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

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.

Figures (24)

• Figure 1: Experimental determination of the LHY correction. Grand-canonical equation of state of a single-component Bose gas of $^7$Li atoms (green squares) and of a gas of weakly-bound bosonic molecules formed by two $^6$Li atoms (red circles). In the figure, $h_S=P/(2P_0)$ is the pressure of the gas normalized by that of an ideal Fermi gas $P_0$, and $\delta=\hbar/\sqrt{2m\tilde{\mu}}a$ is an interaction parameter. The latter is proportional to the scattering length $a$ and depends on the chemical potential $\tilde{\mu}$, from which the energy of the molecular bound state has been subtracted in the fermionic case. The grand-canonical equation of state $P(n)$ can be related to the more widely used canonical one $\mu(n)$ though thermodynamical relations. Figure extracted from NavonPhD2011.
• Figure 2: Mean-field phase diagram of a quantum mixture. The phase diagram of a bosonic mixture where the two components are denoted as $\uparrow$ and $\downarrow$ is controlled by the three coupling constants $g_{\uparrow \uparrow}$, $g_{\downarrow \downarrow}$, and $g_{\uparrow \downarrow}$. For interspin interactions $-\sqrt{g_{\uparrow\uparrow} g_{\downarrow\downarrow}}<g_{\uparrow\downarrow}<\sqrt{g_{\uparrow\uparrow} g_{\downarrow\downarrow}}$, the system remains miscible. The condition $\left|g_{\uparrow\downarrow}\right|=\sqrt{g_{\uparrow\uparrow}g_{\downarrow\downarrow}}$, indicated in the figure by the diagonal dashed lines, marks the instability of the gas towards collapse (for the attractive $g_{\uparrow\downarrow}<0$ case) or phase separation (for the repulsive $g_{\uparrow\downarrow}>0$ case). Figure extracted from CabreraPhD2018.
• Figure 3: Bogoliubov excitation spectrum of a quantum mixture. (a)The excitation spectrum of a two-component Bose gas with contact interactions features two distinct Bogoliubov branches: density excitations, characterized by in-phase motion of the components, and spin excitations, where the components move out of phase. Each branch has a sound velocity and a healing length associated to it (dashed lines and vertical gray lines in (b) and (c), respectively). Here we depict the simpler case of symmetric intraspin interactions $g_{\uparrow\uparrow}= g_{\downarrow\downarrow}=\bar{g}$. (b) When interspin interactions are repulsive $g_{\uparrow\downarrow}>0$, the Bogoliubov branch of lowest energy corresponds to spin excitations. For $c_s<0$ the energy branch becomes imaginary and the system becomes unstable towards phase separation. (c) The situation is reversed in the case of attractive interspin interactions ($g_{\uparrow\downarrow}<0$), for which the lowest energy branch is the density one. In this case the instability, marked by $c_n<0$, is towards collapse.
• Figure 4: Properties of the self-bound quantum liquid droplets in a mixture of Bose-Einstein condensates. (a) Droplet wavefunction $\phi$ as a function of the radial coordinate in renormalized units $\tilde{r}$. For sufficiently large atom numbers the wavefunction in the center of the droplet reaches a saturation value, leading to a flattop density profile characteristic of a liquid. For small atom numbers, its shape is dominated by surface effects. (b) Excitation spectrum of a droplet as a function of the renormalized atom number $\tilde{N}$ compared to the critical one $\tilde{N_{\mathrm{c}}}$. The blue solid line $-\tilde{\mu}$ corresponds to the particle emission threshold that separates continuum and discrete excitations. At low atom numbers, the droplet has a monopole mode. At high atom numbers, the system exhibits surface ripplon excitations characteristic of a classical liquid. In the intermediate atom number regime, there is a region with no excitations where the system is expected to self-evaporate to zero temperature. Figure extracted from CabreraPhD2018, adapted from PetrovPRL2015.
• Figure 5: Self-bound character of quantum liquid droplets in the different systems. In all cases, the external trapping potential is removed at $t=0$, and the system evolves freely in the absence of confinement afterwards. (a) Dysprosium bosonic atoms with a large magnetic dipole moment. (b) Mixture of potassium Bose-Einstein condensates with competing mean-field interactions of overall strength $\delta a$ (top and middle rows), and single-component BEC of scattering length $a$ (bottom row). In this experiment, a vertical confinement is maintained along the vertical direction to prevent the atoms from falling out of the depth of focus of the imaging system. The self-bound character is therefore only probed along the transverse directions. (c) Ground state polar NaCs molecules, with large electric dipole characterized by the molecular dipolar length $a_{\mathrm{d}2}$ (see ZhangArXiv2025). Figure adapted from SchmittNature2016CabreraPhD2018ZhangArXiv2025.