We analyze the role played by quantum fluctuations on a Raman spin-orbit-coupled system in the stripe phase under attractive interspin interactions. We show that beyond-mean-field effects stabilize the collapse predicted by mean-field theory for attractive enough interspin interactions 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 dropletlike solutions that feature internal structure through a striped pattern. We estimate the critical binding number of these droplets and show that their value is experimentally accessible. We report an approximate energy functional in order to ease the evaluation of the Lee-Huang-Yang correction in practical terms.
We analyze the role played by quantum fluctuations on a Raman spin-orbit-coupled system in the stripe phase under attractive interspin interactions. We show that beyond-mean-field effects stabilize the collapse predicted by mean-field theory for attractive enough interspin interactions 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 dropletlike solutions that feature internal structure through a striped pattern. We estimate the critical binding number of these droplets and show that their value is experimentally accessible. We report an approximate energy functional in order to ease the evaluation of the Lee-Huang-Yang correction in practical terms.
We study the quantum-droplet state in a three-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–Huang–Yang 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.
A Bose gas under the effect of Raman spin-orbit coupling (SOC) is analyzed using the discrete spin T-moves diffusion Monte Carlo method. Upon computing the energy as well as the static structure factor and the superfluid fraction of the system, the emergence of an energetically favorable supersolid stripe state is observed, which is in agreement with recent observations. A significant enhancement of the stability of the stripe phase with respect to the mean-field prediction is observed when the strength of the interatomic correlations is increased. We also quantify and characterize the degree of superfluidity of the stripes and show that this quantity is mostly determined by the ratio between the Raman coupling and the square of the momentum difference between the pair of SOC-inducing laser beams.
We theoretically investigate a one-dimensional quantum droplet with spin-orbit (SO) coupling and periodically modulated Raman coupling. Using a Floquet approach, we derive an effective time-independent model that reveals a controlled phase diagram. Crucially, the interplay of quantum fluctuations encoded in the Lee-Huang-Yang correction and SO coupling suppresses the plane-wave phase that is typical of mean-field SO-coupled condensates. The ground state is instead restricted to two distinct phases: a stripe phase, with periodic density modulations, and a zero-momentum phase, with a flat-top density profile. The plane-wave droplet is a metastable state of the system. We demonstrate that the boundary between these phases can be dynamically manipulated by tuning the driving parameters. Furthermore, we explore the system's response to quantum quenches, uncovering a wealth of nonequilibrium phenomena, including the dynamical formation of stable, solitonlike structures. Our work establishes periodic driving as a powerful tool for the dynamical control of self-bound quantum matter and elucidates the fundamental role of quantum fluctuations in shaping the phase diagram of SO-coupled droplets.
We study the static and traveling droplet phases of spin-orbit coupled, one-dimensional Bose-Bose mixtures with a Lee-Huang-Yang correction. For the static droplets, employing a variational analysis for an infinitely extended mixture, we show that the system can feature two ground-state droplet phases: a stripe droplet phase up to a critical Rabi coupling strength, at which it undergoes a first-order transition to a zero-momentum droplet phase. The variational results for the ground-state properties are shown to be in good agreement with numerical solutions for droplets in the thermodynamic limit as well as finite-size droplets in the "flat-top regime". For finite-size droplets, the stripe phase can feature multiple stationary states with different numbers of peaks in their stripe density patterns. The calculation of the excitation spectrum for these stationary states confirms their stability. To study the moving droplets, we investigate the ground-state phases of the system in the droplet's comoving frame. We show that in the comoving frame, droplets acquire a finite longitudinal magnetization. As a consequence, the phase diagram features two distinct phases: a traveling stripe droplet and a plane-wave droplet. The real-time evolution of these droplets in the space-fixed frame, complemented by the Bogoliubov analysis in the comoving frame, illustrates the stability of these traveling droplet phases.
We investigate the ground state and dynamics of one-dimensional spin-orbit coupled (SOC) quantum droplets within the extended Gross--Pitaevskii approach. As the SOC wave number increases, stripe droplet patterns emerge, with a flat-top background, for larger particle numbers. The surface energy decays following a power-law with respect to the interactions. At small SOC wave numbers, a transition from Gaussian to flat-top droplets occurs for either a larger number of atoms or reduced intercomponent attraction. The excitation spectrum shows that droplets for relatively small SOC wave numbers are stable, otherwise stripe droplets feature instabilities as a function of the particle number or the interactions. We also witness rich droplet dynamical features using velocity imprinting and abrupt changes in the intercomponent interaction or the SOC parameters. Characteristic responses include breathing oscillations, expansion, symmetric and asymmetric droplet fragmentation, admixtures of single and stripe droplet branches, and erratic spatial distributions suggesting the triggering of relevant instabilities. Our results reveal the controlled dynamical generation and stability properties of stripe droplets that should be detectable in current cold-atom experiments.
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.