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Journal of Physics B Atomic Molecular and Optical Physics

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

G. M. Kavoulakis

Published Feb 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1361-6455/ae4307

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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.
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.
We investigate dynamical effects of a bright soliton in Bose–Einstein condensed (BEC) systems with local and smooth space variations of the two-body atomic scattering length. It includes a discussion about the possible observation of a new type of standing nonlinear atomic matter wave in cigar-type traps. A rich dynamics is observed in the interaction between the soliton and an inhomogeneity. By considering an analytical time-dependent variational approach and also full numerical simulation of one-dimensional and three-dimensional Gross–Pitaevskii equations, we study processes such as trapping, reflection and transmission of the bright matter soliton due to the impurity. We also derive conditions for the collapse of the bright solitary wave, considering a quasi-one-dimensional BEC with attractive local inhomogeneity.
We experimentally study the excitation modes of bright matter-wave solitons in a quasi-one-dimensional geometry. The solitons are created by quenching the interactions of a Bose-Einstein condensate of cesium atoms from repulsive to attractive in combination with a rapid reduction of the longitudinal confinement. A deliberate mismatch of quench parameters allows for the excitation of breathing modes of the emerging soliton and for the determination of its breathing frequency as a function of atom number and confinement. In addition, we observe signatures of higher-order solitons and the splitting of the wave packet after the quench. Our experimental results are compared to analytical predictions and to numerical simulations of the one-dimensional Gross-Pitaevskii equation.
We discuss the possible observation of a new type of standing nonlinear atomic matter wave in the condensate: the nonlinear impurity mode. It is investigated dynamical effects of a bright soliton in Bose-Einstein-condensed (BEC) systems with local space variations of the two-body atomic scattering length. A rich dynamics is observed in the interaction between the soliton and an inhomogeneity. Processes as trapping, reflection and transmission of the bright matter soliton due to the impurity are studied considering an analytical time-dependent variational approach and also by full numerical simulation. A condition is obtained for the collapse of the bright solitary wave in the quasi-one-dimensional BEC with attractive local inhomogeneity.
We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at a finite temperature. We consider both the non-relativistic case, described by the Gross-Pitaevskii equation, and the relativistic one, described by the cubic nonlinear Klein-Gordon equation. We analyze similarities and differences in the two equations and, in particular, in the phase and amplitude modes (i.e., Goldstone and Higgs modes) of the bosonic matter field. We show that the coupling between phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in the non-relativistic case. Instead, in the relativistic case the spectrum has two branches: one is gapless and the other is gapped. In the non-relativistic limit we find that the relativistic spectrum reduces to the Bogoliubov one. Finally, as an application of the above analysis, we consider the Bose-Hubbard model close to the superfluid-Mott quantum phase transition and we investigate the elementary excitations of its effective action, which contains both non-relativistic and relativistic terms.