基研研究会「熱場の量子論とその応用」
31 August 2015
Yukawa Institute, Kyoto University

冷却原子系による量子シミュレーション

Kyoto University

Yoshiro Takahashi

OutlineI) Preparation of Quantum Gas
Laser cooling and trapping, evaporative cooling, Bose-Einstein condensate,Fermi Degenerate GasII) Ultracold Atoms in a Harmonic Trap
Feshbach resonance, Cooper paring, BEC-BCS crossover, unitary gas,
spin-orbit interactionIII) Ultracold Atoms in an Optical Lattice
Superfluid-Mott insulator transition, quantum-gas-microscope, Higgs mode,
Non-standard lattices(frustrated magnetism, flat band), Fermi-Hubbard model,
Bose-Fermi mixtureOutlineI) Preparation of Quantum Gas
Laser cooling and trapping, evaporative cooling, Bose-Einstein condensate,
Fermi Degenerate Gas

II) Ultracold Atoms in a Harmonic Trap
Feshbach resonance, Cooper paring, BEC-BCS crossover, unitary gas,
spin-orbit interactionIII) Ultracold Atoms in an Optical Lattice
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“optical trap”
anti-Helmholtz coils

Vint   p  E

U pot (r )  

E (r ) 2CCD

2Six laser beams
for laser cooling

500 mm“magnetic trap”• Number: 107• Density: 1011/cm310mm• Temperature: 10µK

“Magneto-optical Trap”

Vint   m  B

Cooling to Quantum Degeneracy
Pressure ~10-12 torr
Lifetime > 10 sec“Evaporative cooling”

BEC Formation(Yb atom)

T:low

T:highMomentum distribution

T~100 nK N~ 105

Cooling to Quantum Degeneracy“Boson versus Fermion”“Bose-Einsten Condensation”“Fermi Degeneracy”
6Li and 7Li

87RbFermiPressureMomentum Distribution[E. Cornell et al, (1995)]

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

Experimental Setup for Cold AtomOutlineI) Preparation of Quantum Gas
Laser cooling and trapping, evaporative cooling, Bose-Einstein condensate,Fermi Degenerate Gas

II) Ultracold Atoms in a Harmonic Trap
Feshbach resonance, Cooper paring, BEC-BCS crossover, unitary gas,
spin-orbit interactionIII) Ultracold Atoms in an Optical Lattice
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 interactionas   l / kCollision is in Quantum Regime
It is described by s-wave scattering length as 0  4 f 0  4 as

Coupling between “Open Channel” and “Closed Channel”

Potential

Control of Interaction(as)as ( B)  abg (1 22B)
B  B0

Molecular State

F2+F2

Two Atoms-C6/R3

F1+F1“Magnetic field tunes the energy difference
between closed channel and open channel”[C. Regal and D. Jin, PRL90, 230404(2003)]

BEC – BCS Crossover
Leggett, ...
TBCS  0.3TF exp( “molecular”“Unitary Gas”
01/(kFa)2k F a s)

BEC – BCS Crossover: experiments

Zwierlein et al., PRL(2004)

6Li

Inada et al., PRL(2008)

6Li

Regal et al., PRL(2004)

40K

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: SOI   y k x
Y. –J. Lin, et al., Nature 471, 83(2011)“lasers for Raman Transition”
xe“pseudo-spin +1/2”:

E=E2cos(k2x-ω2t)
E=E1cos(k1x-ω1t)

1/ 2ħω0  0  (1  2 ) : detuning: momentum transfer
Q  k1  k 2

1/ 20   1 0  z  0 
22  0  1Qx z)after the local pseudo-spin rotation with θ=Qx about z-axis U  exp(i22 222  k     Qk x  ER ER H
1y 
 z  Q
y 
12m
2m
222m
4:Recoil Energy
BzSOI

Spin-Orbit Interaction in Cold Atoms:
α=β: SOI   y k x
Y. –J. Lin, et al., Nature 471, 83(2011)“Boson: 87Rb”

P. Wang et al., (2012)
L. W. Cheuk et al., (2012)“Fermion: 6Li, 40K”

Topological Superfluids, Majorana Edge stateby Spin-Orbit Interaction + Strong s-Wave Interaction[M. Sato et al., PRL103,020401(2009), PRB82, 134521(2010)]
δ/2π=-8kHz

1S0kx/Q

kｙ/Q

kｙ/Q

kｙ/Q
kx/Q

δ/2π=8kHz

δ/2π=0kHz

kx/Q

Feshbach Resonance:1S0+3P2(171Yb )Number of atoms(arb. units)

SOI between 1S0+3P2(174Yb )Magnetic Field (G)Stern-Gerlach-Separated Images
3P2ℏQAtoms(1S0): 𝜇 = 0,
M=MaAtoms(3P2): 𝜇 = 𝜇𝑎(3P2),
M=Ma

Feshbach Molecules(1S0 +3P2) ∶𝜇 ≈ 𝜇𝑎(3P2), M=2Ma

T/TF ~ 0.25

OutlineI) Preparation of Quantum Gas
Laser cooling and trapping, evaporative cooling, Bose-Einstein condensate,
Fermi Degenerate Gas

II) Ultracold Atoms in a Harmonic Trap
Feshbach resonance, Cooper paring, BEC-BCS crossover, unitary gas,
spin-orbit interactionIII) Ultracold Atoms in an Optical Lattice
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 Systemλlattice
Vo ( x)  Vo sin 2 (k L x)

λlatticeλlatticeλlattice/2“ultracold atoms in an optical lattice”
ideal Quantum Simulator of Hubbard Model:

H   J  c c j  U  ni nii , j iJi-thj-thi

UNice Features of ultracold Atoms in an Optical latticeH   J  ai a j 
i , j Uni (ni  1)
2 i

V  Vo sin 2 (kx)1)Macroscopic Quantum System：typically ～1052)Clean (no impurity, no lattice defects)
3)High controllability of Hubbard parameters
U/J
Small

Weakly-interacting“Superfluid”

Large

Strongly-correlated“Mott Insulator”

Phase Diagram of Bose-Hubbard Model (T>0)
H   J  ai a j
i , j U  ni (ni  1)2 i
   i nii“An interference fringe isthe direct signature of the phase coherence”

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

Phase Diagram of Bose-Hubbard Model (T>0)
H   J  ai a j
i , j U  ni (ni  1)2 i
   i nii“An interference fringe isthe direct signature of the phase coherence”:weight:width

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

“amplitude-(Higgs-)mode”
The `Higgs' Amplitude Mode at the TwoDimensional Supefruid-Mott Insulator Transition
M. Endres et al (2012)

Lattice modulation spectraSpectroscopy of Superfluid-Mott Insulator Transition
V=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.

Nice Features of ultracold Atoms in an Optical latticeH   J  ai a j 
i , j Uni (ni  1)
2 i4) Powerful Measurement Methods:

in situ imagingby Quantum Gas Microscope

V  Vo sin 2 (kx)

87Rb

Fluorescence Imaging

ObjectiveLens
ultracold atoms~ 500 nm[WS. Bakr, et al Nature 462,5 (2009)]

Single Site Resolved Detection ofSF-MI Transition[WS Bakr, et al., Science 329, 547(2010)]
87Rb

In Situ-image

after analysis

TOF-image

SFMINice Features of ultracold Atoms in an Optical latticeH   J  ai a j 
i , j Uni (ni  1)
2 i

V  Vo sin 2 (kx)4) Powerful Measurement Methods:

Single-site manipulation
by Quantum Gas Microscope

87Rb[C. Ewitenberg et al, Nature 471, 319(2011)]

Ytterbium Quantum Gas Microscope“Observation of Single Yb Atoms in an Optical Latticewith Hubbard Regime”“dual molasses“

174Yb (boson)

399nm BP filter
EMCCD camera
AndoriXonEM Blue

M=150f = 600mm
Optical parallels

Objective (NA=0.75)
Working distance 6.23mm
Focal length f = 4mm(Mitutoyo Corporation)
3mm

399 nm
556 nm399 nm

556 nm
Yb atom
Glass cell

Lattice constant: d=266 nm

psf: 𝜎𝐴 =487 nm~30 μmYtterbium Quantum Gas Microscope“Observation of Single Yb Atoms in an Optical Latticewith Hubbard Regime”~10 μm“dual molasses“

174Yb (boson)

399nm BP filter
EMCCD camera
AndoriXonEM Blue

M=150
f = 600mm
Optical parallels

Objective (NA=0.75)
Working distance 6.23mm
Focal length f = 4mm(Mitutoyo Corporation)
3mm

399 nm
556 nm399 nm

556 nm
Yb atom
Glass cell

Lattice constant: d=266 nm

psf: 𝜎𝐴 =487 nm~30 μmNice Features of ultracold Atoms in an Optical latticeH   J  ai a j 
i , j Uni (ni  1)
2 iV  Vo sin 2 (kx)5)Various Lattice Geometries：
Dimensionality:

Standard and Non-standard Lattices:
Cubic(Square) Honeycomb(hexagonal)

KagomeLieb

Honeycomb

KagomeMany interesting materials(high-Tc Cuprate, graphene, ..)have non-standard lattice structures

Several Non-Standard
Optical Lattices are realized

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

Triangular

http://extreme.phys.sci.kobe-u.ac.jp/extreme/student/2007/tomoo/Crystal_gallery.html
http://hiroi.issp.u-tokyo.ac.jp/data/crystal_gallery/crystal_gallery-Pages/Image31.html

LiebQuantum Simulation ofFrustrated Magnetism in a Triangular Lattice[J. Struck et al, Science (2011)]

Phase Modulation of Optical Lattice：

Kcos(𝜔𝑡)

𝐽 → 𝐽 × 𝐽0 𝛽

:Zero-th order
Bessel Function
𝛽=K/𝜔Novel Band Structures
Honeycomb

Dirac Cone:linear dispersion

E(k)=c k:massless Dirac particle

Square(standard)

Cosine Band:

E(k)=Jcos(kd)[from Katsura & Maruyama(2014)]

KagomeFlat Band:

E(k)=const.
: infinite mass/localization

Novel Band Structures
Honeycomb“Creating, moving, and merging Dirac points
with a Fermi gas in a tunable honeycomb
lattice”, Tarruell, et al Nature (2011)

Kagome“Ultracold atoms in a Tunable Optical Kagome Lattice”
Gyu-Boong Jo, et al, PRL (2012)

“Ultracold atoms in a flat band”(case of Kagome lattice)resonating hexagon:“localized eigenstate”

closed packingat 𝜈 c=1/9

𝜈 > 𝜈cSpatial overlap
between hexagons

Exotic strongly
correlated state:huge degeneracy !

Super-Solid

Prediction of super-solidfor bosons in Kagome lattice
Huber & Altman PRB 82, 184502 (2010)𝜈 − 𝜈c[Discussion with S. Furukawa]What is Super-Solid ?
“superfluid”(Off-Diagonal Long-Range Order)→ 𝑛𝟎“Solid order”(Density Long-Range Order/ density wave)
𝛿𝑛 x = 𝛿𝑛(x + 𝑑)

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

Band Structure of Lieb LatticedOperators in k-space:(Tight-Binding Approximation)

Hˆ TB   aˆk† , Akaˆk† , B aˆk , A  aˆ i,Saˆk† ,C T Lieb  aˆk , B  1 N eik x aˆ
kk ,S aˆ  k ,C 



i ,S

S=A,B,CJJ0
 2 J cosk x d / 2   2 J cosk z d / 2 T Lieb   2 J cosk x d / 2 00  2 J cosk d / 2 00zBy diagonalization, we obtain 3 eigenvalues;E   2 J cos 2 (k x d / 2)  cos 2 (k z d / 2) ,E0  0,Flat
Band( tanθk = cos(kxd/2)/cos(kzd/2) ) k , A  sin  k , B  cos  k , C ,k , 1st 

1k , 3rd 

1k2k , 2nd  cos  k k , B  sin  k k , C ,2k k , A  sin  k , B  cos  k , C ,kk

Ultracold Atoms in a Flat Band
In order to study interesting physics of flat band, flat band:|𝐁 − |𝐂 (q=0)
we need to load ultracold atoms into flat band.
X=A, B, C

initially|𝐁 + |𝐂 (q=0)

after Phase Imprinting|𝐁 − |𝐂“Phase Imprinting Method”

Brillouin Zones(q=0)

Ultracold Atoms in a Flat Band
We could observe Unique Dynamics inFlat band by measuring A-site population.

“Usual band”|𝐁 + |𝐂(q=0)“Flat band”|𝐁 − |𝐂

This is a direct confirmationof(q=0)“localized state” in a flat bandOther Bosons in a Lieb latticePhotonic WaveguideExciton-Polariton Condensate

D. Guzman-Silva et al.,NJP16, 063061(2014)arXiv:1505.05652, F. Baboux et al.,Fermions in an Optical Lattice

Fermi-Hubbard Model:
40K “A Mott insulator of 40K atoms in an optical lattice”[R. Jördens et al., Nature 455, 204 (2008)] [U. Schneider, et al., Science 322,1520(2008)]

Fermions in an Optical Lattice

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

Spin-structure factor:
6Li T~ 1.4T (=0.5t)
N

optical Bragg scatteringFermionic quantum gas macroscope
40K

EIT-cooling

40K Raman-sideband cooling
L. W. Cheuk et al.,arXiv:1503.02648v1,PRL(2015)

E. Haller et al., arXiv:1503.02005v2

6Li

Raman-sidebandcooling

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

SU(N=6) Fermions in an Optical Lattice
173Yb(I=5/2)

SU(N=6) system

SU(N) Heisenberg model:
22tH
U S (i)S ( j )mnnm

Nuclear spin permutation operators:

S  c cm  n mmn

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

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

Neel order, dimerization,
Valence-Bond-Solid,
Chiral Spin Liquid, …
*Two-orbital SU(N) fermions is successfully created[MPQ, LENS,..]

Bose-Fermi Mixture in a 3D optical latticeU BB
H  t B  a a j n(n1)
tc
BiBi
F  i c j  U BF  nBi nFi
2 i
i , j 
i , j i
i“ 40K(Fermion)-87Rb(Boson)”aBF = -10.9 nm[K. Günter, et al, PRL96, 180402 (2006)][S. Ospelkaus, et al, PRL96, 180403 (2006)]“Role of interactions in Rb-K Bose-Fermi
mixtures in a 3D optical lattice”[Th. Best, et al, PRL102, 030408 (2008)]

Bose-Fermi Mixture in a 3D Optical Lattice“Phase Separation”

Boson:174YbBoson

Fermion

NF=2 ×104

Fermion: 173Yb“Mixed Mott Insulator”
NF=1 ×104

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

Bose-Fermi Mixture in a 3D Optical Lattice
Boson:174Yb“Phase Separation”
Boson

Fermion

NF=2 ×104

Fermion
Theory:“Mixed Mott Insulator”
173: Ehud
Yb Altman, Eugene Demler, Achim Rosch“Mott criticality and pseudogap in Bose-Fermi mixtures”PRL(2013)
NF=1 ×104
I. Danshita and L. Mathey“Counterflow superfluid of polaron pairs in Bose-Fermi mixturesin optical lattices”PRA(2013)
T/TF=0.17 [S. Sugawa, et al., NP.7, 642(2011)]

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

Mielke LatticeBose-Hubbard Model with Off-Site Interaction：UVH   J  a a j   ni (ni  1)   ni n j2 i
2 {i , j}
i , j i

On-site inteaction（yellow circles）

Off-site Interaction（red＋blue lines)
Higgs Phase:exp(-mR)/RDensity Fluctuation

hopping（red＋blue lines)

Coulomb Phase：
1/R
Confinement Phase:
RPosition RSummaryI) Preparation of Quantum Gas
Laser cooling and trapping, evaporative cooling, Bose-Einstein condensate,
Fermi Degenerate Gas

II) Ultracold Atoms in a Harmonic Trap
Feshbach resonance, Cooper paring, BEC-BCS crossover, unitary gas,
spin-orbit interactionIII) Ultracold Atoms in an Optical Lattice
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 AugustMount Daimonji at Kyoto