Excitation spectrum of vortex-lattice modes in a rotating condensate with a density-dependent gauge potential

TL;DR

This work analyzes how a density-dependent gauge potential producing nonlinear rotation reshapes the collective excitations of a quasi-two-dimensional Bose-Einstein condensate. Using Bogoliubov–de Gennes theory and a variational Gaussian approach, the authors show that the dipole mode frequency depends on the nonlinear rotation, signaling a violation of Kohn's theorem, and they derive an analytical dipole frequency along with a breathing-mode frequency largely independent of the nonlinear rotation. They also quantify vortex-displacement (Tkachenko) modes and provide a hydrodynamic expression for the Tkachenko frequency that agrees with BdG results in certain regimes, while revealing roton-like surface excitations whose behavior shifts with the nonlinear rotation. The study highlights the rich interplay between density-dependent gauge fields, vortex lattice dynamics, and collective excitations, with implications for synthetic gauge potentials in ultracold gases.

Abstract

We investigate the collective excitation spectrum of a quasi-2D Bose-Einstein condensate trapped in a harmonic confinement with nonlinear rotation induced by a density-dependent gauge field. Using a Bogoliubov-de Gennes(BdG) analysis, we show that the dipole mode frequency depends strongly on the nonlinear interaction strength, violating Kohn's theorem. Further utilizing the variational analysis, we derive analytical expressions for the dipole and breathing modes, which suggests a strong dependence of the condensate's width on the nonlinear rotation resulting from the density-dependent gauge potential. We identify four different vortex displacement modes -- namely Tkachenko, circular, quadratic, and rational-whose frequencies are sensitive to the nonlinear rotation. In addition to the numerical analysis, we also derive an analytical expression for the Tkachenko mode frequency using a Hydrodynamic approach that agrees well with the frequencies obtained by the Fourier analysis of the transverse and longitudinal vortex dynamics induced by a Gaussian perturbation as well as the frequencies from the BdG excitation spectrum. Our findings also reveal that the excitation spectrum remain symmetric around the angular quantum number $l=0$, with modified energy splitting between $l$ and $-l$ as the nonlinear rotation changes from negative to positive values. Finally, we demonstrate that the surface mode excitation frequency increases (decreases) with an increase in the positive (negative) nonlinear rotation strength.

Figures (11)

• Figure 1: Pseudo-color representation of the condensate density for (a1) $\tilde{C} = -30$, (a2) $\tilde{C} = -20$, (a3) $\tilde{C} = 0$, (a4) $\tilde{C} = 10$, and (a5) $\tilde{C} = 20$ at $\Omega = 0.6$ and $\text{g}=400$. Panels (b1)–(b5) show the phase profiles of density illustrated in (a1)-(a5).
• Figure 2: Variation of different components of the total energy, total energy $E_{\rm T}$, kinetic energy $E_{\rm K}$, potential energy $E_{\rm pot}$, and interaction energy $E_{\rm int}$ as a function of $\tilde{C}$. Right vertical axis shows the variation of angular momentum $\langle L_{z} \rangle$. $E_{\rm int}$ and $E_{\rm pot}$ remains almost unchanged with $\tilde{C}$, while $E_{\rm K}$ show linear increment with $\tilde{C}$. Liner increment of $\langle L_{z} \rangle$ with $\tilde{C}$ indicates in the increase in the number of vortices upon increasing$\tilde{C}$.
• Figure 3: Collective excitation spectrum for various angular quantum numbers $l$ with radial quantum number $n_r=0$ as a function of nonlinear rotation strength $\tilde{C}$ at $\Omega=0.6$ are presented. The triangles and squares denote the Tkachenko and dipole modes, respectively. The pentagon, hexagon, and circles represent the higher order surface modes.
• Figure 4: Real space condensate density at different times portraying the vortex displacement that arises due to the application of a linear perturbation on the ground state which excites mainly the dipole mode for different values of $\tilde{C}$ at $\Omega=0.6$. (a1)-(a5): for fixed $\tilde{C}=-30$ and $t=(0, 10, 20, 30,40)$; (b1)-(b5): for fixed $\tilde{C}=0$ and $t=(0, 10, 20, 30,40)$; (c1)-(c5): for fixed $\tilde{C}=20$ and $t=(0, 10, 20, 30,40)$. The real-time dynamics of the condensate reveal its precessional motion around the trap center with time.
• Figure 5: Panels (a)-(c), shows the center of mass oscillation $x_{\rm cm}(t)$ of the condensate as a function of time for $\tilde{C}=-30, 0, 20$, respectively. (d)-(f) illustrates the Fourier transform of $x_{\rm cm}$. The dominant peaks observed at $\omega_{\rm d}=0.68, 39$, and $0.19$ are the dipole modes obtained via a linear perturbation to the condensate. (g) dipole mode frequency as a function of nonlinear rotation strength $\tilde{C}$ for $\Omega=0.6$. The black dashed line, blue crosses ($\textcolor{blue}{\times}$), and the red squares ($\textcolor{red}{\Box}$) represent the dipole modes obtained using the variational analysis, linear perturbation to the condensate, and from the Bogoliubov de-Gennes equations respectively.