Stationary Properties and Dynamical Response of One-Dimensional Multi-Component Quantum Droplets

Abstract

Ultracold atomic gases provide versatile platforms for exploring quantum many-body (MB) phenomena, while offering an unprecedented level of control of the external geometry, inter-atomic interactions and number of particles. In particular, several fundamental aspects of quantum MB physics such as collective excitations, correlation effects in both the ground state and non-equilibrium dynamics, as well as dimensional crossovers and quantum phase transitions, can be cleanly induced and observed in the laboratory. From the theoretical side, understanding these phenomena and novel correlated phases of matter beyond the standard Mean-Field (MF) approach, is a formidable task.

The present cumulative thesis aims to contribute to the understanding of the influence of quantum fluctuations in weakly interacting bosonic mixtures, by studying the recently discovered self-bound quantum droplets in one spatial dimension (1D). They are studied in terms of their ground state configurations and in terms of their dynamical response following a quench. To this end, a combination of numerical and analytical techniques is employed. The time-dependent MB Schr\"odinger equation is numerically solved utilizing the sophisticated {\it ab-initio} method: Multi-Layer Multi-Configuration Time-Dependent Hartree Method for Atomic Mixtures (ML-MCTDHX). Moreover, the perturbative Bogoliubov theory is employed to derive analytical results. Finally, effective models which provide qualitative and quantitative insights into the observed phenomenology are derived and their predictions are compared to suitable numerical simulations.

The first part of this thesis is focused on exact numerical simulations of the ground state and non-equilibrium quantum dynamics of 1D two-component bosonic mixtures, using the ML-MCTDHX approach, in regimes slightly outside the formal validity region of the standard Lee-Huang-Yang (LHY) theory for quantum droplets. Namely, a weak harmonic trap is imposed to emulate usual experimental conditions and test the robustness of droplet configurations beyond flat geometries. Indeed, droplet configurations are revealed to persist in both homonuclear and heteronuclear settings, while characteristic patterns are found in higher-order observables such as their two-body correlation functions and inter-component entanglement. Quenches of the position and frequency of the harmonic trap result in a dipole motion independent of the system parameters and an interaction-dependent breathing mode respectively. The latter leads to delocalization characterized by enhanced inter-component entanglement and long-range two-body intra-component correlations, as well as correlation-induced dephasing for species selective quenches. Moreover, droplet configurations of the bath were found to persist also in highly particle imbalanced mixtures, while their phenomenology was shown to be adequately explained by an effective model derived within the context of the LHY theory. Interestingly, this particle imbalanced setting exhibits droplet to soliton transitions, while it indicates a strong connection between the phenomenology of quantum droplets and that of few interacting impurities.

The second part of this thesis is devoted on the analytical derivation, and subsequent study, of the equations of motion for multi-component 1D droplets, which are valid across the entire MF stability regime. Motivated by the robustness of the LHY phenomenology, in regimes where the theory is not formally valid as discussed in the first part of this thesis, the perturbative Bogoliubov theory for 1D bosonic mixtures was applied, while focusing on requiring minimal assumptions. This allowed the derivation of the LHY energy and generalized extended Gross-Pitaevskii Equations (eGPEs) for quantum droplets consisting of two, three, and four components. The original eGPEs for two-component droplets were shown to be obtained from these new exact equations as the first order approximation at the edges of the MF stability regime. For repulsive interactions a previously unseen early onset of phase-separation was found to occur for both homonuclear and heteronuclear two-component mixtures. The investigation was thereafter concentrated on three‑component particle imbalanced settings, revealing a multitude of mixed droplet configurations characterized also by distinct elementary excitation spectra.