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Quantum Simulation of Hubbard Model Using ... Quantum Simulation Magnetism, Superconductivity Hubbard Model: n p ! i i i i j H J c i c U n n j, i-th j-th J U Quantum Simulation Hubbard

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  • 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

    MM GV 

  • 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 i

    i ji

    i nncc UJH j ,

    i-th j-th

    J

    U

  • Quantum Simulation

    Hubbard Model:

     

       i i

    i ji

    i nncc UJH j ,

    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 i

    i ji

    i nncc UJH j ,

    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

    1 EU 

    “Optical Trap”

    satI

    I

     

    8

    2

    “Optical Lattice”

    )(sin)( 2 xkVxV Loo 

     

     3

    1

    2 3

    1

    2 )(sin)(sin)( j

    jLo

    j

    jLojo xkVxkVV x R

    L R

    E

    V s

    m

    k E 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 i

    i ji

    i nncc UJH j ,

    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

    lkr Pil

     

    “out-going” “in-coming”

     

      

      

    )) 2

    (exp()) 2

    (exp( 2

    )(cos )12(

    0

      l

    kri l

    kri ikr

    P il

    l

    ll

    With atom-atom interaction

    )exp()exp( ikr r

    f ikzSC  f :scattering amplitude

     

     

    0

    )2/sin( )(cos)12(

    l

    l l

    l

    SC kr

    lkr Pil

     

    :phase shift l

     

      

      

    )) 2

    (exp()2exp() 2

    (exp( 2

    )(cos )12()exp(

    0

     

     l

    krii l

    kri ikr

    P ili l

    l

    ll

    l

    00S

  • Atomic Scattering Theory

     

     0

    )(cos)12( l

    ll fPlf  k i

    ik

    i f ll

    l l

    )sin( )exp(

    2

    1)2exp(  

     

     with

    )(sin )12(4

    )12(4 2 2

    2

    lll k

    l fl 

     

     

    ik

    S fl

    2

    100 

    “At low temperature, only s-wave (l=0) scattering is important”

    k a ls

      0k

    saf 0 22

    00 44 saf  

  • Q. What is Scattering Length ?

    as

    as: positive

    R

    E

    Internuclear distance

    kR

    aRk

    kR

    kR R sSC

    ))(sin()sin( )( 0

     

     

     

    R

    E

    Internuclear distance

    as

    as: negative

    R

    )( 4

    21

    2

    int rr m

    a V s

      

      

    4 )(3

    2

    )( 4 is xxwxd

    m

    a U

    

  • Analytical Expression of Scattering Length

    6

    6)( R

    C RV  

      

      )

    8 tan(1

     ss aa

     

    

    0

    )(2 1

    r

    dRRVm 

     

      

     

     

     

    )4/5(

    )4/3(

    4

    2 )

    4 cos(

    2/1

    6

    C as

    m

    as

    0

    [Gribakin & Flambaum

    PRA, 48 546(1993)]

    ) 2

    1 (

    8  Dv

     

    R

    E

    Internuclear distance

    6

    6)( R

    C RV 

    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

    C rV

    m

      

     

    

    0

    )(2 1

    r

    dRRVm 

  • Q. What is Feshbach Resonance ? P

    o te

    n ti

    al

    -C6/R 3

    Two Free Atoms

    Molecular State

    Coupling between “Open Channel” and “Closed Channel”

    Control of as )1()( 0BB

    B aBa 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)

    E n

    er g

    y

    U (R

    )

    R

    S+P

    S+S

    Optical

    Excitation Two Atoms

    Molecular State

    2/2/

    2/2/ 00

    ii

    ii S

    S

    S

    

     

    2

    fVb lasS 

     :spontaneous decay rate

     :detuning from the PA resonance

    22

    2

    00 )2/2/(

    )1( 

    

    

     

    S

    S PA

    k S

    k K for 1/  S0

    inE

    refE 

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