Quantum Brownian Motion Seminar1

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    16/08/54 1,2,3 1

    Quantum Brownian Motion

    Samak Boonphan51641256

    By

    26 August 2009Department of Physics, Faculty of Science, Kasetsart University

    Physics Seminar 420597

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    Out line

    - Classical Brownian Motion

    - Quantum Brownian Motion

    - Density matrix for a single system- Time evolution of the system

    - Caldeira - Leggett Model

    - Influence functional- Reduced density matrix

    - Master equation- Conclusions

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    Classical Brownian Motion

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    How the environment affects on the Brownian particle?

    Consider a Brownian particle, which is the free particle

    interacting with the environment.

    )t(f)t()t(m K!

    System

    Environment

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    Consider a Brownian particle, which is the free particle,

    interacting with the environment.

    )t(f)t(x)t(xm K

    Friction force

    System

    Environment

    How the environment affects on the Brownian particle?

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    Consider a Brownian particle, which is the free particle,

    interacting with the environment.

    )t(f)t(x)t(xm K

    Random force

    System

    Environment

    How the environment affects on the Brownian particle?

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    Quantum Brownian Motion

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    Quantum Brownian Motion

    System

    Environment

    Interaction

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    |pat

    sall v r

    [ (t)]i

    ab et),,(J

    )L( ,t[ (t)]t

    !

    X

    t

    xa

    xb

    Feynman path Integration

    Feynman had showed that the path integral give as

    where

    is action

    a

    tHi

    b

    S[x(t)]i

    ab xexeD[x],0)xt;,K(xJJ

    ||

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    Density matrix for a single system

    |1//.

    .

    2222

    1111

    0

    0

    ]]

    ]]

    C

    C

    kkC ]]V ! kk

    )Atr(A !

    Define

    Expectation Value

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    Time evolution of the system

    Coordinate representation

    t)i

    exp(( )t)i

    exp(-(t)JJ

    !

    xe(0)exx(t)xt),x(x,tH

    itH

    i

    d!d|d

    JJ

    iii xxdx iii xxxd ddd

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    xexx(0)xxexxdxd

    xe(0)ext),x(x,

    tHi

    iiii

    tHi

    ii

    tHi

    tHi

    dddd

    dd

    JJ

    JJ

    Time evolution of the system

    Coordinate representation

    t)i

    exp(( )t)i

    exp(-(t)JJ

    V!

    )xK(x, i )x,x(K idd

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    Time evolution of the system

    Coordinate representation

    t)i

    exp(( )t)i

    exp(-(t)JJ

    V!

    t),x,x(K,0x,xt),xK(x,xdxd

    xexx(0)xxexxdxd

    xe(0)ext),x(x,

    iiiiii

    tHiiiii

    tHiii

    tHi

    tHi

    dddd!

    dddd!

    d!d

    JJ

    JJ

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    Time evolution of the system

    t),x,x(K,x,xt),xK(x,xdxdt),x(x, iiiiii dddd!d

    bxa

    x

    axd

    bxd

    x

    t

    t),xK(x, i

    t),x,x(K idd

    ,0x,x ii d

    t),x(x, d

    t

    x

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    Caldeira-Leggett Model

    A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149, 374(1983)

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    Modelof system + environment

    M

    m

    m

    m

    m

    m

    SystemorBrownian particle

    Environment

    Interaction

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    IBAt t LLLL !

    ! !

    !N

    1n

    N

    1nnn

    2n

    2n

    2n

    n222tot )xqC()qq(

    2

    m)xx(

    2

    ML

    Lagrangian of system +environment

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    ! q][x,[q][x]iexpD[x]D[q])q,x;q,K(x IBA

    x

    x

    q

    qaabb

    b

    a

    b

    a J

    Action and Propagatorof system+environment

    !! !

    t

    1

    1

    )xqC()qq()xx(dsq][x,

    X, q

    t

    System Environment+Interaction

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    Reduced Density matrix

    -Elimination environment coordinate out of density matrix

    q)(x, ? Atrq q(t)qdq

    q)(x, ? Atrq

    -So that after trace out of environment we have

    t),x'(x,r

    Reduce density matrix

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    Time evolution operator of density matrix

    ),x,x(, )x,xt;x,(xxdxdt),x,(x abAababrababr dddddd! 0

    - Time evolution of reduced density matrix

    ? A ? A ? Axx,FxSxSi

    ]expxD[D[x],0)x,xt;x,(xJb

    a

    b

    a

    x

    x

    x

    xababr d

    dd!dd

    d

    d J

    where

    Influence functional

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    Time evolution operator of density matrix

    t

    t

    bx

    x

    ax

    axd

    bxd

    x

    ]xA[x,]xS[S[x]]xA[x, dd!d

    ((

    7(!d

    t

    0 022121

    t

    0

    s

    022121

    1

    1

    )()()(dd

    )()()(dsds],[s

    ssssssi

    ssssxxA

    R

    LH

    2

    1

    2

    d

    d

    dd!ddt),x,(x

    ,)x,(xiir

    ii

    ]xA[x,i

    ]expxD[x]D[, )x,xt,x,(xJJ

    xx d!(

    2

    xx d!

    Effective action

    Influence effective action

    |

    |

    Relative path

    Center of relative path

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    )()cosI(d

    kernelndissipatio)s-(s

    0

    21

    21 ss !

    !

    g

    [[

    L

    Memory effect

    2s!

    !

    g

    1n

    0

    21

    s)cos2

    cot(Id

    kernelnoies)s-(s

    J

    R

    - memory kernel

    1s 2s Spectral density

    of environment !

    N

    1n nn

    2n

    n 2m

    C)(

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    Master Equation

    B.L Hu, Juan Pablo Paz, and Yuhong Zhang, Phys.Rev. D 45, 2843(1992)

    Subhasis Sinha and P.A. Sreeram ,Phys.Rev. E 79,051111(2009)

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    Master equationtime evolution of reduced density matrix

    ,0xx,(t,0)Jt),x'(x, rrr d!

    t),x(x,)x(x21)

    xx(

    2Mt),x(x,

    t r222

    2

    2

    2

    22

    r d ddxxxx

    !dxxJ

    Without Environment

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    Master equationtime evolution of reduced density matrix

    (t,0)Jr

    ,0xx,(t,0)Jt),x'(x, rrr d!

    dt,0)(tJr

    .)???.......(.........t),x(x,

    t

    r !d

    x

    x

    Environment

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    Derivation of the master equation

    step1

    ]~,~,,[]~,~[],[]',[ xxxxAxxAxxAxxAi dddd!

    path integral

    path

    mx

    step2

    dt

    ts)x(xx(s)x~ mfm

    !

    dt,0)(tJr

    (t,0)Jr

    xix fxmx x(s)

    t

    s

    dtt

    dts-t

    ]x[ ]x~[

    t

    ]x~[

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    Derivation of the master equation

    dt, )(tJr

    dt(t, )J-dt, )(tJ

    t(t, )J rrr !x

    x

    step3 step4 step5

    ,0)x,(x iird

    Multiply by

    And integrateover coordinate

    Subtract by

    and take limitdt 0(t,0)Jr

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    Frequencyshift

    Dissipation

    Diffusion andDecoherence

    The Master equation

    t),x(x,xx

    xx(t)f(t)t),x(x,xx(t) (t)iM

    t),x(x,xx

    )x(t)(xit),x(x,xx(t)M21

    t),x(x,xxM2

    1

    xx2Mt),x(x,

    ti

    rr2

    rr222

    r222

    2

    2

    2

    22

    r

    d

    dxx

    xx

    ddd+

    d

    dxx

    xx

    ddd

    d

    d

    dx

    x

    xx

    !dxx

    J

    J

    JJ

    H

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    Frequency shift

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    (frequency shift)

    p;

    ohmic(n=1, a) subohmic(n=1/2, b)

    Supraohmic(n=3, c)2000,3.00

    !0!K

    (t)(t) 222p !

    masterequation

    spectral densityA.J. Leggett[4]

    !

    2

    21

    exp

    2I

    0= cut-offrequency

    A.J. Leggett et al., Rev. Mod. Phys. 59, 1(1987)

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    Dissipation

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    dissipation,

    2000,3.00

    !0!K

    )(t+

    Dissipation constant,ohmic(n=1, a)subohmic(n=1/2, b)Supraohmic(n=3,c) 200030

    0!0! ,.K

    )(

    )()(ds(t)

    t