Quantum Simulation of Hubbard Model Using …...Quantum Simulation Magnetism, Superconductivity...

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 10@5.20,3.40

=60m

m

=70mm

AGBGB zx 10@95.8,7.11

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