Dynamics of Stripe Patterns in Supersolid Spin-Orbit-Coupled Bose Gases

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Dynamics of Stripe Patterns in Supersolid Spin–Orbit-Coupled Bose Gases

Kevin T. Geier,1, 2, 3, ∗Giovanni I. Martone,4, 5, 6, ∗Philipp Hauke,1, 2 Wolfgang Ketterle,7, 8 and Sandro Stringari1, 2

1Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento, 38123 Trento, Italy

2Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy

3Institute for Theoretical Physics, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

4Laboratoire Kastler Brossel, Sorbonne Université, CNRS,

ENS-PSL Research University, Collège de France; 4 Place Jussieu, 75005 Paris, France

5CNR NANOTEC, Institute of Nanotechnology, Via Monteroni, 73100 Lecce, Italy

6INFN, Sezione di Lecce, 73100 Lecce, Italy

7MIT-Harvard Center for Ultracold Atoms, Cambridge, Massachusetts 02138, USA

8Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Dated: July 13, 2023)

Despite ground-breaking observations of supersolidity in spin–orbit-coupled Bose–Einstein con-

densates, until now the dynamics of the emerging spatially periodic density modulations has been

vastly unexplored. Here, we demonstrate the nonrigidity of the density stripes in such a supersolid

condensate and explore their dynamic behavior subject to spin perturbations. We show both ana-

lytically in inﬁnite systems and numerically in the presence of a harmonic trap how spin waves aﬀect

the supersolid’s density proﬁle in the form of crystal waves, inducing oscillations of the periodicity

as well as the orientation of the fringes. Both these features are well within reach of present-day

experiments. Our results show that this system is a paradigmatic supersolid, featuring superﬂuidity

in conjunction with a fully dynamic crystalline structure.

Supersolidity is an intriguing phenomenon exhibited

by many-body systems, where both superﬂuid and crys-

talline properties coexist as a consequence of the simul-

taneous breaking of phase symmetry and translational

invariance [1–6]. After unsuccessful attempts in solid he-

lium [7, 8], supersolidity was ﬁrst experimentally realized

in Bose–Einstein condensates (BECs) with spin–orbit

coupling (SOC) [9, 10] or inside optical resonators [11].

More recently, the supersolid phase has been identiﬁed

in a series of experiments with dipolar Bose gases, where

phase coherence, spatial modulations of the density pro-

ﬁle, as well as the Goldstone modes associated with the

superﬂuid and crystal behavior have been observed [12–

19].

Since the experimental realization of spin–orbit-

coupled Bose–Einstein condensates (SOC BECs) [20, 21],

this platform has emerged as a peculiar candidate of su-

persolidity because the spin degree of freedom is cou-

pled to the density of the system [22–29]. Without SOC,

a two-component BEC has already two broken symme-

tries, one for the absolute phase and one for the relative

phase between the two BEC order parameters. The ad-

dition of weak SOC mixes the spatial and spin degree of

freedom, resulting in a stripe phase where the relative

phase between the two condensates breaks the transla-

tional symmetry of space—the deﬁning property of a su-

persolid. The Goldstone modes associated with the rela-

tive phase are spin excitations, whose dispersion relations

as a function of the Raman coupling have been explored

in Refs. [28, 30], but a connection to the crystal dynamics

of the stripes has so far only been established for their

rigid zero-frequency translational motion [28, 29].

The rigidity of the stripe pattern has been controver-

sially discussed in the literature. For supersolids induced

by coupling a BEC to two single-mode cavities [11, 31],

Spin

Momentum

Raman

coupling

Interference with wave vector

Figure 1. Illustration of the interference eﬀects that lead to

the appearance and dynamics of stripe patterns. The Ra-

man process responsible for spin–orbit coupling turns the two-

component Bose–Einstein condensate (two big circles) into a

system with a four-component wave function. Components

with the same spin form a spatial interference pattern.

the wave vector of the density modulations is determined

by the cavity light and the associated Goldstone mode

is suppressed for nonzero wave vectors [32]. Since the

spin–orbit eﬀect is induced by Raman laser beams, it

has been widely believed that the stripe pattern in SOC

BECs is also externally imposed by the light and thus

rigid. Up to now, conclusive evidence for the nonrigidity

of the stripe pattern has been lacking, as previous studies

of stripe dynamics have mainly focused on the inﬁnite-

wavelength limit [28, 29]. In this Letter, we elucidate the

lattice-phonon nature of the spin Goldstone mode at ﬁ-

nite wavelengths and thus demonstrate that the stripes

form a fully dynamic crystal that is by no means rigid.

Speciﬁcally, we show how spin perturbations can excite

oscillations of both the spacing and the orientation of

arXiv:2210.10064v2 [cond-mat.quant-gas] 11 Jul 2023

the density fringes, establishing SOC BECs as paradigm

examples of supersolidity.

Origin of stripe dynamics.—We consider the common

scenario where SOC is generated in a binary mixture of

atomic quantum gases by coupling two internal states

using a pair of intersecting Raman lasers [34–36]. In con-

trast to quantum mixtures with a simple coherent cou-

pling of radio frequency or microwave type, the Raman

coupling involves a ﬁnite momentum transfer −2ℏk0=

−2ℏk0ˆ

ex, which we assume to point in the negative xdi-

rection. In the limit of weak Raman coupling, the emer-

gence of the stripes and their dynamics can simply be

explained as a spatial interference eﬀect within a wave

function which has four components (Fig. 1): spin-up

condensate at zero momentum with a SOC admixture of

spin down at wave vector 2k0, and spin-down conden-

sate at zero momentum with a SOC admixture of spin

up at wave vector −2k0. Because of SOC, there is now a

spatial interference pattern between the two spin-up and

two spin-down components with wave vector K= 2k0.

The spontaneously chosen relative phase between the two

condensates determines the origin of the stripe pattern.

If there is a chemical potential diﬀerence between the

two components, the relative phase of the two conden-

sates will oscillate and therefore also the position of the

stripes. One can add a spin current to the system, e.g.,

an out-of-phase or relative motion between the two con-

densates, which thus obtain the momenta ±ℏkrel/2. The

four components of the wave function are now at krel/2,

−krel/2−2k0for spin up and −krel/2,krel/2 + 2k0for

spin down, and the spatial interference pattern has now

the wave vector K= 2k0+krel. If the spin current is

oscillating, the wave vector of the stripe pattern will os-

cillate at the same frequency. When krel is parallel to k0,

the fringe spacing oscillates. When they are perpendicu-

lar, the angle of the fringes oscillates. In what follows, we

conﬁrm and extend this intuitive picture using rigorous

perturbative calculations and numerical simulations.

Theoretical framework.—After transforming to a spin-

rotated frame, the single-particle Hamiltonian of the sys-

tem takes the time-independent form [21]

HSOC =1

2m(p−ℏk0σz)2+ℏΩ

2σx+ℏδ

2σz+V(r),(1)

where mis the atomic mass, σxand σzare Pauli matrices,

Ωis the strength of the Raman coupling, δis the eﬀective

detuning, and V(r)is a single-particle potential. In inﬁ-

nite systems (V≡0), the Hamiltonian is translationally

invariant and allows for a spontaneous breaking of this

symmetry, which, in combination with the broken U(1)

symmetry in the BEC phase, gives rise to supersolidity.

Since quantum depletion of a SOC BEC is typically

small under realistic conditions [37, 38], interactions be-

tween atoms are well described by mean-ﬁeld theory via

the Gross–Pitaevskii (GP) energy functional [39]

E=ZdrΨ†HSOCΨ + gnn

2n2+gss

2s2

z+gnsnsz.

(2)

Here, the order parameter is given by a two-component

spinor Ψ = (Ψ↑,Ψ↓)⊺with complex wave functions

Ψ↑and Ψ↓for the individual spin states. The last

three terms in Eq. (2) describe density–density, spin–

spin, and density–spin interactions, respectively, where

n=|Ψ↑|2+|Ψ↓|2denotes the total particle density and

sz=|Ψ↑|2− |Ψ↓|2is the spin density. The correspond-

ing interaction constants gnn = (g↑↑ +g↓↓ + 2g↑↓)/4,

gss = (g↑↑ +g↓↓ −2g↑↓)/4, and gns = (g↑↑ −g↓↓)/4are

obtained from suitable combinations of the coupling con-

stants gij = 4πℏ2aij /m, determined by the s-wave scat-

tering lengths aij of the respective spin channels with

i, j ∈ {↑,↓}. We focus our analysis on symmetric in-

traspecies interactions, assuming gns = 0 and δ= 0 from

now on.

At the critical Raman coupling ℏΩcr =

4Erp2gss/(gnn + 2gss)[23, 25], where Er= (ℏk0)2/2m

is the recoil energy, the system undergoes a ﬁrst-order

transition between the supersolid (stripe) phase and

the superﬂuid (but not supersolid) so-called plane-wave

and single-minimum phases (see, e.g., Ref. [27]). The

latter are characterized by a strong Raman coupling

that is responsible for the locking of the relative phase

between the two spin components [40], resulting from

the competition between the spin (gss) and density

(gnn) interaction components of the mean-ﬁeld energy

functional (2) [41]. Consequently, there is only a single

spin–density-hybridized Goldstone mode above Ωcr.

Conversely, in the supersolid phase, below Ωcr, the

spontaneous breaking of both phase and translational

symmetry implies the existence of two Goldstone modes

of predominantly density and spin nature with distinct

sound velocities (see Supplemental Material (SM) [42]

for further details) [26, 29].

A major question to be addressed in what follows is

how the spin degree of freedom can induce dynamics in

the stripe patterns and in particular how the excitation

of a spin wave results in the excitation of a crystal wave

aﬀecting the time dependence of the density proﬁle.

Perturbation approach in inﬁnite systems.—A useful

scenario to probe this question consists in suddenly re-

leasing at time t= 0 a small static spin perturbation of

the form −λErσzcos(q·r), with 0< λ ≪1. The wave

vector qis assumed to be small in order to explore the

relevant phonon regime, where a major eﬀect of the re-

lease of the perturbation is the creation of a spin wave

propagating with velocity cs. Here, we are mainly inter-

ested in its eﬀect on the dynamic behavior of the stripes

characterizing the density distribution. Starting from the

results of Ref. [29] for the Bogoliubov amplitudes of the

phonon modes in the long-wavelength limit, and neglect-

ing the small contributions of the gapped modes of the

Bogoliubov spectrum, the space and time dependence of

the density can be written in the form

n(r, t) = ¯n+

+∞

¯m=1

˜n¯mcos[ ¯mχ(r, t)] .(3)

Here, ¯nis the average density and

χ(r, t)=2k1x+ϕ+δϕ(t) cos(q·r)(4)

the relative phase between the two condensates in the

spin-rotated frame. The sum over the integer index ¯m

reﬂects the presence of higher harmonics in the density

proﬁle (3) characterizing the stripe phase, whose coeﬃ-

cients are denoted by ˜n¯m. Equations (3) and (4) explicitly

reveal that the perturbed density fringes are a combined

eﬀect of the equilibrium modulations, ﬁxed by the wave

vector 2k1= 2k1ˆ

ex(which diﬀers from 2k0at ﬁnite Ra-

man coupling [25, 29]), and those induced by the external

perturbation, characterized by the wave vector q. The

perturbative expression for k1is reported in Ref. [29] and

for convenience in the SM [42]. The phase ϕrepresents

the spontaneously chosen oﬀset of the stripe pattern in

equilibrium. The time dependence of the function δϕ is

ﬁxed by the sound velocities cn,s of the density and spin

phonons as well as by the Raman coupling Ω.

For q≪k1, the relative phase (4) varies very slowly

over a large number of equilibrium density oscillations.

Consequently, in a region of space |r−r0| ≪ q−1around

a given point r0, one can approximate χby its ﬁrst-order

Taylor expansion,

χ(r, t)≃K(r0, t)·r+ Φ(r0, t).(5)

This expression features a local time-dependent stripe

wave vector, whose structure

K(r0, t) = ∇χ(r0, t)=2k1−δϕ(t) sin(q·r0)q(6)

conﬁrms the intuitive scenario of Fig. 1 (where k1has

been approximated by k0), upon identifying krel with the

second term in Eq. (6). In addition, Eq. (5) contains the

phase shift

Φ(r0, t) = χ(r0, t)−r0· ∇χ(r0, t)

=ϕ+δϕ(t)[cos(q·r0)+(q·r0) sin(q·r0)] ,

(7)

which is responsible for the time modulation of the oﬀset

of the stripe pattern.

Carrying out a perturbative analysis of the order pa-

rameter of the condensate up to second order in ℏΩ/4Er

(see Refs. [29, 44]) yields the result

δϕ(t) = −2k1vs

csqsin(csqt),(8)

where we have introduced the velocity

vs=λℏk0

m"1

2−βℏΩ

4Er2#,(9)

with β=Er¯n[2Ergnn +2(2Er+¯ngnn)gss +¯ng2

ss]/[2(2Er+

¯ngnn)2(2Er+ ¯ngss)], and the expression for csis reported

in Ref. [29] and for convenience in the SM [42]. At the

leading order Ω2, only the spin sound velocity csenters

Eq. (8), while a second term oscillating at the density

phonon frequency cnqappears at order Ω4[44]. For

Ω=0, the velocity vsﬁxes the time variation rate of

the relative phase of the quantum mixture, without any

consequence for the density distribution since the con-

trast of fringes exactly vanishes in this limit [25, 29] (see

also SM [42]).

If cos(q·r0) = ±1, the initial static spin perturbation

has a peak (antinode) at r0, and close to this point it

becomes of the form ∓λErσz. After releasing the spin

perturbation, there is a spin imbalance at r0and the

diﬀerence in chemical potentials causes an oscillation of

the relative phase of the two condensates. From Eqs. (7)

and (8) one sees that, at times satisfying the condition

t≪(csq)−1(which is easily fulﬁlled for the small qof

interest here), the stripes show a displacement at veloc-

ity ±vs, i.e., χ(r, t)≃2k1(x∓vst)+ϕ, in excellent agree-

ment with the numerical ﬁndings of Ref. [28] (see SM [42]

for further details). At q= 0, the spatial translation of

stripes corresponds to the zero-frequency limit of the spin

Goldstone branch.

Far from the antinodes, after the spin quench there is

an oscillating spin current, which makes also the stripe

wave vector (6) vary in time. The strongest oscillations

occur when sin(q·r0) = ±1, i.e., r0is a node of the initial

perturbation, which is thus antisymmetric under inver-

sion with respect to r0and locally behaves as ±λErq·

(r−r0)σz. In particular, if q=qˆ

ex, the local stripe

wavelength 2π/|K(r0, t)|= [1 ∓(vs/cs) sin(csqt)]π/k1

oscillates around its equilibrium value. By contrast, if

q=qˆ

ey, the stripes rotate by an angle ±(vs/cs) sin(csqt)

about the zaxis. This eﬀect occurs in combination with

the fringe displacement seen above, unless r0coincides

with a maximum or minimum of the equilibrium density

distribution.

The above discussion shows that a spin perturbation

applied to the stripe conﬁguration can cause a rigid mo-

tion of the stripes as well as a periodic change in either

magnitude or orientation of their wave vector, depending

on the local behavior of the perturbation. Although the

analytic results (8) and (9) have been derived by carry-

ing out a perturbative analysis up to order Ω2, we have

veriﬁed that they provide a rather accurate description

of the dynamics of stripes, as compared to a numerical

solution of the time-dependent linearized GP equation in

inﬁnite systems, also for fairly large values of the Raman

coupling.

Numerical simulations in a harmonic trap.—Having

understood how spin perturbations aﬀect the dynamics

of the stripe pattern in inﬁnite systems, we now illus-

trate similar eﬀects taking place in ﬁnite-size conﬁgura-

tions, namely, in the presence of a harmonic trapping

potential V(r) = m(ω2

xx2+ω2

yy2+ω2

zz2)/2with an-

gular frequencies ωiand corresponding oscillator lengths

ai=pℏ/mωi,i=x, y, z. To this end, we numerically

solve the full time-dependent GP equations, which can be

derived by applying the variational principle iℏ∂tΨ↑/↓=

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摘要：

DynamicsofStripePatternsinSupersolidSpin–Orbit-CoupledBoseGasesKevinT.Geier,1,2,3,∗GiovanniI.Martone,4,5,6,∗PhilippHauke,1,2WolfgangKetterle,7,8andSandroStringari1,21PitaevskiiBECCenter,CNR-INOandDipartimentodiFisica,UniversitàdiTrento,38123Trento,Italy2TrentoInstituteforFundamentalPhysicsandApplica...

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Dynamics of Stripe Patterns in Supersolid Spin-Orbit-Coupled Bose Gases

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