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Dynamics and thermodynamics of a driven quantum system Juzar Thingna 5 July, 2017 University of Luxembourg, Luxembourg
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  • Dynamics and thermodynamics of a drivenquantum system

    Juzar Thingna5 July, 2017

    University of Luxembourg, Luxembourg

  • Outline

    Mechanisms for relaxation Dense reservoir: Redeld regime Sparse reservoir: Landau-Zener regime

    Thermodynamics First law (Markovian and non-Markovian eects) Second law

    Summary

    1

  • Model

    Free fermionic Hamiltonian

    H (t ) = tdd +

    Ln=1

    ncncn +

    2

    Ln=1

    (dcn + H.c.

    ) HS (t ) + HR +V

    HS

    Finite number ofreservoir levels

    L

    Exactly solvable Decoupled initial condition tot (0) = (0) e

    (HRNR )

    ZR

    2

  • Redeld-like kinetics: Dense reservoir

    t

    1

    L

    Bare Reservoir

    3

  • Redeld-like kinetics: Dense reservoir

    t

    1

    L

    Bare Reservoir

    Bare System

    3

  • Redeld-like kinetics: Dense reservoir

    t

    1

    L

    Bare Reservoir

    Bare System

    Assumptions

    Dense reservoir and strong mixing Ln+1n 1 Weak system-reservoir coupling 1

    M. Esposito and P. Gaspard, Phys. Rev. E 68, 066112 (2003).

    3

  • Redeld-like kinetics: Dense reservoir

    Time-dependent Redeld master equation

    dtnn =i

    Liinnii

    dtnm =i, j

    Li jnmi j n ,m, i , j

    with

    L1122 = L1111 = T 1221 (t ) +T 1221 (t ) L1212 = [T 1221 (t ) +T

    2112 (t )

    ]+ it

    L2211 = L2222 = T 2112 (t ) +T 2112 (t ) L2121 = [T 2112 (t ) +T

    1221 (t )

    ] it

    Populations and coherence decouple

    H. Zhou, J. Thingna, P. Hnggi, J.-S. Wang, and B. Li, Scientic Reports 5, 14870 (2015).

    4

  • Redeld-like kinetics: Dense reservoir

    Non-Markovian Transition rates (Real + Imaginary parts)

    T 2112 (t ) =

    t0

    dt ei t 0 t d

    d

    2[1 f ( )]eit

    Bath Correlators

    T 1221 (t ) =

    t0

    dt ei t 0 t d

    d

    2 f ( )eit

    5

  • Redeld-like kinetics: Dense reservoir

    Non-Markovian Transition rates (Real + Imaginary parts)

    T 2112 (t ) =

    t0

    dt ei t 0 t d

    d

    2[1 f ( )]eit

    Bath Correlators

    T 1221 (t ) =

    t0

    dt ei t 0 t d

    d

    2 f ( )eit

    Occupied state population p (t ) = dd

    dtp (t ) = T+ (t )[1 p (t )] T (t )p (t )

    with the transition rates T + (t ) = 2Re[T 1221 (t )] andT (t ) = 2Re[T 2112 (t )].

    5

  • Redeld-like kinetics: Dense reservoir

    102

    0 10 20 30 40 50Time t

    0

    0.2

    0.4

    0.6

    0.8

    1

    p(t)

    0 20 40 60 80Time t

    0 20 40 60 80 100Time t

    0.6

    0.7

    0.8

    0.9

    1

    p(t)

    0 150 3000.6

    0.7

    0.8

    0.9

    1

    p(t)

    Strong mixing

    Weak mixing

    Autonomous

    Driven

    Autonomous

    6

  • Landau-Zener kinetics: Sparse reservoir

    n

    n+1

    L

    tn tn+1 tL

    Filled

    Empty

    Assumptions (Linear driving t = t )

    Sequential crossing: n = n+1 n

    Landau-Zener validity time:

    cn

    lz 1| |

    max(1,

    | |

    )7

  • Landau-Zener kinetics: Sparse reservoir

    Landau-Zener Markov Chain

    p (tn+1) =

    System Gain R p (tn ) +

    Reservoir Loss (1 R) f (n )

    Fermi

    Landau-Zener transition probability: R = exp[ 22

    ]

    8

  • Landau-Zener kinetics: Sparse reservoir

    Landau-Zener Markov Chain

    p (tn+1) =

    System Gain R p (tn ) +

    Reservoir Loss (1 R) f (n )

    Fermi

    Landau-Zener transition probability: R = exp[ 22

    ]

    Continuous time Landau-Zener master equationFast driving regime 2 = R 1

    2

    2 and coarse graining intime

    dtp (t ) = T+ ()[1 p (t )] T ()p (t )

    Markovian version of Redeld-like equation

    F. Barra and M. Esposito, Phys. Rev. E 93, 062118 (2016).

    8

  • Landau-Zener kinetics: Sparse reservoir

    102

    p(t) 0 20 40 60 80

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    0 20 40 60 800.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    0 0.2 0.4 0.6 0.80.992

    0.994

    0.996

    0.998

    1

    0 20 40 60 800

    0.2

    0.4

    0.6

    0.8

    1

    0 0.2 0.4 0.6 0.8Time t

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    0 0.2 0.4 0.6 0.80

    0.2

    0.4

    0.6

    0.8

    1

    L = 100, L = 100, = 0.1

    ; c lz ; c = lz ; c lz

    = ; c = lz = ; c lz ; c lz

    2 = 1 = 1

    2 = 1 = 100

    2 = 100 = 100

    2 = 104

    = 102 2 = 102

    = 1 2 = 102

    = 100

    9

  • Take home message # 1

    Driving a system can help it dissipate (via the Landau-Zenermechanism) compensating for the sparseness of the reservoir

  • Thermodynamics: Identities

    First lawdtU = Q + W

    Rate of change in Internal energydtU = Tr [dt (t ) {HS (t ) +V }] + Tr [ (t )dtHS (t )]

    Heat currentQ = IE INEnergy currentIE = Tr [dt (t )HR]Particle currentIN = Tr [dt (t )NR]

    Rate of workW = Wmech + IN

    Rate of mechanical workWmech = Tr [dtHS (t ) (t )]

    M. Esposito, K. Lindenberg and C. Van den Broeck, New J. Phys. 12, 013013 (2010).

    11

  • Thermodynamics: Redeld regime

    Energy currentIE = T + (t )[1 p (t )] T (t )p (t )Particle currentIN = T

    + (t )[1 p (t )] T (t )p (t )

    Rate of mechanical workWmech = tp (t )

    Modied (non-Markovian) transition rates

    T + (t ) = 2Re[ t

    0dt ei

    t 0 t d

    d

    2 f ( )eit

    ]

    T (t ) = 2Re[ t

    0dt ei

    t 0 t d

    d

    2[1 f ( )]eit

    ]

    J. Thingna, J. L. Garca-Palacios, and J.-S. Wang, Phys. Rev. B 85, 195452 (2012).

    12

  • Thermodynamics: Landau-Zener regime

    Markov chainEnergy changeE (tn+1) = n[p (tn+1) p (tn )]Particle changeN (tn+1) = p (tn+1) p (tn )

    Mechanical workWmech (tn+1) = (n+1 n )p (tn+1)

    13

  • Thermodynamics: Landau-Zener regime

    Markov chainEnergy changeE (tn+1) = n[p (tn+1) p (tn )]Particle changeN (tn+1) = p (tn+1) p (tn )

    Mechanical workWmech (tn+1) = (n+1 n )p (tn+1)

    Continuous time Landau-ZenerEnergy currentIE = t [T + ()[1 p (t )] T ()p (t )]Particle currentIN = T

    + ()[1 p (t )] T ()p (t )

    Rate of mechanical workWmech = tp (t )

    Identical to Traditional thermodynamics with Markovian QMEsH. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press.

    13

  • Thermodynamics

    0 20 40 60 80Time t

    54

    13

    22

    W

    41.5

    1

    3.5

    Q

    11.5

    4

    6.5

    W

    0.51

    2.5

    4Q

    0 20 40 60 80Time t

    0

    8

    16

    24

    12840

    311

    3

    1.50.50.5

    1.5

    0 0.1 0.2 0.3 0.4 0.5t

    63.511.54

    0 0.1 0.2 0.3 0.4 0.5t

    31.5

    01.53

    Non-Markovian

    Markovian

    Redeld Landau-Zener

    14

  • Take home message # 2

    Dynamics is insucient to capture thermodynamics

  • Periodic driving

    Second lawEntropy production

    iS = D[ (t ) | | (t ) eqR ] =Shanon entropy

    S Q

    Fast driving Slow driving

    Markovchain

    LZQMEdtiS < 0

    16

  • Periodic driving

    Second lawEntropy production

    iS = D[ (t ) | | (t ) eqR ] =Shanon entropy

    S Q

    Fast driving Slow driving

    Markovchain

    LZQMEdtiS < 0

    16

  • Multiple Reservoirs

    HamiltonianHS (t ) = td

    d ; H =n

    ncncn

    V =

    2

    n

    (dcn + H.c.

    ) (0) = d (0)

    e (HN )

    Z

    n

    n+1

    ...

    L

    tn tn+1 tL

    hc

    0 20 40 60 80Time t

    0

    0.2

    0.4

    0.6

    0.8

    1

    p(t)

    dtp (t ) =

    T + ()[1 p (t )] T ()p (t )

    TT =

    = 0.1 Linear response

    TT =

    = 0.5

    TT =

    = 0.9

    17

  • Summary

    Strong mixing between the system and reservoir = Redeldregime.

    Driving a system can relax the strong mixing condition =Landau-Zener regime.

    Thermodynamics requires more information than the dynamicswhich is most evident in the non-Markovian Redeld regime.

    The Landau-Zener regime exists even for multiple reservoirsallowing us to explore transport due to nite reservoirs.

    18

  • Acknowledgments

    Collaborators

    Assoc. Prof. Felipe Barra (University of Chile)

    Prof. Massimiliano Esposito (University of Luxembourg)

    Questions