Droplet Arrays in Doubly-Dipolar Bose-Einstein condensates Ratheejit Ghosh1Chinmayee Mishra1 2Luis Santos3and Rejish Nath1 1Department of Physics Indian Institute of Science Education and Research Pune Pune 411 008 India

2025-05-03 0 0 2.48MB 8 页 10玖币

侵权投诉

Droplet Arrays in Doubly-Dipolar Bose-Einstein condensates

Ratheejit Ghosh,1Chinmayee Mishra,1, 2 Luis Santos,3and Rejish Nath1

1Department of Physics, Indian Institute of Science Education and Research Pune, Pune 411 008, India

2Indian Institute of Technology Gandhinagar, Gandhinagar 382 355, India

3Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstrasse 2, DE-30167 Hannover, Germany

(Dated: October 28, 2022)

Gases of doubly-dipolar particles, with both magnetic and electric dipole moments, oﬀer intriguing novel pos-

sibilities. We show that the interplay between doubly-dipolar interactions, quantum stabilization, and external

conﬁnement results in a rich ground-state physics of supersolids and incoherent droplet arrays in doubly-dipolar

condensates. Our study reveals novel possibilities for engineering quantum droplets and droplet supersolids,

including supersolid-supersolid transitions and the realization of supersolid arrays of pancake droplets.

I. INTRODUCTION

The anisotropic and long-range nature of the dipole-dipole

interactions leads to a rich physics in dipolar quantum gases,

qualitatively diﬀerent than that of their non-dipolar counter-

parts [1–3], including anisotropic superﬂuidity [4–6], roton-

like excitations [7,8], and the recent realization of quantum

droplets [9–12]. The latter result from the interplay between

contact and dipolar interactions and the stabilization provided

by quantum ﬂuctuations [13]. Interestingly, the external con-

ﬁnement may result in the formation of arrays of droplets,

which under proper conditions may remain mutually coher-

ent, building a dipolar supersolid [14–17], whose properties

have recently been the focus of major attention [17–26].

Experiments on dipolar Bose-Einstein condensates have

been realized so far with atoms with large permanent magnetic

moments such as chromium [27,28], erbium [29], and dys-

prosium (Dy) [30]. Interestingly, a pair of quasi-degenerate

states with opposite parity oﬀers the possibility of inducing an

additional electric dipole moment in Dy atoms using an elec-

tric ﬁeld [31]. Recently, doubly-dipolar atoms and molecules

possessing both electric and magnetic dipole moments [31–

39] have attracted a large deal of interest due to their poten-

tial applications in quantum simulation [40], computing [41],

tests of fundamental symmetries [42], and for the tuning of

collisions and chemical reactions [43]. Interestingly, the elec-

tric and magnetic moments may be oriented in diﬀerent di-

rections, opening novel possibilities for doubly-dipolar con-

densates [44]. Self-bound quantum droplets may undergo a

dimensional crossover when varying the angle between the

dipole moments without modifying the external conﬁnement.

In this paper, we show that the control of the relative angle

between the two dipole moments opens new intriguing sce-

narios for quantum droplet arrays in doubly-dipolar conden-

sates, including a density-modulated single droplet ground-

state, supersolid-supersolid transitions, and the possibility of

realizing an array of pancake-shaped quantum droplets.

The paper is structured as follows. In Sec. II, we re-

view a particular realization of a doubly-dipolar system us-

ing dysprosium atoms, employed in the rest of the paper. In

Sec. III, we discuss the anisotropic properties of the dou-

bly dipolar potential. In Sec. IV, we introduce the extended

Gross-Pitaevskii equation for a doubly-dipolar condensate, in-

corporating beyond-mean-ﬁeld corrections. The properties of

a single self-bound droplet are brieﬂy discussed in Sec. V.

Section VI is devoted to analyzing quantum droplet arrays in

doubly-dipolar condensates. Finally, we summarize our con-

clusions in Sec. VII.

II. DOUBLY-DIPOLAR DYSPROSIUM ATOMS

In this section, we discuss a particular realization of a

doubly-dipolar system using Dy atoms, brieﬂy reviewing the

proposal of Ref. [31]. However, other realizations, e.g. using

molecules, should result in a similar physics.

In addition to its permanent magnetic moment, an electric

moment may be induced in Dy atoms by an external electric

ﬁeld owing to a pair of quasi-degenerate states with oppo-

site parity. These states, |ai(odd parity) and |bi(even par-

ity), have total angular momenta {Ja=10,Jb=9}, and en-

ergies {Ea=17513.33 cm−1,Eb=17514.50 cm−1}. Within

the electric-dipole approximation, the line-widths of the states

are Γa≈0 (metastable) and Γb=2.98 ×104s−1, respec-

tively. We assume that the Dy atoms are in uniform mag-

netic and electric ﬁelds. The magnetic ﬁeld, B=Bˆz, is

directed along z, setting the quantization axis and splitting

the degeneracy of the energy levels Eaand Eb. The elec-

tric ﬁeld, E=Eˆumixes the Zeeman sublevels of the states,

{|Ma=−Jai, ..., |+Jai,|Mb=−Jbi, ..., |+Jbi}, inducing an

electric dipole moment along ˆu. We assume that ˆulies on

the xz plane forming an angle αwith the z-axis. This relative

angle plays a crucial role in the physics discussed below.

Restricting to the subspace of both Eaand Eb, the Hamilto-

nian for a Dy atom is ˆ

H=ˆ

HB+ˆ

Hstark with

HB=EaX

Ma|MaihMa|+EbX

Mb|MbihMb|+µBB(gaMa+gbMb),

(1)

where ga=1.3 and gb=1.32 are the Land´

egfactors. The

term ˆ

Hstark accounts for the interaction of the electric ﬁeld

with the Dy atom. The electric ﬁeld strength is such that the

lowest eigenstate of the atom is |Si=c0|Ma=−10i+P0

ici|ii

with P0

i|ci|2/|c0|21, where the sum P0

iis taken over all the

magnetic sublevels except |Ma=−10i, and ciis the probabil-

ity amplitude for ﬁnding the atom in the state |ii. The summa-

tion P0

ihas two contributions, one from the sublevels of |ai

and the other from those of |bi. Since Γa≈0, only the con-

tributions from the sublevels {|Mbi} determine the lifetime of

arXiv:2210.01093v2 [cond-mat.quant-gas] 27 Oct 2022

0.5

0.1 0.2 0.3 0.4

00.40.30.2 0.5

0.1

Figure 1. (Color online) (a) Electric dipole moment deand (b) life-

time τof the |Sistate of a Dy atom, as a function of the electric ﬁeld

strength Eand the angle αbetween the electric and magnetic ﬁelds

for B=100 G.

the stretched state |Si,τ=(nbΓb)−1, where nbis the total pop-

ulation in {|Mbi} sublevels. The magnetic and electric dipole

moments of a Dy atom in |Siare, respectively:

dm=−µB





Ma=−Ja|cMa|2Ma+gb

Mb=−Jb|cMb|2Mb





(2)

de=−1

Ma,Mb

c∗

MacMbhMa|ˆ

Hstark|Mbi+c.c.(3)

with

hMa|ˆ

Hstark|Mbi=−s4π

3(2Ja+1) ha|| ˆ

d||biE ×

Y∗

1,Ma−Mb(α, 0) CJaMa

JbMb,1,Ma−Mb,(4)

where, ha|| ˆ

d||bi=8.16 Debye is the reduced transition

dipole moment, Yl,m(θ, φ) are the spherical harmonics and

CJaMa

JbMb,1,Ma−Mbare the Clebsch-Gordan coeﬃcients.

Figure 1depicts, for B=100 G, deand dmfor the state

|Si, as a function of Eand α. When α=0, the spherical

harmonics, Y∗

1,Ma−Mb(α, 0) are non-zero only when Ma=Mb

and thus, the electric ﬁeld couples pairs of sublevels with

Ma=Mb. Since the state |Ma=−10ihas no counterpart

in the {|Mbi} subspace, the former is unaﬀected by the electric

ﬁeld. Hence, the electric dipole moment of the Dy atom in

the state |Sivanishes for α=0. When αgrows, the electric

ﬁeld couples |Ma=−10iwith other |Mbisublevels, reach-

ing a maximum mixing for α=π/2 (see Fig. 1(a)). There-

fore, for a given E,deincreases with αuntil α=π/2. This

comes at the cost of decreasing the lifetime of |Si, as shown

in Fig. 1(b). For a range of experimentally realistic E=0-4

Figure 2. (Color online) Doubly-dipolar atoms. Both electric (de)

and magnetic (dm) dipoles are assumed polarized on the xz plane,

forming an angle αbetween them. The angle θm(θe) is the angle

between dm(de) and the vector joining the atoms, r.

kV/cm, devaries from 0 to 0.16 Debye and the magnetic mo-

ment remains constant, dm'13 µB(higher than ground-state

Dy atoms), whereas the lifetime of |Sivaries from 28 s (con-

sidering electric-quadrupole and magnetic-dipole transitions)

to 10 ms [31].

III. DOUBLY DIPOLAR POTENTIAL

The doubly-dipolar interaction between two atoms is

Vd(r)=µ0d2

4π

(1 −3 cos2θm)

|r|3+d2

4π0

(1 −3 cos2θe)

|r|3(5)

where µ0(0) is the vacuum permeability (permittivity) and

θm(θe) is the angle formed by the magnetic (electric) dipole

moment with the vector rjoining the atoms (Fig. 2). Whereas

Vd(r) is always repulsive along the y-axis, it is anisotropic

on the xz-plane. This anisotropy is well characterized by the

angular part of the dipolar potential on the xz-plane:

Vy=0

d(r, θ)∝"1−3cos2θ+γ(cos θcos α+sin αsin θ)2

1+γ#,(6)

where θis the polar angle, and γ=(de/dm)2/(µ00) char-

acterizes the relative strength between the electric and mag-

netic dipole moments. The ratio γcan be varied indepen-

dently of αby tuning E. In Fig. 3, we depict Vy=0

d(r) for

diﬀerent αand γ. When α=0, we have the usual dipolar

potential, attractive along zand repulsive along x[3]. As

αincreases up to π/2, the dependence of the potential on

γbecomes more signiﬁcant. For α=π/2, when γgrows

the potential inverts eventually its anisotropy (last column of

Fig. 3). If −4γ2+γ−4≤9γ, there exists a critical angle

2cos−1h(−4γ2+γ−4)/9γiabove which the xz potential be-

comes purely attractive (see Figs. 3(g), (h), and (l)), but re-

mains anisotropic except when α=π/2 and γ=1. For that

case Vy=0

d(r, θ)=−1/r3.

Despite this nontrivial anisotropy, we may deﬁne an eﬀec-

tive polarization axis, depicted by arrows in Fig. 3, given by

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

DropletArraysinDoubly-DipolarBose-EinsteincondensatesRatheejitGhosh,1ChinmayeeMishra,1,2LuisSantos,3andRejishNath11DepartmentofPhysics,IndianInstituteofScienceEducationandResearchPune,Pune411008,India2IndianInstituteofTechnologyGandhinagar,Gandhinagar382355,India3Institutf¨urTheoretischePhysik,Leibn...

展开>> 收起<<

Droplet Arrays in Doubly-Dipolar Bose-Einstein condensates Ratheejit Ghosh1Chinmayee Mishra1 2Luis Santos3and Rejish Nath1 1Department of Physics Indian Institute of Science Education and Research Pune Pune 411 008 India.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Droplet Arrays in Doubly-Dipolar Bose-Einstein condensates Ratheejit Ghosh1Chinmayee Mishra1 2Luis Santos3and Rejish Nath1 1Department of Physics Indian Institute of Science Education and Research Pune Pune 411 008 India

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: