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arXiv:2508.06240

Bogoliubov analysis of Higgs mode in trapped Fermi superfluids with spatial inhomogeneity

Kakimoto, Kensuke, Takahashi, Junichi, Y. Yamanaka

Published Aug 8, 2025• 0 citations
arXivHTMLPDF

Abstract

The Higgs mode is a key component in the spontaneous breaking of a continuous symmetry along with the Nambu-Goldstone mode, and has been studied extensively for homogeneous systems. We consider it for inhomogeneous systems, using the superfluid of harmonically trapped ultracold Fermi atomic gas. The Fermionic field operators are expanded in a complete set of wave functions corresponding to inhomogeneous situation. Within the Hatree-Fock approximation, we derive integral equations from the Bogoliubov-de Gennes equations, which lead to the frequencies of the collective modes, including the Higgs and Nambu-Goldstone modes. The results show that the frequency of the Higgs mode equals twice the absolute value of the order parameter at the center of trap. This feature is robust against variations in the interaction strength, trap potential, and temperature. These results are consistent with previous theoretical and experimental studies.

DOI
10.48550/arxiv.2508.06240

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The Higgs mode is a key component in the spontaneous breaking of a continuous symmetry along with the Nambu-Goldstone mode, and has been studied extensively for homogeneous systems. We consider it for inhomogeneous systems, using the superfluid of harmonically trapped ultracold Fermi atomic gas. The Fermionic field operators are expanded in a complete set of wave functions corresponding to inhomogeneous situation. Within the Hatree-Fock approximation, we derive integral equations from the Bogoliubov-de Gennes equations, which lead to the frequencies of the collective modes, including the Higgs and Nambu-Goldstone modes. The results show that the frequency of the Higgs mode equals twice the absolute value of the order parameter at the center of trap. This feature is robust against variations in the interaction strength, trap potential, and temperature. These results are consistent with previous theoretical and experimental studies.
In quantum many-body systems with spontaneous breaking of continuous symmetries, Higgs modes emerge as collective amplitude oscillations of order parameters. Recently, Higgs mode has been observed in the ultracold Fermi gas. In the present paper, we use the time-dependent Bogoliubov-de Gennes equations to investigate Higgs amplitude oscillations of the superfluid order parameter in a Fermi gas induced by a rapid change of the s{\it s}s-wave scattering length. In particular, we investigate the Higgs mode with different values of the initial scattering length. We find that the energy of the Higgs mode coincides with the threshold energy of the pair-breaking excitation, and exponent of the power-low decay of the Higgs mode γγγ continuously changes between γ=−1/2γ=-1/2γ=−1/2 and γ=−3/2γ=-3/2γ=−3/2 through the Bardeen-Cooper-Schrieffer-Bose-Einstein condensation (BCS-BEC) crossover. Moreover, we propose the optimal ramp speed of the scattering length for observing the clearest Higgs oscillations.
Abstract Experiments with ultracold gases on the lattice give the opportunity to realize superfluid fermionic mixtures in a trapping potential. The external trap modifies the chemical potential locally. Moreover, this trap also introduces non-homogeneity in the superconducting order parameter. There are, among other approaches, two methods which can be used to describe the system of two-component mixtures loaded into an optical lattice: the Local Density Approximation (LDA) and the self-consistent Bogoliubov–de Gennes equations. Here, we compare results obtained within these two methods. We conclude that the results can be distinguishable only in the case of a small value of the pairing interaction.
We study collective modes of superfluid Bose gases in optical lattices at commensurate fillings.We focus on the vicinity of the quantum phase transition to the Mott insulator, where there exists the Higgs amplitude mode in addition to the Nambu-Goldstone phase mode associated with the spontaneous U(1) symmetry breaking.We analyze finite-temperature effects on the damping of the collective modes by using an effective spin-1 model and field-theoretical methods based on the finite-temperature Green's function.We calculate the damping rates up to 1-loop order and evaluate them analytically and numerically.We show that the damping rate of the Higgs mode increases with increasing temperature but it remains underdamped up to the typical temperature achieved in experiments.Moreover, we find that the Nambu-Goldstone mode attenuates via a Landau damping process resulting from interactions with the Higgs mode and it can be overdamped at the typical temperature in a certain parameter region..............
We consider the low energy collective mode spectrum of a superfluid Fermi gas in a spherical trap in the collisionless regime. Using a self-consistent random-phase approximation, the effects of superfluidity on modes of dipole and quadrupole symmetries are systematically examined. The spectrum is calculated for varying pairing strength and temperature, and we identify several spectral features such as the emergence of Goldstone modes that can be used to detect the onset of superfluidity. Our analysis is relevant for present experiments aimed at observing a superfluid phase transition in trapped Fermi gases.
A new field of research, which gained interest in the past years, is the field of non- equilibrium physics of strongly interacting fermionic systems. In this thesis we present a novel apparatus to study an ultracold strongly interacting superfluid Fermi gas driven out of equilibrium. In more detail, we study the excitation of a collective mode, the Higgs mode, and the dynamics occurring after a rapid quench of the interaction strength. Ultracold gases are ideal candidates to explore the physics of strongly interacting Fermi gases due to their purity and the possibility to continuously change many different system parameters, such as the interaction strength. The novel apparatus enabling the experiments outlined below is built from scratch. The setup is described and characterized in detail. We produce a fermionic superfluid of 4 × 10^6 6Li atoms at a temperature of T/T_F = 0.07 ± 0.02, where T_F is the Fermi temperature. This is achieved within 23 s by a combination of laser cooling, radio frequency evaporation of 23Na and simultaneous sympathetic cooling of 6Li in an optical plugged magnetic trap and subsequent evaporative cooling of 6Li in a dipole trap. In this thesis we provide the first experimental evidence of the Higgs mode in a strongly interacting Fermi gas in the crossover from a BCS superfluid to a molecular Bose-Einstein condensate. We develop a novel excitation method, which directly couples to the amplitude of the order parameter. This is achieved by continuously changing the population of one spin component by driving a radio frequency transition to a previously unoccupied third hyperfine state. This effectively modulates the interaction strength and the amplitude of the order parameter. We spectroscopically observe a resonance behavior at twice the gap frequency. For strong coupling, the peak width broadens and eventually the Higgs mode disappears when the Cooper pairs turn into tightly bound dimers signaling the instability of the Higgs mode. It has been suggested that the Higgs mode frequency is a precise measure of the superconducting gap in the BEC-BCS crossover, where the exact value of the gap is yet unknown and numerical calculations are challenging. Hence, our novel method provides a unique technique to determine the superconducting gap. Moreover, we perform rapid variations of the interaction parameter and vary both the initial interaction strength and the amplitude of the quench. The rapid quenches are performed by rapidly transferring one spin component of a two-component spin mixture into a third state with a different interaction strength by using a radio frequency transition. Using this novel method, we can perform the quenches in half the Fermi time, which is experimentally very close to a sudden change and to date faster than any other research group performing interaction quenches with ultra cold gases. In the experiment, we observe a fast relaxation to a zero order parameter for large quenches, whereas for small quenches we observe, after a sudden drop, a revival of the order parameter and equilibration to a long-term superfluid steady state. Our measurement provides the first evidence of the collapse and subsequent revival of order in a strongly interacting fermionic system.
We study the Bogoliubov collective excitations of harmonically trapped superfluid Fermi gases in the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluid to Bose-Einstein condensate (BEC) beyond Thomas-Fermi (TF) limit. Starting from a generalized Gross-Pitaevskii equation and an equation of state valid for the whole crossover, we derive Bogoliubov--de Gennes (BdG) equations for low-lying collective modes at zero temperature. We use a Fetter-like variational ground state wave function to remove the noncontinuity of slope at the boundary of condensate, which appears in the TF limit. We solve the BdG equations analytically and obtain explicit expressions for all eigenvalues and eigenfunctions, valid for various crossover regimes and for traps with spherical and axial symmetries. We discuss the feature of these collective excitations in the BCS-BEC crossover and show that the theoretical result obtained agrees with available experimental data near and beyond the TF limit.