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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

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

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

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

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