Exciting the long-lived Higgs mode in superﬂuid Fermi gases with particle removal Guitao Lyu1Kui-Tian Xi2 3 1 Sukjin Yoon4 5 6Qijin Chen7 8 9 and Gentaro Watanabe1 10 1School of Physics and Zhejiang Institute of Modern Physics

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Exciting the long-lived Higgs mode in superﬂuid Fermi gases with particle removal

Guitao Lyu,1, ∗Kui-Tian Xi,2, 3, 1, †Sukjin Yoon,4, 5, 6 Qijin Chen,7, 8, 9, ‡and Gentaro Watanabe1, 10, §

1School of Physics and Zhejiang Institute of Modern Physics,

Zhejiang University, Hangzhou, Zhejiang 310027, China

2College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

3Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing 211106, China

4Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea

5Asia Paciﬁc Center for Theoretical Physics, Pohang, Gyeongsangbuk-do 37637, Korea

6Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea

7Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences,

University of Science and Technology of China, Hefei, Anhui 230026, China

8Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics,

University of Science and Technology of China, Shanghai 201315, China

9Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China

10Zhejiang Province Key Laboratory of Quantum Technology and Device,

Zhejiang University, Hangzhou, Zhejiang 310027, China

(Dated: March 2, 2023)

Experimental evidence of the Higgs mode in strongly interacting superﬂuid Fermi gases had not been ob-

served until recently [Behrle et al., Nat. Phys. 14, 781 (2018)]. Due to the coupling with other collective modes

and quasiparticle excitations, generating stable Higgs-mode oscillations is challenging. We study how to excite

long-lived Higgs-mode oscillations in a homogeneous superﬂuid Fermi gas in the BCS-BEC crossover. We

ﬁnd that the Higgs mode can be excited by time-periodically modulating the scattering length at an appropriate

amplitude and frequency. However, even for a modulation frequency below twice the pairing-gap energy, quasi-

particles are still excited through the generation of higher harmonics due to nonlinearity in the superﬂuid. More

importantly, we ﬁnd that persistent Higgs-mode oscillations with almost constant amplitude can be produced by

removing particles at an appropriate momentum, and the oscillation amplitude can be controlled by the number

of removed particles. Finally, we propose two ways to experimentally realize of particle removal.

I. INTRODUCTION

The Higgs mode is an amplitude oscillation of the Higgs

ﬁeld which plays an important role in mass generation for el-

ementary particles via spontaneous symmetry breaking [1–3].

As an analogy, the Higgs mode in condensed-matter systems

refers to an amplitude oscillation of the order parameter re-

lated to the spontaneous breaking of a continuous symme-

try [4–6]. While the excitation and detection of the Higgs

mode in high-energy experiments require extremely large-

scale facilities such as the Large Hadron Collider at CERN,

they are feasible with tabletop experiments for condensed-

matter systems [7–17]. Therefore, the latter is an important

test bed for studying the Higgs mode and has been drawing

attention especially after the detection of the Higgs boson in

high-energy experiments in 2012 [18,19].

The ﬁrst work on the Higgs mode in condensed matter was

done by Volkov and Kogan around a half century ago [20],

although it was not called the Higgs mode at that time. They

found that the frequency of the Higgs mode in the supercon-

ductor is '2∆0/¯

h(∆0is the pairing gap in equilibrium) and

the amplitude decays with time tin a power law of t−1/2.

However, in the context of the BCS-BEC crossover in super-

ﬂuid Fermi gases, it was predicted that the amplitude decay

∗guitao@zju.edu.cn

†xiphys@nuaa.edu.cn

‡qjc@ustc.edu.cn

§gentaro@zju.edu.cn

follows a different power law of t−3/2in the BEC regime [21],

unlike the t−1/2behavior in the BCS regime. The earliest evi-

dence for the existence of the Higgs mode was found in super-

conductors [7], where unexpected peaks in Raman scattering

were revealed to be the Higgs mode [22,23]. Since the re-

alization of degenerate atomic Fermi gases [24], there have

been many theoretical studies on the Higgs mode in super-

ﬂuid Fermi gases with various types of external drive, such as

a sudden quench [21,25–35], ramping [36–38], time-periodic

modulation of the interaction strength [36,37,39–41], and so

on [42,43]. The visibility of the Higgs mode in fermionic su-

perﬂuids has also been studied [44–46]. Regarding the small

oscillations around the equilibrium state after quench, a de-

cay is inevitable in homogeneous three-dimensional systems

in the BCS-BEC crossover since the Higgs mode is coupled

with the Goldstone mode due to the absence of particle-hole

symmetry [47]. On the other hand, when the initial pertur-

bation is sufﬁciently large or for some particular nonequi-

librium initial states, persistent Higgs-mode oscillations have

been predicted to be possible [26,28–30]. Generating stable

Higgs-mode oscillations is important for future practical ap-

plications. For example, persistent Higgs-mode oscillations

enable us to probe the information about the material phase,

such as the interband couplings in multiband superconductors

(see, e.g., Refs. [48–50]) and the superconductivity in pho-

toinduced states (see, e.g., Refs. [51–54]). A long-lived Higgs

mode is also possible in trapped two-dimensional Fermi gases,

where the trapping conﬁnement can stabilize the Higgs mode

by making its decay channels discrete [55]. However, ex-

perimental evidence for the Higgs mode in atomic superﬂuid

arXiv:2210.09829v2 [cond-mat.quant-gas] 1 Mar 2023

Fermi gases was observed only in recent years [17] because

generating stable Higgs-mode oscillations against decay has

been challenging.

In the present paper, we propose a scheme for generating

long-lived Higgs-mode oscillations in a homogeneous super-

ﬂuid Fermi gas in the BCS-BEC crossover. We approach

this problem by numerical simulations of the time-dependent

Bogoliubov-de Gennes (BdG) equations. First, we consider

time-periodic modulations of the scattering length in the BCS-

BEC crossover. A basic idea behind this scheme is to sus-

tain the Higgs-mode oscillations against decay by a contin-

uous drive. Although this scheme can generate Higgs-mode

oscillations, other unwanted excitations are also created due

to the nonlinearity of the BdG equations. As an alternative

scheme to excite the Higgs mode, we propose an unconven-

tional type of quench by removing particles around certain

momentum. This scheme can generate stable and persistent

Higgs-mode oscillations whose lifetime and amplitude are

much larger than those observed in a recent experiment [17].

By tuning the momentum of the removed particles at the peak

of the pair wave function Fk, the long-lived Higgs mode can

be excited. Furthermore, the amplitude of the oscillations can

be controlled by the number of removed particles. The long

lifetime and large amplitude are strong advantages for future

experimental detections and investigations. Particularly, the

amplitude of the oscillations excited by this scheme remains

almost constant for a long time instead of experiencing prompt

power-law decay.

This paper is organized as follows. In Sec. II, we present

the formulation of our simulations. In Sec. III, we discuss

the scheme using the time-periodic modulation of the scatter-

ing length and its resulting higher-harmonic excitations. The

scheme of removing particles will be discussed in Sec. IV. In

Sec. V, we summarize our work and discuss how to experi-

mentally realize the particle removal.

II. FORMULATION

We consider a homogeneous superﬂuid Fermi gas at zero

temperature in the BCS-BEC crossover. Our system con-

sists of attractively interacting unpolarized (pseudo)spin-1/2

fermionic atoms. The time evolution of the pairing gap

∆(t)is studied by numerically solving the time-dependent

Bogouliubov-de Gennes (TD-BdG) equations [56–58]:

i¯

dt uk(t)

vk(t)=H0∆(t)

∆∗(t)−H0uk(t)

vk(t),(1)

where H0≡¯

h2k2/(2m)−µ=εk−µis the single-particle

Hamiltonian with wave number k=|k|and εk≡¯

h2k2/2m,

mis the mass of an atom, and µis the chemical poten-

tial of the reference equilibrium state introduced for con-

venience to remove the fast rotation of the overall phase.

(In our simulations, the initial equilibrium state is taken as

the reference state.) The time-dependent quasiparticle am-

plitudes uk(t)and vk(t)satisfy the normalization condition:

|uk(t)|2+|vk(t)|2=1. The pairing gap reads

∆(t) = −g

V∑

uk(t)v∗

k(t) = −g

V∑

Fk(t),(2)

where gis the coupling constant of the contact interaction and

Vis the volume of the system. Here,

Fk(t)≡uk(t)v∗

k(t)(3)

is the pair wave function. The number density nof atoms is

n=2

V∑

|vk(t)|2=2

V∑

nk(t),(4)

with

nk(t)≡|vk(t)|2(5)

being the momentum distribution of atoms (either spin up or

spin down). The energy density Eis given by

E=2

V∑

εk|vk(t)|2+1

g|∆(t)|2.(6)

All the summations here are restricted to the energy range

0≤εk≤Ec, where Ec≡¯

h2k2

c/2mis the cutoff energy (kcis

the cutoff wave number), and the coupling constant gis renor-

malized as [59–61]

g=m

4π¯

h2a−1

V∑

k≤kc

2εk

4π¯

h2a−mkc

2π2¯

h2,(7)

where ais the s-wave scattering length.

In our numerics, we ﬁrst solve self-consistently the gap

equation (2) and the number equation (4) for the equilibrium

solution for {uk}and {vk}, by tuning the chemical potential

in an iterative way. Then we treat the equilibrium solution as

the initial state, and solve the TD-BdG equations (1) in mo-

mentum space using the ﬁfth-order Adams–Bashforth back-

ward predictor-corrector method. In each step of the time in-

tegration, the corrector step is iterated until the absolute error

of all ukand vkis less than 10−6. The step size δkof the

momentum grid is taken to be δk=0.001kF. The time step

δtis typically taken to be δt=0.0005¯

h/EF, but is allowed to

be adjusted to reach convergence.

III. HIGHER HARMONIC EXCITATIONS FROM

TIME-PERIODIC MODULATION

First, we consider time-periodic modulation of the coupling

constant g, which can be experimentally achieved by modulat-

ing the magnetic ﬁeld near a Feshbach resonance [62–64] (see

also Ref. [65] and references therein). The coupling constant

gis periodically modulated around the initial value g0as

g(t) = g0(1+Asinωt),(8)

where Aand ωare the modulation amplitude and frequency,

respectively.

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4FIG.3.QuasiparticlespectraEkandpositionsoftheresonancepeaksforthefundamentalandhigherharmonics.Thisplotisshownfor1=kFa=1andw=1EF=¯hasanexample.growingpeaks(anddips)inthemomentumdistributionnkandthepairwavefunctionFkduringthetimeevolution.Fur-thermore,evenifthemodulationfrequencyissmallerthanthethr...

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Exciting the long-lived Higgs mode in superﬂuid Fermi gases with particle removal Guitao Lyu1Kui-Tian Xi2 3 1 Sukjin Yoon4 5 6Qijin Chen7 8 9 and Gentaro Watanabe1 10 1School of Physics and Zhejiang Institute of Modern Physics.pdf

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Exciting the long-lived Higgs mode in superﬂuid Fermi gases with particle removal Guitao Lyu1Kui-Tian Xi2 3 1 Sukjin Yoon4 5 6Qijin Chen7 8 9 and Gentaro Watanabe1 10 1School of Physics and Zhejiang Institute of Modern Physics

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