Quantum Simulation Using Ytterbium Atoms in an Optical Lattice
Takeshi Fukuhara1 , Seiji Sugawa1, Takuya Tsujimoto1, Yosuke Takasu1, and Yoshiro Takahashi1,2
The cold fermionic atoms in an optical lattice aredescribed by the Hubbard Hamiltonian:H = − t ∑ c †i,σ c j,σ + U ∑ n i,↑ n i,↓

Simulation of Quantum Systemi, j ,σdifficult

Classical computer

Quantum computer(Controllable quantum system)ξ ≡ (Vo /E R )1/4

Unique features• A variety of isotopes (5 Bosons and 2 Fermions)

8 3ξπ

H. P. Büchler and G. Blatter, PRL. 91, 130404 (2003),
M. Lewenstein, et al., PRL. 92, 050401 (2004),
L. Mathey et al., PRL. 93, 120404 (2004),
D.-W. Wang, M. D. Lukin, E. Demler, PRA. 72, 051604(R ) (2005),
L. Pollet, et al., PRL. 96, 190402 (2006).

W. Hofstetter, et al.,
PRL. 89, 220407 (2002).

Hubbard model:t• Tunneling• Onsite interaction : U: <n>• Filling factor:T• Temperature

Nuclear spin I=0 for bosonic isotopes

Spin-3/2 Fermionic Atoms in Optical Lattices

Atomic system
Lattice potential by the laser intensity( Optical ) Feshbach Resonance
Atom density
Temperature of atoms

Experimental Procedure

Cooling to Quantum Degeneracy

1P1S0ν 3P1

atomicbeamYb

2

1S -1P
01

1.5

174

Horizontal FORT
Beam waist: 14μm
Potential depth: up to 650 μK

172
173 (F’=3/2)
173 (F’=7/2)

1

171 (F’=3/2)

176

171
(F’=1/2)
170

173(F’=5/2)

0.50-1000-500

0500
Frequency (MHz)

1000

1500

Laser Wavelength: 532 nm
Laser Power: up to 〜5.5 W

gravity3(b)171(F’=1/2)

1.5

173Yb

4Sympathetic Cooling with 174Yb-10123(b)MOTAtom Number : 〜10Temperature: 15〜40µK
Atom Density: 1012cm-3

Thermalizing elastic collisions
are important to perform
efficient evaporative cooling

5172Yb

Fukuhara, et al., PRL 98, 030401.

Sympathetic Cooling with 174Yb

(c)
-50

BEC

050
Position (um)

174Yb

100

176Yb

170Yb

4Frequency (GHz)70

N173 = 6×103 T173 = 140nK （T/TF〜0.45）
N174 = 3×104 T174 = 80nK < Tc BEC

TOF=10ms“BEC-BEC mixture”

0-2-5Fermi velocity vF=4.4μm/ms

00.5-3T/TF〜0.4

BEC-FDG Mixture

170

13(a)

-100

N〜1×104173 (F’=3/2)
173 171 (F’=3/2)(F’=5/2)

176

22Velocity (um/ms)

172

173
(F’=7/2)

1(a)

174

1S -3P01

4

2.5

T/TF〜0.9
0Evaporation time [sec]

(c)

3.5

171Yb

1240 m

Vertical FORT
Beam waist: 85μm
Potential depth: 18 μK

54.5

Fermi-Dirac distribution

174

TOF=10ms

Optical Trap
(FORT)

Isotope shifts

CCD camera

106 in FORT

171

100.1

107 atoms in MOT

556nm

Fluorescence (arb. units)556-nmlaser for MOT

399nm

Fluorescence (arb. units)399-nmZeeman slowing
laserer

168Yb

Integrated optical density (arb. units)

n slow
Zeema3P
1

Evaporative Coolingin the Crossed FORT

Temperatures [μK]100ν 1P1 1

Our group previously obtained BEC:
Takasu, et al., PRL 91 040404.

H.-H. Tu, et al., PRB. 76, 014438 (2007).

Laser Cooling and Trapping
anti-Helmholtz
coils

Fermion: 171Yb (14.3%, I=1/2)
173Yb (16.2%, I=5/2)
Boson: 168Yb (0.13%)
170Yb (3.05%)
172Yb (21.9%)
174Yb (31.8%)
176Yb (12.7%)

Bose-Fermi Mixtures in an Optical Lattice

E R ≡ (hk) 2 /2m

Controllable Parameters

Excellent controllability of various parameters
Lack of undesirable impurities

Quantum Simulation of high-Tc SC using
Cold Fermionic Atoms in an Optical Lattice

U = ER a sk

as : Scattering lengthk : Wave number of the lattice beams
V0 : Lattice potential depth

Cold Fermionic Atoms
In an Optical Lattice

analogy4 3
ξ exp(−2ξ 2 )π

Distribution (arb. units)

Electrons
In a Crystal Latticet = ER• Ultranarrow optical transition: 1S0-3P0 1S0-3P2
possibility to detect weak interaction: superconducting gap

Onsite interactionof atoms

Transmission

efficient• Only nuclear spin in the ground state: 1S0 (S=0)
less sensitive to stray magnetic field: small decoherence

high-Tc cuprate superconductors.

1

Interesting quantum system

Widely recognized as a model of strongly
correlated electron systems such asi

Tunneling between
adjacent sites

Ytterbium (Yb)

BEC

Integrated optical density (arb. units)

Quantum Simulation

Kyoto University1, JST-CREST2

BEMB-50050

Position (um)160μm

N〜1×105

120 mN176=1×104
PSD176>2.6
TOF=8ms

Optical LatticeObservation of 1st Brillouin Zone

“Number Squeezed State”

Mapping the quasi momentum to the free particle momentum
by adiabatically decreasing the lattice depth.

When the lattice depth was increased, the interference pattern was lost.

TOF=10ms

After Producing BEC〜20 ER

86μm

Lattice
depth

Optical lattice parameter
Laser wavelength: 532 nm → Period of lattice sites: 266 nm
Beam waist: 20 μm (Spatial mode is cleaned by using optical fiber)
Laser Power: up to 100 mW → Lattice potential depth: up to 40 ER

100 msFree expansion of an array of BECs

adiabatically

2hktm

1000

0-100-50

050

100Position (um)gravity

We observed the diffraction of a 174Yb BEC
by a pulsed standing light wave.

restoredlostTOF
12 ms
-100

-50

050

100

-100

-50

050

100

-100

-50

050

100

-100

-50

050

100

-100

-50

050

0.8

The number of atoms decreased probably
due to three-body recombination lasses.

0μs
7μs

The loss of coherence was mainly due to suppressed number fluctuation.
→ Number Squeezed state

Optical lattice

The fraction of condensate atoms
in the first diffraction order

Pulse duration

100

280μm

14μs

N1/N

Optical density [arb. units]

mirrorInterference patterns

86μm

2000

Pulsed Optical Lattice

Crossed-beam
optical trap

Ntotal〜3×104〜 20 sites

rapidlyTime

200 ms

TOF
12 ms

BEC in a 1D Optical Lattice
After creating 174Yb BEC in the crossed-beam opticaltrap, we loaded the BEC into one-dimensional opticallattices by ramping up the intensity of the lattice beam.

3000

Optical density (arb. units)

A standing wave created by counterpropagating beams causes a periodic
potential for atoms.

0.400

Optical density (arb. units)

TOF=12 ms

2 ms

4 ms

6 ms

8 ms

10 ms

12 ms

14 ms

We investigate change in the interference pattern of a released 174Yb BEC
by mixing fermionic 173Yb impurity atoms into the BEC.

28μs

-50

0Position (um)

50

35μs

100

k : Wave number of the lattice beams
m : Atomic masst : Expansion time

49μs
-50

0Position (um)50

Yb

A variety of isotopes (5 Bosons and 2 Fermions)100

NB=105
NF=103ω=2π×500 Hz
T=0 K

Quadrupolar Oscillations

50

After the lattice beam is suddenly turned on,
almost all the atoms are distributed over statesof the two lowest-energy even bands.
The oscillation frequency corresponds to theenergy difference between the two band.

We can precisely determine
the lattice depth by using thispulsed lattice technique.

• Loading 174Yb BEC into 1D optical lattices
Interference patterns
Number squeezing
Mixing Fermionic inpurity• Observation of 1st Brillouin Zone
• Calibration of the lattice potential by pulsed lattice

N174=3×104 (Half of atoms is condensed.)
N173=1×104 T=100nk• Investigating collective oscillations in a Bose-Fermi mixture

Thomas-Fermi radius of 174Yb BEC after TOF of 12ms.
Fermion60●174Yb only
△174Yb with 173Yb

40
Radius [μm]Radius [μm]

5040
30

Future plan
• Realizing BEC and degenerate Fermi gas in 3D optical latticeand observing the Mott-insulator transition

30
delay

200100200
Time [ms][Kitagawa, et al., arXiv:0708.0752]

40• Production of a variety of degenerate gases and mixtures ofYb isotopes

We excited quadrupolar oscillations in a Bose-Fermi mixture of 174Yb-173Yb
by changing trapping potential and observed oscillations of 174Yb BEC
with/without 173Yb atoms.

Boson

Scattering lengths for Yb isotopic combinations are evaluatedby photoassociation spectroscopy and mass scaling.

30

Summary

120μm

Ex. Bose-Fermi Mixture(Phase separation)

20

Pulse duration [s]Significant change has not been observed.

Quantum degenerate mixture
Ultracold atomic gases system is quite suitable for understanding
quantum many-body system and one unique feature of this systemis that a variety of mixtures can be investigated.

42μs〜 8×103 173Yb impurity atoms
N173/ N174 〜0.4-10010↓

TOF=12ms-100

TOF 0 ms

21μs

Optical density (arb. units)

gravity

2hk±tmBose-Fermi Mixture in Optical Lattices

300

400

20200

300

Time [ms]When we mixed 173Yb atoms, a delayof the oscillations was observed.

• Further cooling in optical lattices (Sideband cooling)400• Investigating behavior of Bose-Fermi mixture in optical lattices• Understanding mixtures of quantum many-body system
Stability (collapse), Structure (phase separation), and Dynamics (collective mode)