to quantum simulation of bosonic systems in the continuum, focusing on weakly interacting Bose-Bose mixtures with competing mean-field interactions. When the values of such interactions are fine-tuned to almost completely cancel the mean-field energy, quantum fluctuations become apparent and dominate the behavior of the system, stabilizing an ultradilute quantum liquid phase. An analogous situation appears in single-component dipolar quantum gases. We review the mechanism that gives rise to this exotic quantum liquid, which can form droplets that are self-bound in the absence of any external confinement, and discuss their properties and dynamics in both the mixture and the dipolar cases. In dipolar gases, arrays of dipolar droplets stabilized by quantum fluctuations can establish global phase coherence and form supersolids. In bosonic mixtures, supersolidity can emerge already at the mean-field level through spin-orbit coupling. We discuss the properties of such spin-orbit-coupled supersolids, comparing them to their dipolar counterparts. Specifically, we focus on their periodic density modulation, phase coherence, and peculiar excitation spectrum, which hosts both superfluid and crystal excitations. Finally, we conclude by discussing open research directions in the areas of quantum liquid droplets and spin-orbit-coupled supersolids, in particular at the interface of the two research topics.

Quantum fluctuations can stabilize Bose-Einstein condensates (BEC) against the mean-field collapse. Stabilization of the condensate has been observed in quantum degenerate Bose-Bose mixtures and dipolar BECs. The fine-tuning of the interatomic interactions can lead to the emergence of two new states of matter: liquid-like self-bound quantum droplets and supersolid crystals formed from these droplets. We review the properties of these exotic states of matter and summarize the experimental progress made using dipolar quantum gases and Bose-Bose mixtures. We conclude with an outline of important open questions that could be addressed in the future.

Recently achieved two-component dipolar Bose-Einstein condensates open exciting possibilities for the study of mixtures of ultradilute quantum liquids. While nondipolar self-bound (without external confinement) mixtures are necessarily miscible with an approximately fixed ratio between the two densities, the density ratio for the dipolar case is free. Therefore, 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 immiscible droplets are possible due to mutual nonlocal intercomponent attraction, which results in the formation of a droplet molecule. Moreover, our analysis of the impurity regime shows that quantum fluctuations in the majority component crucially modify the miscibility of impurities. Our work opens intriguing perspectives for the exploration of spinor physics in ultradilute liquids, which should resemble to some extent that of ^{4}He-^{3}He droplets and impurity-doped helium droplets.

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Abstract The simultaneous presence of two competing inter-particle interactions can lead to the emergence of new phenomena in a many-body system. Among others, such effects are expected in dipolar Bose–Einstein condensates, subject to dipole–dipole interaction and short-range repulsion. Magnetic quantum gases and in particular Dysprosium gases, offering a comparable short-range contact and a long-range dipolar interaction energy, remarkably exhibit such emergent phenomena. In addition an effective cancellation of mean-field effects of the two interactions results in a pronounced importance of quantum-mechanical beyond mean-field effects. For a weakly dominant dipolar interaction the striking consequence is the existence of a new state of matter equilibrated by the balance between weak mean-field attraction and beyond mean-field repulsion. Though exemplified here in the case of dipolar Bose gases, this state of matter should appear also with other microscopic interactions types, provided a competition results in an effective cancellation of the total mean-field. The macroscopic state takes the form of so-called quantum droplets. We present the effects of a long-range dipolar interaction between these droplets.

Supersolidity---a coexistence of superfluidity and crystalline or amorphous density variations---has been vividly debated ever since its conjecture. While the initial focus was on helium-4, recent experiments uncovered supersolidity in ultracold dipolar quantum gases. Here we propose a self-bound supersolid phase in a binary mixture of Bose gases with short-range interactions, making use of the nontrivial properties of spin-orbit coupling. We find that a first-order phase transition from a self-bound supersolid stripe phase to a zero-minimum droplet state of the Bose gas occurs as a function of the Rabi coupling strength. These phases are characterized using the momentum distribution, the transverse spin polarization, and the superfluid fraction. The critical point of the transition is estimated in an analytical framework. The predicted density-modulated supersolid stripe and zero-minimum droplet phase should be experimentally observable in a binary mixture of

^{39}\mathrm{K}

with spin-orbit coupling.

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.