Excitation spectrum of a bright solitary wave in a Bose-Einstein condensate and its connection with the Higgs and the Goldstone modes

TL;DR

The paper studies a quasi-1D Bose-Einstein condensate confined on a torus with attractive interactions, revealing a spontaneous symmetry-breaking transition at $\gamma_c = -\tfrac{1}{2}$ from a homogeneous to a localized density. Using analytic mean-field and many-body diagonalization in a truncated basis, it derives two main excitation regimes: a homogeneous phase with a Bogoliubov-type spectrum $\Delta E_n(L) = (2n+L)\omega_{\rm hom} - \gamma L^2/N$ and a localized phase with a breathing-mode frequency $\omega_{\rm loc}$ and an effective mass $m_{\rm eff}$ giving $\Delta E_n(L) = n\omega_{\rm loc} + \frac{L^2}{N}\frac{1}{2m_{\rm eff}}$, where $\omega_{ m hom}=\sqrt{2\gamma+1}$, $\omega_{ m loc}=\sqrt{\tfrac{4}{7}(3\gamma-2)(1+2\gamma)}$, and $m_{\rm eff}=\tfrac{2}{\gamma}\tfrac{2\gamma+1}{1-5\gamma}$. The results reveal Higgs-like amplitude (breathing) modes and Goldstone-like translational (rotational) modes, with mode frequencies softening near the transition and good agreement between analytic predictions and numerical diagonalization. The framework provides experimentally accessible signatures in toroidal traps and clarifies how symmetry breaking shapes the excitation spectrum in this clean, controllable setting.

Abstract

We consider the problem of Bose-Einstein condensed atoms, which are confined in a (quasi) one-dimensional toroidal potential. We focus on the case of an effective attractive interaction between the atoms. The formation of a localized blob (i.e., a ``bright" solitary wave) for sufficiently strong interactions provides an example of spontaneous symmetry breaking. We evaluate analytically and numerically the excitation spectrum for both cases of a homogeneous and of a localized density distribution. We identify in the excitation spectrum the emergence of the analogous to the Goldstone and the Higgs modes, evaluating various relevant observables, gaining insight into these two fundamental modes of excitation.

Figures (8)

• Figure 1: The energy $E_{\rm MF}/N$ of Eq. (\ref{['emffg']}) (in units of $e_1$) for $L/N = 0$ and for two values of $\gamma$, as function of Re$(d_{-1})$ and Im$(d_{-1})$. In the upper plot $\gamma = -0.1 > \gamma_c$ and the minimum of the potential is at the center, where $d_{-1} = d_1 = 0$. As a result, the density distribution of the atoms is homogeneous. In the lower plot $\gamma = -1 < \gamma_c$, $|d_{-1}| = |d_1| \neq 0$, and we have the formation of a localized blob (i.e., a "bright" solitary wave).
• Figure 2: The density $n(\theta) = |\Psi(\theta)|^2$ of Eq. (\ref{['densityy']}) (in units of $1/R$) for $L/N = 0$ and for three values of $\gamma > -1/2$ (straight, horizontal line), $\gamma = -0.55$ (dashed curve) and $\gamma = -0.6$ (solid curve). Here $\varphi_c = 0$.
• Figure 3: Blue stars: The amplitudes $f_{k}^2$, Eq. (\ref{['mbs']}), for $k = 0, 1$, and 2 (on a logarithmic $y$ scale), for $\gamma = -0.1$, which result from the numerical diagonalization of the many-body Hamiltonian, for $N = 1000$ and $L = 0$. Dashed curve: The analytic expression $f_k^2 = (|\gamma|/2)^{2k}$.
• Figure 4: Blue dots: The amplitudes $f_{k}^2$, Eq. (\ref{['mbs']}), for $k = 0, 1, 2, \dots, 180$ which result from the numerical diagonalization of the many-body Hamiltonian, with $\gamma = -1$, $N = 1000$ and $L = 5$. Solid curve: The analytic expression, Eq. (\ref{['fk22']}). The difference between the dots and the curve is hardly visible.
• Figure 5: The effective mass $m_{\rm eff}$ (in units of $2 M$) from Eq. (\ref{['effmass']}) (solid curve), along with the values that we extract from the diagonalization of the many-body Hamiltonian (squares), as function of $\gamma$. Here we considered $N = 1000$ atoms in the diagonalization of the Hamiltonian.