冷却原子系による量子シミュレーション Kyoto …...Outline I) Preparation of Quantum...

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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

Transcript of 冷却原子系による量子シミュレーション Kyoto …...Outline I) Preparation of Quantum...

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

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

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

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

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

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

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

Nice Features of ultracold Atoms in an Optical lattice

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:

Nice Features of ultracold Atoms in an Optical lattice

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

Nice Features of ultracold Atoms in an Optical lattice

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

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

Group Members

Thank you very much for attention

16 August Mount Daimonji at Kyoto