Spin and density excitations of one-dimensional self-bound Bose-Bose droplets

Abstract

We study density and spin excitations of one-dimensional self-bound Bose-Bose droplets within Bogoliubov theory, and show that spin excitations come alive, especially as the interspecies coupling is made less attractive. We argue that spin excitations are particularly relevant in the one-dimensional droplet regime, where droplets are realized within the mean-field stability regime, as has been confirmed by the Quantum Monte Carlo simulations. As the interspecies coupling strength increases within the mean-field stability regime, spin modes ultimately fall below the particle-emission threshold, thus becoming observable in the droplet spectrum. We analyze the Bogoliubov model for both pseudospinor and population-imbalanced scalar mixtures, encompassing both the density and spin sectors, and corroborate our findings through variational analysis of density and spin breathing modes, as well as real-time dynamics. Additionally, we compare our results with Petrov's "original" theory, which considers the Lee-Huang-Yang (LHY) correction at the attractive edge of the mean-field stability regime and a beyond-LHY description of Bose-Bose mixtures.

Figures (11)

• Figure 1: Density ratio $\beta = n_1/n_2$ obtained by a minimization of the energy per particle $\mathcal{E}(n,\beta)/n$ as a function $g_{11}/g_{22}$ for $g_{12} = -0.6 g_{22}$ for Bogoliubov and Petrov's theories. Symbols denote the density ratio at the center of the finite-sized self-trapped solutions with $N = 20$ and $60$ obtained by numerically solving the GP Eqs. (\ref{['eeGPE']}) for Bogoliubov and (\ref{['oeGPE']}) for Petrov’s theory. For comparison, the simpler estimate of density ratio $\sqrt{g_{22}/g_{11}}$, which minimizes the meanfield energy density for $g_{12} = -\sqrt{g_{11}g_{22}}$ is also shown.
• Figure 2: The density profiles of the self-bound ground-state solutions of the Bogoliubov model, viz. Eqs. (\ref{['eeGPE']}), for equal intraspecies coupling strengths, $g_{11} =g_{22} = g$. Component density for (a) $N = 20$ and (b) $g_{12} = -0.6 g$, where $n_1(x) = n_2(x)$.
• Figure 3: (a)-(d) Solid circles are excitation energies calculated by solving BdG Eqs. (\ref{['bdg']}) for the Bogoliubov model. (a) and (b): Excitation energies as a function of $g_{12}/g$ for (a) $N = 20$ and (b) $N = 80$. (c) and (d): Excitation energies as a function of $N$ for (c) $g_{12} =-0.6$ and (d) $g_{12} =-0.5$. Dark (dark-blue) and light (yellow) solid circles correspond to spin and density modes, respectively. 'Cross' and 'star' symbols are the energies of density- and spin-breathing modes, respectively, obtained from the variational analysis. The energies are in units of particle emission threshold $|\mu|$, and only the energies below the emission threshold are shown.
• Figure 4: The quasiparticle amplitudes of the three lowest lying density and spin modes for $N = 60$ and $g_{12} = -0.6 g$ [see Fig. \ref{['sp_br']}(c)]. Left panel: (a) first, (c) second and, (e) third density mode. Right panel: (b) first, (d) second, and (f) third spin mode. Density modes correspond to in-phase oscillations with $u_1 = u_2$ and $v_1=v_2$, whereas spin modes correspond to out-of-phase oscillations with $u_1 = -u_2$ and $v_1 = -v_2$.
• Figure 5: Comparison of density- and spin-breathing mode energies obtained from Petrov's and Bogoliubov's models. (a) Excitation energies as a function of $g_{12}/g$ for $N = 80$. (b) Excitation energies as a function of $N$ for $g_{12}/g=-0.6g$. Numerical BdG results are shown together with variational calculations (continuous lines in matching colors). The blue dotted (green dash-dotted) curve denotes the density-breathing mode obtained from the Bogoliubov (Petrov) model, while the orange dashed (red solid) curve corresponds to the spin-breathing mode from the Bogoliubov (Petrov) model. Inset in (a): comparison of variational results for $N=20$ within the Bogoliubov model, where the LHY term is expanded up to first, second, and third order in $(\sigma_1(t)-\sigma_2(t))/2$.