Stochastic Quantum Molecular Dynamics - TDDFT Stochastic Quantum Molecular Dynamics: a functional theory

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  • Stochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei

    dynamically coupled to an environment

    Heiko Appel

    University of California, San Diego

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 1 / 38

  • Outline

    Open Quantum Systems

    I Different approaches to deal with decoherence and dissipation

    I Stochastic current density-functional theory

    I Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex

    Stochastic Quantum Molecular Dynamics: Theory and Applications

    I Including nuclear motion: Stochastic Quantum Molecular Dynamics

    I Application: Stochastic quantum MD of 4-(N,N-Dimethylamino)benzonitrile

    Outlook: future prospects of SQMD

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 2 / 38

  • Outline

    Open Quantum Systems

    I Different approaches to deal with decoherence and dissipation

    I Stochastic current density-functional theory

    I Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex

    Stochastic Quantum Molecular Dynamics: Theory and Applications

    I Including nuclear motion: Stochastic Quantum Molecular Dynamics

    I Application: Stochastic quantum MD of 4-(N,N-Dimethylamino)benzonitrile

    Outlook: future prospects of SQMD

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 2 / 38

  • Outline

    Open Quantum Systems

    I Different approaches to deal with decoherence and dissipation

    I Stochastic current density-functional theory

    I Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex

    Stochastic Quantum Molecular Dynamics: Theory and Applications

    I Including nuclear motion: Stochastic Quantum Molecular Dynamics

    I Application: Stochastic quantum MD of 4-(N,N-Dimethylamino)benzonitrile

    Outlook: future prospects of SQMD

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 2 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Why Open Quantum Systems?

    General aspects:

    I Cannot have perfectly isolated quantum systems

    I Dissipation and Decoherence

    I Every measurement implies contact with an environment One actually needs to bring a system into contact with an environment (i.e. measurement apparatus), in order to perform a measurement → environment as continuos measurement of the system.

    Research fields:

    I Quantum computing/Quantum information theory

    I (time-resolved) transport and optics

    I Driven quantum phase transitions

    I Quantum measurement

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 3 / 38

  • Open Quantum System

    S +B : ĤS ⊗HB ,Ψ, ρ̂

    S : ĤS ,ΨS , ρ̂S

    System

    B : ĤB ,ΨB , ρ̂B

    Environment

    Hamiltonian of combined system

    Ĥ = ĤS ⊗ ÎB + ÎS ⊗ ĤB + ĤSB

    Unitary time evolution

    i∂tΨ(t) = Ĥ(t)Ψ(t) d

    dt ρ̂(t) = −i

    h Ĥ(t), ρ̂(t)

    i

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 4 / 38

  • Open Quantum System

    S +B : ĤS ⊗HB ,Ψ, ρ̂

    S : ĤS ,ΨS , ρ̂S

    System

    B : ĤB ,ΨB , ρ̂B

    Environment

    Hamiltonian of combined system

    Ĥ = ĤS ⊗ ÎB + ÎS ⊗ ĤB + ĤSB

    Unitary time evolution

    i∂tΨ(t) = Ĥ(t)Ψ(t) d

    dt ρ̂(t) = −i

    h Ĥ(t), ρ̂(t)

    i Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 4 / 38

  • Reduced system dynamics

    S +B : ĤS ⊗HB ,Ψ, ρ̂

    S : ĤS ,ΨS , ρ̂S

    System

    B : ĤB ,ΨB , ρ̂B

    Environment

    Tracing over bath degrees of freedom

    ρ̂S = TrB ρ̂

    d

    dt ρ̂S(t) = −iTrB

    h Ĥ(t), ρ̂(t)

    i

    I ρS(t) represents in general no pure state.

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 5 / 38

  • Reduced system dynamics

    S +B : ĤS ⊗HB ,Ψ, ρ̂

    S : ĤS ,ΨS , ρ̂S

    System

    B : ĤB ,ΨB , ρ̂B

    Environment

    Tracing over bath degrees of freedom

    ρ̂S = TrB ρ̂

    d

    dt ρ̂S(t) = −iTrB

    h Ĥ(t), ρ̂(t)

    i I ρS(t) represents in general no pure state.

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 5 / 38

  • Nakajima-Zwanzig projection operator technique

    Projection Operators

    P̂ ρ = trB{ρ} ⊗ ρB ≡ ρS ⊗ ρB Q̂ρ = ρ− P̂ ρ

    Properties

    P̂ + Q̂ = I

    P̂ 2 = P̂

    Q̂2 = Q̂

    P̂ Q̂ = Q̂P̂ = 0

    S. Nakajima, Progr. Theor. Phys., 20, 948-959 (1958). R. Zwanzig, J. Chem. Phys., 33, 1338-1341 (1960).

    Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 6 / 38

  • Nakajima-Zwanzig projection operator technique

    Total Hamiltonian

    Ĥ = ĤS + ĤB + αĤSB

    Liouville von Neumann equation in interaction picture