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[3] A. Griffin, T. Nikuni,
E. Zaremba, Bose-condensed gases at finite 1
・Developmentof a harmonic
turbulent
that
results fromtrapping. Here we eliminat∫ 2
temperatures (Cambridge University Press, 2009).p(θ) =n(k) (m3)

T

Main them

4

2. ModelB cindicated by the vertical grey band.

2016a ∂ψ(r,lt)

10
ψ(r,1 t) : macroscopiclae wave functionℏ2a 2
Cavendish Laboratory, University of Cambridge, J. J. Thomson
Avenue, Cam−∇ + V(r, t) + g | ψ(r, t) |2 ψ(r, t)
102
g : interaction between particles
2m
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diss(r) ao-V(r,(t) = Vbox(r) + VG
Vdiss(r) =iVdiss(r) : imaginary
potentialtaen(k)(µm3)

Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

{ 60

n
|z| ≤
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Anisotropyof the three-dimensional
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3p(k, θk, φk) =N(k) =
n(k)k 2sinθkdθkdφk∫0 ∫0
he route to turbulence.
a, b, Small-amplitudeN(k)tG
P oscillations
3of the centre
2π π
1 for N = 10(1) ×104. a, Free oscillation1
ss (along thesztheaxis,
see Fig.
anisotropy: A(k) 1a)=p(k, θk, φk) − piso 3
sinθkdθkdφk n piso l
=

106

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a 20-ms kick with ∆t
U/kB aHorizontal
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Bogoliubov
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anisotropy
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104
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3

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Numerical
results
n oscillations, with driving amplitudeN- ∆(U/kB = 0.3 nK and ts = 2 s (blueG・Visualizing the dynamics
turbulent
states). The solid blueline is a fit (see Methods). Error bars in a and b
ΔU/μ = 1.36 (strong excitaion)andsent 1σ errors.Realc, Small-amplitudeωres versus N for free (orange)
spacen(r) = | ψ(r, t) |n (blue) oscillations with the same ∆U and ts values as in a and b.

104
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10

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n(k)(µm3)

Zres/(2π) (Hz)Avenue, Cambridge CB3 0HE, UK.

r
r1
by D.
studying
turbulencea homogeneous[4]
Proment, S.
Nazarenko, M.in
Onorato,Phys. Rev. A 80,quantum
051603 N̄ gas
μ : chemical potential
cascade2π(2009).basic idea of our experiment is outlined in1Fig. 1
9The[5] M. T. Reeves, T. P. Billam, B. P. Anderson, A. S. Bradley, k-3.5
103
APhys.
= 87 atots (s)
0.3
quasi-pureBose–Einstein
Rev. Lett. 114,155302 (2015). condensate (BEC) of 2Rb
∫0
−
γ
7
0.16
0.2
drical∼ drive
3.5 it out of equilibrium withn(k)∼opticalk box γ, and
0.41
0.1
0.61
form, oscillating force that primarily couples to the low101–π
π 1.22
–π
π –π
π
0
0
0
mode. Ourdistribution
box has length L = 27(1) µm and radiu
2.44n(k) axial: momentum
Figure 1 | From unidirectional sloshing to isotropic turbulence.
3.29
a, Gross–Pitaevskii
simulationsofashaken,box-trapped
Bose
gas.
The
(here
and
elsewhere,
errors represent
1σ uncertainties).
-1
Numerical
in6blue shading10indicatesthe gas density; the red lines indicate vortices.
5
0.5
1
2
4
8
atom number N ≈ 10 , the initial, equilibrium BEC hb–d, Experimental absorption images taken along x after 100 ms of TOF-1 corresponding
and theexpansion, with N ≈ 8 × 10 atoms (upper panels),
turbulent
cas
k (m
)
kD kD : dissipation
potential µ/kBwave≈ 2 nKnumber(where kB is theBoltzmann constana p(θ), averagedz over 20 images
t-1underaeangular distributions
taken
identical
conditions (lower panels). b, Initial BEC; c, after shaking for 2 s at 8 Hzenergy per particle Eint/kB ≈ 1 nK and negligible kinetic
with amplitude∆U/µ≈ 1.2; and d, after the turbulent
cloud wasFig.S3:
Momentum-space
dynamics
in allowed
numerical
oM1ovsimulations.
to relax for 1.5 s. The dashed circle in c corresponds
to an expansion
the critical temperature for Bose–Einstein condensation
UDthe=redkBlines⇥ 120
nK andvarious shaking times ts .
In the lower forpanels,correspondto theenergy ofn(k)k T /2.computed
・Momentum
diamond-likeisotropic distributions
depictedin the insets.
The andpower-law
momentum
distributionn(k) ⇠ k , with ⇡ 3.5,
The driving force is provided by a magnetic
field gradie
develops in the wake of the cascade front. A steady state is estab108
3 lished once the cascade front reaches kD ; in this example kD = 8/⇠, a potential U(r) = ∆Uz/L, where the coordinate
z is alot N liG
3 106
Gross-Pitaevskii
model
y

p(T)

c

acting Bose gas, which is
vskii equation (GPE) forr = (x, y, z) is the spatial
is widely used to model
merical results have been
lly, qualitative evidence
es22–25, but quantitative
inhomogeneous density
eliminate this problem
ntum gas.
d in Fig. 1. We prepare aof 87Rb atoms in a cylinum with a spatially unio the lowest, dipole-likeand radius R = 16(1) µm
tainties). For our typicalm BEC has a chemicalnn constant), interaction
ble kinetic energy, while
densation is Tc ≈ 50 nK.
eld gradient that creates
ate z is along the axis ofy≈-of-flight (TOF)xexpansion, t0sTOF ≥ 100 ms.
(ts ≤ 4 s)s
4.0simages we
0.2s From the
Isot
time
ct the centreMomentumof massspaceand momentum distribution of the cloud. strongη of > 0.5.
rapn(k) = | ψ̃(k, t) |excitation
position of the centre of mass reflects the axial in-trap sloshing,
To see hhe evolution ofkthe kmomentum distribution reveals the cascade tially unifo
Aniscitations from small
to large wavevectors k—the so-called direct weak
spectroscokexcitationde.
In contrasγ∼n(k , k ) = n(k)dk∫ a qualitative experimental signature of turbuFig. 1b–d we showby the trap
[1] N.Navon, A. L. Gaunt,kwith three key examplesof TOF images (upper panels) and the [2] Navon
interaction
C.Eigen, J.Zhanksponding angular distributions of atoms p(θ) (lower panels). The arXiv:1807.07564v1.
shaking amnll BEC (Fig. 1b) shows an anisotropic expansion, which is driven
We first
e conversion of interaction into kinetic energy, and reflects the ious N, usi
e of the container26. In sharp contrast, after sufficiently long shak- pulse on ahe expansion is isotropic (Fig. 1c), with p(θ) exhibiting small fluc- in the trap
ons around 1/(2π ). This is the first qualitative signature of a the centre2

log10 n(k)

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同様に頻度と最小サイズの関係を見てみると…

t1 1 / (2R0 ) 1.03
<latexit sha1_base64="ZkgNtpZ6bUsZWVX8W4kSLGPGsFk=">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</latexit>

t1 1 / (2R0 ) 0.886<latexit sha1_base64="DqKYe/KNt9a5x1Z74bxV6fEOwcY=">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</latexit>

同様のべき則が見えた！！

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