冷却原子系による量子シミュレーション Kyoto …...Outline I) Preparation of Quantum Gas II) Ultracold Atoms in a Harmonic Trap Laser cooling and trapping, evaporative

Kyoto University

冷却原子系による量子シミュレーション

31 August 2015

基研研究会「熱場の量子論とその応用」

Yoshiro Takahashi

Yukawa Institute, Kyoto University

Outline I) Preparation of Quantum Gas

II) Ultracold Atoms in a Harmonic Trap

Laser cooling and trapping, evaporative cooling, Bose-Einstein condensate,

Fermi Degenerate Gas

III) Ultracold Atoms in an Optical Lattice

Feshbach resonance, Cooper paring, BEC-BCS crossover, unitary gas,

spin-orbit interaction

Superfluid-Mott insulator transition, quantum-gas-microscope, Higgs mode,

Non-standard lattices(frustrated magnetism, flat band), Fermi-Hubbard model,

Bose-Fermi mixture










Bose-Fermi mixture

Laser Cooling and Trapping

CCD

anti-Helmholtz coils

10mm

“Magneto-optical Trap”

• Number: 107

• Density: 1011/cm3

• Temperature: 10µK

Six laser beams

for laser cooling

mm 500

“optical trap”

“magnetic trap”

EpV int2

)()(

2rErU pot

BV mint

Cooling to Quantum Degeneracy

Pressure ~10-12 torr

Lifetime > 10 sec

T~100 nK N~ 105

“Evaporative cooling”

BEC Formation

(Yb atom)

Momentum distribution T:high

T:low

Momentum Distribution [E. Cornell et al, (1995)]

87Rb

“Bose-Einsten Condensation”

“Boson versus Fermion”

Cooling to Quantum Degeneracy

Spatial Distribution

6Li and 7Li

Spatial Distribution [R. Hulet et al, (2000)]

“Fermi Degeneracy”

Fermi

Pressure

Experimental Setup for Cold Atom










Bose-Fermi mixture

Feshbach Resonance: ability to tune an inter-atomic interaction

Coupling between “Open Channel” and “Closed Channel”

Control of Interaction(as)

Pote

nti

al

-C6/R3

Two Atoms

Molecular State

F1+F1

F2+F2

)1()(0BB

BaBa bgs

ka ls /22

00 44 saf

Collision is in Quantum Regime

It is described by s-wave scattering length as

[C. Regal and D. Jin, PRL90, 230404(2003)]

“Magnetic field tunes the energy difference

between closed channel and open channel”

BEC – BCS Crossover

1/(kFa)

0

“Unitary Gas”

“molecular”

Leggett, ...

)2

exp(3.0sF

FBCSak

TT

Regal et al., PRL(2004)

BEC – BCS Crossover: experiments

40K

Zwierlein et al., PRL(2004) 6Li

Inada et al., PRL(2008) 6Li

Equation of State for Unitary Gas M. J. H. Ku et al, (2011): MIT Zwierlein Group

Density

6Li

Spin-Orbit Interaction in Cold Atoms:

α=β: xykSOI

Y. –J. Lin, et al., Nature 471, 83(2011)

“pseudo-spin +1/2”:

“lasers for Raman Transition”

E=E1cos(k1x-ω1t)

)( 210 : detuning

10

01

22

00

z

x E=E2cos(k2x-ω2t)

2/1

2/1

ħω0

e

21 kkQ : momentum transfer

after the local pseudo-spin rotation with θ=Qx about z-axis

14222

12

22

R

yx

zy

E

m

kQ

m

kH

m

QER

2

22

:Recoil Energy

)2

exp( z

QxiU

SOI Bz

Spin-Orbit Interaction in Cold Atoms:

“Boson: 87Rb”

Y. –J. Lin, et al., Nature 471, 83(2011)

“Fermion: 6Li, 40K”

P. Wang et al., (2012)

L. W. Cheuk et al., (2012)

α=β: xykSOI

Topological Superfluids, Majorana Edge state by Spin-Orbit Interaction + Strong s-Wave Interaction

[M. Sato et al., PRL103,020401(2009), PRB82, 134521(2010)]

δ/2π=0kHz

kｙ/

Q

kx/Q

1S0

ℏQ

3P2

δ/2π=8kHz

kx/Q

kｙ/

Q

δ/2π=-8kHz

kx/Q

kｙ/Q

SOI between 1S0+3P2(

174Yb ) Feshbach Resonance:1S0+3P2(

171Yb )

Magnetic Field (G)

Nu

mber

of

atom

s

(ar

b. u

nit

s)

Atoms(3P2)

: 𝜇 = 𝜇𝑎(3P2),

M=Ma

Feshbach Molecules

(1S0 +3P2) ∶

𝜇 ≈ 𝜇𝑎(3P2), M=2Ma T/TF ~ 0.25

Atoms(1S0)

: 𝜇 = 0,

M=Ma

Stern-Gerlach-Separated Images










Bose-Fermi mixture

Quantum Simulation of Strongly Correlated Electron System

i

ii

ji

i nncc UJHj

,

i-th j-th

J

U

ideal Quantum Simulator of Hubbard Model:

λlattice λlattice

λlattice

λlattice/2

“ultracold atoms in an optical lattice”

)(sin)( 2 xkVxV Loo

)(sin2 kxVV o

)1(2,

i

i

i

ji

i nnaaU

JHj

1)Macroscopic Quantum System：typically ～105

2)Clean (no impurity, no lattice defects)

Nice Features of ultracold Atoms in an Optical lattice

3)High controllability of Hubbard parameters

“Mott Insulator”

Large

Weakly-interacting Strongly-correlated

U/J Small

“Superfluid”

i

i

i

i

i

i

ji

i

n

nn

aa

U

JHj

)1(2

,

“An interference fringe is

the direct signature of the phase coherence”

Phase Diagram of Bose-Hubbard Model (T>0)

S. Trotzky, et al, Nature Physics 6, 998(2010)

i

i

i

i

i

i

ji

i

n

nn

aa

U

JHj

)1(2

,

S. Trotzky, et al, Nature Physics 6, 998(2010)

“An interference fringe is

the direct signature of the phase coherence”

Phase Diagram of Bose-Hubbard Model (T>0)

:width

:weight

“amplitude-(Higgs-)mode” The `Higgs' Amplitude Mode at the Two-

Dimensional Supefruid-Mott Insulator Transition

M. Endres et al (2012)

Lattice modulation spectra

V=

Spectroscopy of Superfluid-Mott Insulator Transition

S. Kato et al.,

quantum simulation of antiferromagnetic

spin chains in an optical lattice [J. Simon, et al., Nature, 472, 307(2011)]

1D Bose-Hubbard Model: ( )

“E < U” “E > U”

quantum simulation of antiferromagnetic

spin chains in an optical lattice [J. Simon, et al., Nature, 472, 307(2011)]

They successfully observe the transition from paramagnetic spin state to AF ordered state.

)(sin2 kxVV o

)1(2,

i

i

i

ji

i nnaaU

JHj

4) Powerful Measurement Methods:

Objective

Lens

ultracold atoms

~ 500 nm

in situ imaging

by Quantum Gas Microscope


Fluorescence Imaging

[WS. Bakr, et al Nature 462,5 (2009)]

87Rb

Single Site Resolved Detection of

SF-MI Transition [WS Bakr, et al., Science 329, 547(2010)]

SF MI

TOF-image

In Situ-image

after analysis

87Rb

)(sin2 kxVV o

)1(2,

i

i

i

ji

i nnaaU

JHj

4) Powerful Measurement Methods:


Single-site manipulation

by Quantum Gas Microscope

[C. Ewitenberg et al, Nature 471, 319(2011)]

87Rb

3mm

Objective (NA=0.75)

Working distance 6.23mm

Focal length f = 4mm

(Mitutoyo Corporation)

Yb atom

f = 600mm

399nm BP filter

Optical parallels

EMCCD camera Andor

iXonEM Blue

Glass cell

M=150

“Observation of Single Yb Atoms in an Optical Lattice

with Hubbard Regime”

Ytterbium Quantum Gas Microscope

174Yb (boson)

399 nm 399 nm

556 nm 556 nm

“dual molasses“

~30 μm Lattice constant: d=266 nm psf: 𝜎𝐴 =487 nm

3mm

Objective (NA=0.75)

Working distance 6.23mm

Focal length f = 4mm

(Mitutoyo Corporation)

Yb atom

f = 600mm

399nm BP filter

Optical parallels

EMCCD camera Andor

iXonEM Blue

Glass cell

M=150

“Observation of Single Yb Atoms in an Optical Lattice

with Hubbard Regime”

Ytterbium Quantum Gas Microscope

174Yb (boson)

399 nm 399 nm

556 nm 556 nm

“dual molasses“

~30 μm Lattice constant: d=266 nm psf: 𝜎𝐴 =487 nm

~10 μm

)(sin2 kxVV o

)1(2,

i

i

i

ji

i nnaaU

JHj

5)Various Lattice Geometries：

Dimensionality:

Standard and Non-standard Lattices:

Honeycomb

(hexagonal)

Kagome Cubic(Square) Lieb


http://extreme.phys.sci.kobe-u.ac.jp/extreme/student/2007/tomoo/Crystal_gallery.html

Several Non-Standard

Optical Lattices are realized

Honeycomb Kagome

Triangular Lieb http://hiroi.issp.u-tokyo.ac.jp/data/crystal_gallery/crystal_gallery-Pages/Image31.html

http://en.wikipedia.org/wiki/Graphene

Many interesting materials(high-Tc Cuprate, graphene, ..)

have non-standard lattice structures

Quantum Simulation of

Frustrated Magnetism in a Triangular Lattice

Phase Modulation of Optical Lattice：

Kcos(𝜔𝑡)

𝐽 → 𝐽 × 𝐽0 𝛽 :Zero-th order

Bessel Function

𝛽=K/𝜔

[J. Struck et al, Science (2011)]

Novel Band Structures [from Katsura & Maruyama(2014)]

Honeycomb

Dirac Cone:

E(k)=c k :massless Dirac particle

linear dispersion

Kagome

Flat Band:

E(k)=const.

: infinite mass/localization

Square

(standard)

Cosine Band:

E(k)=Jcos(kd)

Novel Band Structures Honeycomb Kagome

“Ultracold atoms in a Tunable Optical Kagome Lattice”

Gyu-Boong Jo, et al, PRL (2012)

“Creating, moving, and merging Dirac points

with a Fermi gas in a tunable honeycomb

lattice”, Tarruell, et al Nature (2011)

“Ultracold atoms in a flat band”

“localized eigenstate”

Huber & Altman PRB 82, 184502 (2010)

𝜈 − 𝜈c

Prediction of super-solid

for bosons in Kagome lattice

Super-Solid

(case of Kagome lattice)

resonating hexagon:

[Discussion with S. Furukawa]

closed packing

at 𝜈c=1/9

huge degeneracy !

𝜈 > 𝜈c

Exotic strongly

correlated state:

Spatial overlap

between hexagons

What is Super-Solid ?

Absence of Supersolidity in Solid Helium in Porous Vycor Glass:2013

“superfluid” (Off-Diagonal Long-Range Order)

→ 𝑛𝟎

𝛿𝑛 x = 𝛿𝑛(x + 𝑑)

“Solid order” (Density Long-Range Order

/ density wave)

002/cos2

002/cos2

2/cos22/cos20

dkJ

dkJ

dkJdkJ

T

z

x

zx

Lieb

k

k

k

k

kkk

C

B

A

Lieb

†

C

†

B

†

ATB

a

a

a

TaaaH

,

,

,

,,,

ˆ

ˆ

ˆ

ˆˆˆˆ

Band Structure of Lieb Lattice

k

k

xk

S

i

Si

aeN

a

Si

,

,

ˆ1

ˆ

,

Operators in k-space:

S=A,B,C

(Tight-Binding Approximation)

,0

,)2/(cos)2/(cos2

0

22

E

dkdkJE zx

,,cos,sin,2

13rd ,

,,sin,cos2nd ,

,,cos,sin,2

11st ,

CBA

CB

CBA

kkkk

kkk

kkkk

kk

kk

kk

( tanθk = cos(kxd/2)/cos(kzd/2) )

By diagonalization, we obtain 3 eigenvalues;

Flat

Band

J

J

d

Ultracold Atoms in a Flat Band

In order to study interesting physics of flat band,

we need to load ultracold atoms into flat band.

“Phase Imprinting Method”

Brillouin Zones

initially

|𝐁 + |𝐂 (q=0) after Phase Imprinting

|𝐁 − |𝐂 (q=0)

flat band:|𝐁 − |𝐂 (q=0)

X=A, B, C

Ultracold Atoms in a Flat Band

|𝐁 − |𝐂

(q=0)

“Flat band”

|𝐁 + |𝐂

(q=0)

“Usual band”

We could observe Unique Dynamics in

Flat band by measuring A-site population.

This is a direct confirmation of

“localized state” in a flat band

Other Bosons in a Lieb lattice

Exciton-Polariton Condensate

arXiv:1505.05652, F. Baboux et al.,

Photonic Waveguide D. Guzman-Silva et al.,

NJP16, 063061(2014)

Fermi-Hubbard Model:

[R. Jördens et al., Nature 455, 204 (2008)]

“A Mott insulator of 40K atoms in an optical lattice”

[U. Schneider, et al., Science 322,1520(2008)]

Fermions in an Optical Lattice

40K

onset of anti-ferromagnetic correlation [R. A. Hart et al., Nature 519, 211 (2015)]

Fermions in an Optical Lattice

T~ 1.4TN(=0.5t)

Spin-structure factor:

optical Bragg scattering

6Li

Fermionic quantum gas macroscope

E. Haller et al., arXiv:1503.02005v2

L. W. Cheuk et al., arXiv:1503.02648v1, PRL(2015)

40K 40K EIT-cooling Raman-sideband cooling

6Li Raman-sideband

cooling M. F. Parsons, et al, PRL(2015)

SU(N=6) system 173Yb

(I=5/2)

mnccmn

m

nS

Nuclear spin permutation operators:

)()(2

,,

2

jSiSU

tH

nmji

n

m

m

n

SU(N) Heisenberg model:

*SU(N=6) Mott Insulator is already created [Taie et al, NP(2012)]

*Two-orbital SU(N) fermions is successfully created[MPQ, LENS,..]

Neel order, dimerization,

Valence-Bond-Solid,

Chiral Spin Liquid, …

Novel magnetic phases: SU(m x k) m: filling

SU(N=6) Fermions in an Optical Lattice

Bose-Fermi Mixture in a 3D optical lattice

Fi

i

BiBF

ji

iFBi

i

BiBB

ji

iB nnccnnaa UtU

tHjj

,,

)1(2

[K. Günter, et al, PRL96, 180402 (2006)]

[S. Ospelkaus, et al, PRL96, 180403 (2006)]

“ 40K(Fermion)-87Rb(Boson)”

“Role of interactions in Rb-K Bose-Fermi

mixtures in a 3D optical lattice” [Th. Best, et al, PRL102, 030408 (2008)]

aBF = -10.9 nm

Bose-Fermi Mixture in a 3D Optical Lattice

Boson

:174Yb

Fermion

: 173Yb

T/TF=0.17 [S. Sugawa, et al., NP.7, 642(2011)]

NF=2 ×104

NF=1 ×104

“Phase Separation”

“Mixed Mott Insulator”

Boson Fermion

Bose-Fermi Mixture in a 3D Optical Lattice

Boson

:174Yb

Fermion

: 173Yb

T/TF=0.17 [S. Sugawa, et al., NP.7, 642(2011)]

NF=2 ×104

NF=1 ×104

“Phase Separation”

“Mixed Mott Insulator”

Boson Fermion

Theory:

Ehud Altman, Eugene Demler, Achim Rosch

“Mott criticality and pseudogap in Bose-Fermi mixtures”PRL(2013)

I. Danshita and L. Mathey

“Counterflow superfluid of polaron pairs in Bose-Fermi mixtures

in optical lattices”PRA(2013)

Quantum Simulation of Lattice-Gauge-Higgs-Model Kasamatsu et al, PRL(2013), Kuno et al, NJP(2015)

Bose-Hubbard Model with Off-Site Interaction：

j

ji

ii

i

i

ji

i nnnnaaVU

JHj

},{, 2)1(

2

hopping

（red＋blue lines)

On-site inteaction

（yellow circles）

Off-site Interaction

（red＋blue lines)

Mielke Lattice

Den

sity

Flu

ctu

ati

on

Position R

Higgs Phase:

exp(-mR)/R

Coulomb Phase：

1/R

Confinement Phase:

R

Summary I) Preparation of Quantum Gas









Bose-Fermi mixture

Group Members

Thank you very much for attention

16 August Mount Daimonji at Kyoto

Documents

冷却原子系による量子シミュレーション Kyoto …...Outline I) Preparation of Quantum Gas II) Ultracold Atoms in a Harmonic Trap Laser cooling and trapping, evaporative