Kyoto University Quantum Simulation of Hubbard Model Using Ultracold Atoms in an Optical Lattice 25 August 2010 Okinawa FIRST Quantum Information Processing Project Summer School 2010 Y. Takahashi
Kyoto University
Quantum Simulation of Hubbard Model
Using Ultracold Atoms
in an Optical Lattice
25 August 2010 Okinawa
FIRST Quantum Information Processing Project
Summer School 2010
Y. Takahashi
Introduction(自己紹介) Name(氏名) :
Yoshiro Takahashi(高橋義朗)
Education(学歴):
Ohta High-School(群馬県太田高校)
Kyoto University, Faculty of Science (京都大学理学部)
Kyoto University, Graduate School of Science
(京都大学大学院理学研究科)
Degree(学位):
Anomalous Behavior of Raman Heterodyne Signal in Pr3+:LaF3
Employment(職歴):
Kyoto University,
Research Associate(助手):Atoms in Superfluid Helium
Lecturer(講師):Photo-excited triplet DNP
Associate Professor(助教授):Laser Cooling
Professor (教授)
Introduction(自己紹介)
Research Topics:
Quantum Information Science Using Cold Atoms
Quantum Simulation (of Hubbard Model)
Spin Squeezing by QND Measurement
Fundamental Physics Using Cold Atoms:
(Searching for Permanent Electric Dipole Moment)
Test of Newton Gravity:
))exp(1(21
r
r
MMGV
Solid-State System
Atomic System
Quantum Computation Quantum Simulation
Quantum Simulation
Many-body
Classical System
“HARD”
Many-body
Quantum System
“Interesting”
Many-body
Quantum System
“Controllable”
Quantum Simulation
Magnetism, Superconductivity
Hubbard Model:
i
ii
ji
i nncc UJHj
,
i-th j-th
J
U
Quantum Simulation
Hubbard Model:
i
ii
ji
i nncc UJHj
,
i-th j-th
J
U
DMFT(動的平均場)
Gutzwiller
QMC(量子モンテカルロ)
DMRG(密度行列繰り込み群)
Exact Diagonalization (厳密対角化)
Numerical Calculation
Quantum Simulation
Exact Diagonalization of Hubbard Model
Earth Simulator: 1D Fermi Hubbard Model:
Quarter Filling: 24 sits
Half Filling: 20 sites
S. Yamada, T. Imamura, M. Machida
Proceedings of the 2005 ACM/IEEE SC05 Conference(SC’05)
Next generation:
Quarter Filling: 32 sites
Half Filling: 26 sites
Quantum Simulation
Hubbard Model:
i
ii
ji
i nncc UJHj
,
i-th j-th
J
U
λ/2
Cold Atoms in Optical Lattice
)(sin2 kxVV o
Optical Trapping and Optical Lattice
mm 500
2
2
1EU
“Optical Trap”
satI
I
8
2
“Optical Lattice”
)(sin)( 2 xkVxV Loo
3
1
23
1
2 )(sin)(sin)(j
jLo
j
jLojo xkVxkVV xR
LR
E
Vs
m
kE 0
2
,2
)(
“to prevent mutual interference, frequency is shifted relative to each other by tens of MHz”
xzyx
Quantum Simulation of Hubbard Model using
“Cold Atoms in Optical Lattice”
filling factor (e- or h-doping) :n
Controllable Parameters
hopping between lattice sites : J
On-site interaction :U Feshbach Resonance :as
lattice potential :V0
atom density :n
i
ii
ji
i nncc UJHj
,
4/3/8 skaEU LsR
Ro EVs / mkE LR 2/)(, 2
)2exp()/2( 4/3 ssEJ R
, as : scattering length
λ/2
Various geometry
[D. Jaksch et al., PRL, 81 , 3108(1998)]
Atomic Scattering Theory
0
)(cos)()12()cosexp()exp(l
ll
l
SC Pkrjilikrikz
R
0
)2/sin()(cos)12(
l
l
l
kr
lkrPil
“out-going” “in-coming”
))2
(exp())2
(exp(2
)(cos)12(
0
l
kril
kriikr
Pil
l
ll
With atom-atom interaction
)exp()exp( ikrr
fikzSC f :scattering amplitude
0
)2/sin()(cos)12(
l
ll
l
SCkr
lkrPil
:phase shift l
))2
(exp()2exp()2
(exp(2
)(cos)12()exp(
0
l
kriil
kriikr
Pili l
l
ll
l
00S
Atomic Scattering Theory
0
)(cos)12(l
ll fPlf k
iik
if l
ll
l
)sin()exp(
2
1)2exp(
with
)(sin)12(4
)12(4 2
2
2
lllk
lfl
ik
Sfl
2
100
“At low temperature, only s-wave (l=0) scattering is important”
ka l
s
0k
saf 0
22
00 44 saf
Q. What is Scattering Length ?
as
as: positive
R
E
Internuclear distance
kR
aRk
kR
kRR s
SC
))(sin()sin()( 0
R
E
Internuclear distance
as
as: negative
R
)(4
21
2
int rrm
aV s
4
)(32
)(4 is xxwxd
m
aU
Analytical Expression of Scattering Length
6
6)(R
CRV
)
8tan(1
ss aa
0
)(21
r
dRRVm
)4/5(
)4/3(
4
2)
4cos(
2/1
6
Cas
m
as
0
[Gribakin & Flambaum
PRA, 48 546(1993)]
)2
1(
8 Dv
R
E
Internuclear distance
6
6)(R
CRV
E=0
v=1
v=2
v=69 v=vD
))
2
1(tan(1 Dss vaa
vD
r0
Reduced mass
R
69 70 68
Binding Energy and Scattering Length : Case of Yb Atom
[M. Kitagawa, et al, PRA77, 012719 (2008)]
Lennard-Jones-like potential:
2
2
6
6
8
8
12
12 1
2
)1()(
r
JJ
r
C
r
C
r
CrV
m
0
)(21
r
dRRVm
Q. What is Feshbach Resonance ? P
ote
nti
al
-C6/R3
Two Free Atoms
Molecular State
Coupling between “Open Channel” and “Closed Channel”
Control of as )1()(0BB
BaBa bgs
[T. Kohler, K. Goral, P. S. Julienne, RMP 78, 1311 (2006)]
7Li
Optical Feshbach Resonance
Advantages for Intercombination Lines
R. Ciurylo, et al. Phys. Rev. A 70. 062710 (2004)
En
erg
y
U(R
)
R
S+P
S+S
Optical
Excitation Two Atoms
Molecular State
2/2/
2/2/00
ii
iiS
S
S
2
fVb lasS
:spontaneous decay rate
:detuning from the PA resonance
22
2
00)2/2/(
)1(
S
SPA
kS
kK for 1/ S0
inE
refE
ii
iiS
E
E
in
ref
tr 11r
lossl :)2/( Lct
)2/( Lcl
[J. Bohn and P. Julienne PRA(1999)]
Nanometer-scale Spatial Modulation
of an Inter-atomic Interaction
2
mm
UModulation Index β= βLS+ βm
2
LSLS
U
inte
nsi
ty
high
low
10-100 µs Pulse
[R. Yamazaki et al ., PRL105, 050405 (2010)]
Q: How Various Geometry ?
[C. Becker et al., New J. Phys. 12 065025(2010)]
An Example: Triangular Optical Lattice
[M. Greiner Group)]
band structure
,kc : annihilation operator of atom
with spin σ for the wavevector k
,where
,,
,,0 )(
k
kk kccH
))(exp()(,
ji
ji
xxikJk
,
,,,
,0 j
ji
i ccJH
)}cos(){cos(2)( dkdkJk yx
1D case:
2D case:
)cos(2)}exp(){exp()( dkJdikdikJk xxx
0
E
0 π -π kxd
4J
8J
(d :lattice constant)
Phase Diagram of High-Tc Cuprate Superconductor
There is controversy in the under-dope region
[in T. Moriya and K. Ueda, Rep. Prog.Phys.66(2003)1299]
(carrier doping) (carrier doping)
SC SC
AF
hole electron hole electron x
experiment theory
Optical Imaging
resonant probe light
Iincident(x,y)
lens
CCD
Itransmission(x,y)
“The atom distribution after certain time from the sudden release of the atoms
corresponds to the momentum distribution”
Time-of-Flight Image:
Cold Atoms
TOFtMPx )/(
Quantum Simulators using Alkali Atoms
[M. Greiner, et al., Nature 415,39 (2002)]
…
“Superfluid - Mott-insulator Transition”
Bose-Hubbard Model: 87Rb
“Formation of Mott-insulator state” Fermi-Hubbard Model:
[R. Jördens et al., Nature 455, 204 (2008)]
[U. Schneider, et al., Science 322,1520(2008)]
40K
87Rb
40K +
Bose-Bose-Hubbard Model:
[K. Günter, et al, PRL96, 180402 (2006)]
[S. Ospelkaus, et al, PRL96, 180403 (2006)]
Bose-Fermi-Hubbard Model:
[Th. Best, et al, PRL102, 030408 (2008)]
[J. Catani, et al, PRA77, 011603(R) (2008)] 87Rb 41K +
Bosons in a 3D optical lattice
i
i
ii
i
i
ji
i nnnaaU
JHj
)1(2,
λlattice λlattice
λlattice
λlattice/2
“Bose-Hubbard Model”
Phase Diagram of Repulsively Interacting Bosons
“Mott
Insulator”
[RMP80,885(2008)]
01
1
NM
i
iSF aM
01
nM
i
iMI a
“Superfluid”
Interference Fringe :
the direct signature of the phase coherence
',
'ˆˆ))'(exp()(
RR
RRaaRRikkG
)()(~)(2
kGkwkn TOFtMkx )/(
“Sudden Release”
kx
Fourier Transform of the Wannier function
NkGaaRRRR )(ˆˆ ',' no long-range order:
)2/(sin
)2/(sin)(1ˆˆ
2
2
'kd
kdNkGaa RR uniform long-range order:
peaks at Lkn2 (n=0,1,2...)
TOFt
Bose-Hubbard Model: “Superfluid - Mott-insulator Transition”
[M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415,39 (2002)]
REV /0No lattice 3 7 10
13 14 16 20
87Rb
“cubic lattice”
[C. Becker et al., New J. Phys. 12 065025(2010)]
s=
“triangular lattice”
[C. Becker et al., New J. Phys. 12 065025(2010)]
s=
“triangular lattice”
Phase Diagram of Repulsively Interacting Bosons
Shell Structure of Mott States
N=3
N=2
N=1
[RMP80,885(2008)]
High-Resolution RF Spectroscopy:
Observation of Mott Shell Structure
[G. K. Campbell et al., Science 313, 649 (2006)]
)1)(( 1112
11
naaa
Uh n
Fermions in a 3D optical lattice
i
i
iii
iji
i nnncc UJHj
,,
,
λlattice λlattice
λlattice
λlattice/2
“Fermi-Hubbard Model”
Phase Diagram of Repulsively and
Attractively Interacting Fermions
[T. Esslinger, Annu. Rev. Condens. Matter Phys. 2010. 1:129-152,
R. Micnas, J. Ranninger, S. Roaszkiewicw, Rev. Mod. Phys. 62, 113(1990)]
Observation of Fermi-Surface of 41K [M. Köhl, et al., PRL 94, 080403(2005)]
“band mapping”
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)]
Fermi-Hubbard Model:
[R. Jördens et al., Nature 455, 204 (2008)]
“A Mott insulator of 40K atoms in an optical lattice”
Modulation Spectroscopy of Mott Gap:
lattice intensity modulation results in creation of doublon
U
Fermi-Hubbard Model:
[N. Strohmaier et al., PRL 104, 080401 (2010)]
“A Mott insulator of 40K atoms in an optical lattice”
Doublon Decay
α~0.8 [ K. Winkler et. al., Nature 441, 853 (2006)]
Isolated Pair
Repulsively Bound Pair in an Optical Lattice
[ K. Winkler et. al., Nature 441, 853 (2006)]
Bose-Fermi Mixture in a 3D optical lattice
Fi
i
BiBF
ji
iFBi
i
BiBB
ji
iB nnccnnaa UtU
tHjj
,,
)1(2
λlattice λlattice
λlattice
λlattice/2
“Bose-Fermi Hubbard Model”
Phase Diagram of Bose-Fermi Mixture [I. Titvinidze, et al., . PRL 100, 100401(2008)]
Spinless Fermion, Repulsive BF interaction, Half Filling, T=0
fermion boson
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
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-Bose Hubbard Model
[J. Catani, et al, PRA77, 011603(R) (2008)]
“ 41K(Boson)-87Rb(Boson)” aBB = +8.6 nm
87Rb only
87Rb
mixed with 41K
[B. Gadway, et al, PRL105, 045303 (2010)]
“87Rb:F=1(Boson)-87Rb:F=2(Boson)”
aBB ~ +5.3 nm
New Technique: Single Site Observation [WS. Bakr, I. Gillen, A. Peng, S. Folling, and M. Greiner, Nature 462(426), 74-77(2009)]
Fluorescence Imaging
87Rb
Single Site Resolved Detection of MI [arXiv1006.3799v1 J. F. Sherson, et al.,]
Light-Assisted Collision
En
erg
y
U(R
)
Internuclear distance R
2S1/2+2P3/2
2S1/2+2S1/2
MOT Light
1) Fine-structure changing collision
2S1/2+2P1/2
2S1/2+2P3/2
2S1/2+2P1/2
2) Radiative Escape
+ K.E.
2S1/2+2P3/2
2S1/2+2S1/2 + K.E.
Single Site Resolved Detection of MI [arXiv1006.0754v1 WS Bakr, et al.,]
SF MI
TOF-image
In Situ-image
after analysis
Quantum Simulators using Alkali Atoms
[M. Greiner, et al., Nature 415,39 (2002)]
…
“Superfluid - Mott-insulator Transition”
Bose-Hubbard Model: 87Rb
“Formation of Mott-insulator state” Fermi-Hubbard Model:
[R. Jördens et al., Nature 455, 204 (2008)]
[U. Schneider, et al., Science 322,1520(2008)]
40K
87Rb
40K +
Bose-Bose-Hubbard Model:
[K. Günter, et al, PRL96, 180402 (2006)]
[S. Ospelkaus, et al, PRL96, 180403 (2006)]
Bose-Fermi-Hubbard Model:
[Th. Best, et al, PRL102, 030408 (2008)]
[J. Catani, et al, PRA77, 011603(R) (2008)] 87Rb 41K +
Yb Atoms
Our Approach
two-electron atom
Kyoto University
Quantum Simulation of Hubbard Model
Using Ultracold Two-Electron Atoms
in an Optical Lattice
25 August 2010 Okinawa
FIRST Quantum Information Processing Project
Summer School 2010
Y. Takahashi
Unique Features of Ytterbium Atoms
168Yb
(0.13%)
170Yb
(3.05%)
171Yb
(14.3%)
172Yb
(21.9%)
173Yb
(16.2%)
174Yb
(31.8%)
176Yb
(12.7%)
Boson Boson Boson Boson Boson Fermion Fermion
Rich Variety of Isotopes
[M. A. Cazalilla, et al., N. J. Phys11, 103033(2009), Hermele, et al., PRL 103, 130351
(2009); A. V. Gorshkov, et al., Nat. Physics, 6, 289(2010)]
novel magnetism
)(4
21
2
int rrM
aH s
SU(6) system 173Yb (I=5/2)
Unique Features of Ytterbium Atoms
Ultra-narrow Optical Transitions
~15 s (10~40 mHz)
~23 s (15 mHz) 507 nm
578 nm
1S0
3P0
3P2
High-resolution laser spectroscopy
as a Local Probe
High-spatial resolution Optical
Magnetic Resonance Imaging
Another Useful Orbital States with
Different Characters
Preparation of Quantum Degenerate Gases
Optical Trap
(FORT)
gravity
Cold
Hot
Bose-Einstein
Condensation
N~105
T~100nK
174Yb
[Y. Takasu et al., PRL 91, 040404 (2003)]
Current Experimental Setup
Quantum Degenerate Yb Gases
174Yb 160µm
TOF:
10ms
30 µm
170Yb 120 µm TOF: 8 ms
176Yb
Boson [T. Fukuhara et al., PRA 76,051604(R)(2007)] [Y. Takasu et al., PRL 91, 040404 (2003)]
168Yb(0.13%)
Fermion [T. Fukuhara et al., PRL. 98, 030401 (2007)]
171Yb(I=1/2) T/TF =0.3 (2-component)
173Yb(I=5/2) T/TF =0.14 (6-component)
539 538 539 540 539 542
300
200
100
100
200
300
Light shift(arb.unit)
f(GHz)
173Yb:1S0-3P1 transition
Nuclear Spin Dependent Light-Shift (calculation)
MF=
-5/2
-3/2
-1/2
+1/2
+3/2
+5/2
F=7/2 F=5/2 F=3/2 :1
3
0
1 PS :1
3
0
1 PS :1
3
0
1 PS
mF= -3/2 -1/2
-5/2
+3/2
+1/2
+5/2
“Optical Stern-Gerlach Effect”
σ+ σ–
+5/2
+3/2
+1/2
-5/2, -3/2, -1/2
-5/2
-3/2
-1/2
+5/2, +3/2, +1/2 g
TOF 7ms
Ultracold 173Yb: Fermi Gas with 6-spin components
1S0–3P1 Δ~2π×4GHz,
10mW, 3.4ms, 90µm
z
EFp F
FF
m
mm
Optical Stern Gerlach Separation:
Optical Pumping Effect
“No Optical Pumping”
MF= -5/2,
-3/2
-1/2
+1/2
+3/2
+5/2
“Optical Pumping”
+5/2
+5/2
+3/2
Other Quantum Gases of Two-Electron Atoms
40Ca:BEC (PTB, 2009) 84Sr :BEC (Rice, Innsbruck, 2009) 87Sr :Fermi-Degeneracy (Rice, 2010)
[T. Fukuhara et al., Phys. Rev. A 79, 021601(R) (2008)]
173Yb(Fermion) +174Yb(Boson)
T/TF=0.2 NB~3×104,
BEC
174Yb(Boson)+ 176Yb(Boson)
NB~6×104,
BEC
NB~2×1
04
Quantum Degenerate Mixtures of Yb
173Yb(Fermion) +170Yb(Boson)
T/TF ~ 0.5 NB~8×103,
BEC
170Yb(10ms) 173Yb(4ms)
171Yb(Fermion) + 173Yb(Fermion) 171Yb(m=+1/2) 173Yb(m=+5/2)
T/TF = 0.3 T/TF = 0.33
SU(2)×SU(6) Symmetry
T/TF = 0.46 (2-component)
T/TF = 0.54 (6-component)
[Theory: D. B. M. Dickerscheid et al ., Phys. Rev. A 77, 053605 (2008)]
)(][ 2117317120173171 rrSSWWH
“Spinor Superfluidity”
171Yb:
173Yb:
N = 8.0×103
T = 95 nK
N = 1.1×104
T = 87 nK
[S. Taie et al ., arxiv:1005.3710]
Boson 174Yb in a 3D optical lattice
i
i
ii
i
i
ji
i nnnaaU
JHj
)1(2,
λlattice= 532 nm λlattice= 532 nm
λlattice= 532 nm
174Yb as=5.55 nm
Superfluid-Mott Transition T. Fukuhara, et al., PRA. 79, 041604R (2009);H. Moritz and T. Esslinger, Physics 2,31(2009)(Viewpoint)
Unique Applications
A. J. Daley et al, PRL. 101, 170504(2008). Dual Lattice Configuration
A. V. Gorshkov et al, PRL. 102, 110503(2009). Few-Qubit Quantum Register
K. Shibata et al, Appl. Phys. B 97, 753(2009). Single-Atom Addressing by MRI
M. Hermele et al, PRL. 103, 135301(2009). Chiral Spin Liquid
F. Gerbier and J. Dalibard, New J. Physics 12, 033007(2010). Gauge fields
Strongly Interacting Two Different Mott Insulators
Bosonic MI Fermionic MI
???
Bose-Fermi Mixture in a 3D optical lattice
174Yb(Boson) +173Yb(Fermion):
aBB = +5.6 nm aFF = +10.6 nm
170Yb(Boson) +173Yb(Fermion):
aBB = +3.4 nm aFF = +10.6 nm
Repulsive Interaction: aBF = +7.3 nm
Attractive Interaction: aBF = -4.3 nm
λlattice= 532 nm λlattice= 532 nm
λlattice= 532 nm VB ~ VF
ωB ~ ωF
ΔzB ~ ΔzF
tB ~ tF
Measurement of Site Occupancy by Photoassociation [T. Rom, et al., PRL93, 073002(2004)]
boson fermion
Bosonic
Double Occupancy
Fermionic
Double Occupancy
Bose-Fermi
Pair Occupancy
Example:
Fermion-Induced Bosonic Double Occupancy
Pure Boson
Bose-Fermi
Mixture
(attractive) Photoassociation Resonance
boson fermion
Fermion (173Yb) in a 3D optical lattice
173Yb(I=5/2) as=10.5 nm
',
,',
, FF
FFj
mmi
imimFF
ji
iF nncc UtH
[M. A. Cazalilla, A. F. Ho, M. Ueda., N. J. Phys11, 103033(2009), ]
SU(6)Mott-state
Single Site Addressing:
Optical Magnetic Resonance Imaging (MRI)
cmG /10
Spatial resolution: 250 nm
kHz1
Magnetic field gradient
Spectral Resolution
Optical absorption line of linewidth 15 mHz
x
z
f
“Optical Spectrum
of 1S0-3P2 transition”
1S0-3P2:
Nagaoka-ferro
Quantum
Computation
507 nm
1S0
3P2 ~15 s
Bmm 3
[K. Shibata et al., App. Phys. B 97, 753(2009)]
Cold Atoms in a Thin Glass Cell
MOT
Transfered
Optical Tweezer
BEC Formation
14 mm
1D lattice
Towards Single Site Addressing in 2DLattice
MOT coil
8 turns, 5 layers
A10@G/cm8.50
z
Bz
Rectangular coil (2)
10 turns, 4 layers
A10@G/cm8.34
y
Bz
Rectangular coil (1)
10 turns, 4 layers
A10@G/cm32.8
x
Bz
=40m
m
I
AGBGB zy [email protected],3.40
=60m
m
=70mm
AGBGB zx [email protected],7.11
Bz compensation
coil
2/1
2
Paramagnetic molecule
SNN 2BHm
N:rotation S:electron spin
Lattice-Spin Model Using Polar Molecule
6Li MOT
[A. Micheli, et al., Nature Physics 2,341 (2006)]
2,
3
0,
18
AHeff
Current Status of YbLi Experiments
N=1.5×108
T~ 280 µK
The First Yb-Li Simultaneous MOT
174Yb 6Li
N=1.4×107
T=60 µK
[M. Okano et al., Appl. Phys.B98,2(2009)]
Summary
Quantum Simulation of Hubbard Model Using Optical Lattice
Review of Experiments using Alkali Atoms
Superfluid-Mott Insulator Transition
Formation of Fermi Mott Insulator
Bose-Fermi and Bose-Bose Mixtures
Single Site Resolved Observation of SF-Mott Insulator Transition
Report of Experiments using Yb Atoms
Superfluid-Mott Insulator Transition
SU(6) Fermi Mott Insulator
Strongly Interacting Bose-Fermi Mott Insulators
Nano-Scale Modulation of Interatomic Interaction in Bose Condenstate
Towards Single Site Addressing Using 3P2 State
Towards Quantum Simulation Lattice Spin Model by YbLi polar Molecule
R. Yamazaki, YT, R. Inoue, K. Shibata, J. Doyle, S. Kato
Y. Yoshkawa, S. Uetake, S. Sugawa, S. Taie, H. Hara, H. Shimizu, R. Yamamoto, I. Takahashi
R. Namiki , H. Yamada, Y. Takasu, R. Murakami, S. Imai, (S. Tanaka. N. Hamaguchi)
NTT:
K. Inaba M.Yamashita
Quantum Optics Group Members