Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional...
Transcript of Stochastic Quantum Molecular Dynamics - TDDFTStochastic Quantum Molecular Dynamics: a functional...
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-linkedzincbacteriochlorin-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-linkedzincbacteriochlorin-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-linkedzincbacteriochlorin-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 environmentOne 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 environmentOne 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 environmentOne 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 environmentOne 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 environmentOne 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 environmentOne 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 environmentOne 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 : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
Hamiltonian of combined system
H = HS ⊗ IB + IS ⊗ HB + HSB
Unitary time evolution
i∂tΨ(t) = H(t)Ψ(t)d
dtρ(t) = −i
hH(t), ρ(t)
i
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 4 / 38
Open Quantum System
S +B : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
Hamiltonian of combined system
H = HS ⊗ IB + IS ⊗ HB + HSB
Unitary time evolution
i∂tΨ(t) = H(t)Ψ(t)d
dtρ(t) = −i
hH(t), ρ(t)
iHeiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 4 / 38
Reduced system dynamics
S +B : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
Tracing over bath degrees of freedom
ρS = TrB ρ
d
dtρS(t) = −iTrB
hH(t), ρ(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 : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
Tracing over bath degrees of freedom
ρS = TrB ρ
d
dtρS(t) = −iTrB
hH(t), ρ(t)
iI 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
P ρ = trB{ρ} ⊗ ρB ≡ ρS ⊗ ρBQρ = ρ− P ρ
Properties
P + Q = I
P 2 = P
Q2 = Q
P Q = QP = 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
H = HS + HB + αHSB
Liouville von Neumann equation in interaction picture
∂
∂tρ(t) = −iα
hHSB , ρ(t)
i≡ αLSB(t)ρ(t)
Apply Projection Operators
∂
∂tP ρ(t) = αP LSB(t)P ρ(t) + αP LSB(t)Qρ(t)
∂
∂tQρ(t) = αQLSB(t)P ρ(t) + αQLSB(t)Qρ(t)
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 7 / 38
Nakajima-Zwanzig projection operator technique
Formally solve for Qρ(t)
Qρ(t) = G(t, t0)Qρ(t0) + α
Z t
t0
dsG(t, s)QLSB(s)Qρ(s),
where
G(t, s) = T exp
»α
Z t
s
QLSB(s′)ds′–, G(s, s) = I
Nakajima-Zwanzig equation
∂
∂tP ρ(t) =αPLSB(t)P ρ(t) +
Source Termz }| {αPLSB(t)G(t, t0)Qρ(t0)
+ α2
Z t
t0
PLSB(t)G(t, s)QLSB(s)P ρ(s)ds| {z }Memory Term
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 8 / 38
Nakajima-Zwanzig projection operator technique
Typical approximations for the Nakajima-Zwanzig equationI Perturbation theory in α, e.g. Born approximation (up to α2).
I Linked cluster/cumulant expansions.
I Markov approximation. Z t
t0
... ρ(s)ds =⇒Z t
t0
... ρ(t)ds
Problem:
Approximations to the Nakajima-Zwanzig equation can lead to unphysical states(loss of positivity)
Similar problems:
Redfield equations, Caldeira-Legget equation, ...
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38
Nakajima-Zwanzig projection operator technique
Typical approximations for the Nakajima-Zwanzig equationI Perturbation theory in α, e.g. Born approximation (up to α2).
I Linked cluster/cumulant expansions.
I Markov approximation. Z t
t0
... ρ(s)ds =⇒Z t
t0
... ρ(t)ds
Problem:
Approximations to the Nakajima-Zwanzig equation can lead to unphysical states(loss of positivity)
Similar problems:
Redfield equations, Caldeira-Legget equation, ...
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38
Nakajima-Zwanzig projection operator technique
Typical approximations for the Nakajima-Zwanzig equationI Perturbation theory in α, e.g. Born approximation (up to α2).
I Linked cluster/cumulant expansions.
I Markov approximation. Z t
t0
... ρ(s)ds =⇒Z t
t0
... ρ(t)ds
Problem:
Approximations to the Nakajima-Zwanzig equation can lead to unphysical states(loss of positivity)
Similar problems:
Redfield equations, Caldeira-Legget equation, ...
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38
Nakajima-Zwanzig projection operator technique
Typical approximations for the Nakajima-Zwanzig equationI Perturbation theory in α, e.g. Born approximation (up to α2).
I Linked cluster/cumulant expansions.
I Markov approximation. Z t
t0
... ρ(s)ds =⇒Z t
t0
... ρ(t)ds
Problem:
Approximations to the Nakajima-Zwanzig equation can lead to unphysical states(loss of positivity)
Similar problems:
Redfield equations, Caldeira-Legget equation, ...
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 9 / 38
Lindblad theorem
Lindblad equation
d
dtρS(t) = LρS(t)
Most general form for the generator of a quantum dynamical semigroup
LρS(t) = −ihH, ρS(t)
i+Xk
γk
„VkρS(t)V †k −
1
2V †k VkρS(t)− 1
2ρS(t)V †k Vk
«On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976).
Semigroup properties:I Q(t) is completely positiveI Q(t) Q(s) = Q(t+ s)I Q(0) = I
Semigroup preserves:I Hermiticity (probabilities are real numbers)I Trace (conservation of norm)I Positivity (probabilities are positive)
Dynamical semigroup only for time-independent Hamiltonians⇒ Problem for TDDFT formulation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38
Lindblad theorem
Lindblad equation
d
dtρS(t) = LρS(t)
Most general form for the generator of a quantum dynamical semigroup
LρS(t) = −ihH, ρS(t)
i+Xk
γk
„VkρS(t)V †k −
1
2V †k VkρS(t)− 1
2ρS(t)V †k Vk
«On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976).
Semigroup properties:I Q(t) is completely positiveI Q(t) Q(s) = Q(t+ s)I Q(0) = I
Semigroup preserves:I Hermiticity (probabilities are real numbers)I Trace (conservation of norm)I Positivity (probabilities are positive)
Dynamical semigroup only for time-independent Hamiltonians⇒ Problem for TDDFT formulation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38
Lindblad theorem
Lindblad equation
d
dtρS(t) = LρS(t)
Most general form for the generator of a quantum dynamical semigroup
LρS(t) = −ihH, ρS(t)
i+Xk
γk
„VkρS(t)V †k −
1
2V †k VkρS(t)− 1
2ρS(t)V †k Vk
«On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976).
Semigroup properties:I Q(t) is completely positiveI Q(t) Q(s) = Q(t+ s)I Q(0) = I
Semigroup preserves:I Hermiticity (probabilities are real numbers)I Trace (conservation of norm)I Positivity (probabilities are positive)
Dynamical semigroup only for time-independent Hamiltonians⇒ Problem for TDDFT formulation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38
Lindblad theorem
Lindblad equation
d
dtρS(t) = LρS(t)
Most general form for the generator of a quantum dynamical semigroup
LρS(t) = −ihH, ρS(t)
i+Xk
γk
„VkρS(t)V †k −
1
2V †k VkρS(t)− 1
2ρS(t)V †k Vk
«On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, 119-130 (1976).
Semigroup properties:I Q(t) is completely positiveI Q(t) Q(s) = Q(t+ s)I Q(0) = I
Semigroup preserves:I Hermiticity (probabilities are real numbers)I Trace (conservation of norm)I Positivity (probabilities are positive)
Dynamical semigroup only for time-independent Hamiltonians⇒ Problem for TDDFT formulation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 10 / 38
Reduced system dynamics
S +B : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
=⇒ Use density operator ρS
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 11 / 38
Reduced system dynamics
S +B : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
=⇒ Use state vector ΨS
I No need to work with composite objects like density matrices
I Can trace out bath directly on the level of state vectors
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 12 / 38
Reduced system dynamics
S +B : HS ⊗HB ,Ψ, ρ
S : HS ,ΨS , ρS
System
B : HB ,ΨB , ρB
Environment
=⇒ Use state vector ΨS
I No need to work with composite objects like density matrices
I Can trace out bath directly on the level of state vectors
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 12 / 38
Feshbach Projection-Operator Method
H = HS + HB + αHSB , HBχn(xB) = εnχn(xB)
Expand total wavefunction in arbitrary complete and orthonormal basis of the bath
Ψ(xS , xB ; t) =Xn
φn(xS ; t)χn(xB)
Projection Operators
P := IS ⊗ |χk 〉〈χk | Q := IS ⊗Xk 6=n
|χk 〉〈χk |
Apply to TDSE
i∂tPΨ(t) = P HPΨ(t) + P HQΨ(t)
i∂tQΨ(t) = QHQΨ(t) + QHPΨ(t)
P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 13 / 38
Feshbach Projection-Operator Method
H = HS + HB + αHSB , HBχn(xB) = εnχn(xB)
Expand total wavefunction in arbitrary complete and orthonormal basis of the bath
Ψ(xS , xB ; t) =Xn
φn(xS ; t)χn(xB)
Projection Operators
P := IS ⊗ |χk 〉〈χk | Q := IS ⊗Xk 6=n
|χk 〉〈χk |
Apply to TDSE
i∂tPΨ(t) = P HPΨ(t) + P HQΨ(t)
i∂tQΨ(t) = QHQΨ(t) + QHPΨ(t)
P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 13 / 38
Feshbach Projection-Operator Method
H = HS + HB + αHSB , HBχn(xB) = εnχn(xB)
Expand total wavefunction in arbitrary complete and orthonormal basis of the bath
Ψ(xS , xB ; t) =Xn
φn(xS ; t)χn(xB)
Projection Operators
P := IS ⊗ |χk 〉〈χk | Q := IS ⊗Xk 6=n
|χk 〉〈χk |
Apply to TDSE
i∂tPΨ(t) = P HPΨ(t) + P HQΨ(t)
i∂tQΨ(t) = QHQΨ(t) + QHPΨ(t)
P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 13 / 38
Feshbach Projection-Operator Method
Effective equation for P Ψ (still fully coherent)
i∂tPΨ(t) =P HP PΨ(t) +
Source Termz }| {P HQe−iQHQtQΨ(0)
−iZ t
0
dτP HQeiQHQ(t−τ)QHP PΨ(τ)| {z }Memory Term
=⇒ Formal similarity to quantum transport formulation of Kurth and Stefanucci et. al.
Non-Markovian Stochastic Schrodinger equation
I perturbative expansion to second order in αHSBI random phase approximation, dense bath spectrum, bath in statistical equilibrium
i∂tψ(t) =HSψ(t) + αXα
ηα(t)Vαψ(t)
− iα2
Z t
0
dτXαβ
Cαβ(t− τ)| {z }Bath correlation functions
V †αe−iHS(t−τ)Vβψ(τ) +O(α3)
P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 14 / 38
Feshbach Projection-Operator Method
Effective equation for P Ψ (still fully coherent)
i∂tPΨ(t) =P HP PΨ(t) +
Source Termz }| {P HQe−iQHQtQΨ(0)
−iZ t
0
dτP HQeiQHQ(t−τ)QHP PΨ(τ)| {z }Memory Term
=⇒ Formal similarity to quantum transport formulation of Kurth and Stefanucci et. al.
Non-Markovian Stochastic Schrodinger equation
I perturbative expansion to second order in αHSBI random phase approximation, dense bath spectrum, bath in statistical equilibrium
i∂tψ(t) =HSψ(t) + αXα
ηα(t)Vαψ(t)
− iα2
Z t
0
dτXαβ
Cαβ(t− τ)| {z }Bath correlation functions
V †αe−iHS(t−τ)Vβψ(τ) +O(α3)
P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 14 / 38
Feshbach Projection-Operator Method
Effective equation for P Ψ (still fully coherent)
i∂tPΨ(t) =P HP PΨ(t) +
Source Termz }| {P HQe−iQHQtQΨ(0)
−iZ t
0
dτP HQeiQHQ(t−τ)QHP PΨ(τ)| {z }Memory Term
=⇒ Formal similarity to quantum transport formulation of Kurth and Stefanucci et. al.
Non-Markovian Stochastic Schrodinger equation
I perturbative expansion to second order in αHSBI random phase approximation, dense bath spectrum, bath in statistical equilibrium
i∂tψ(t) =HSψ(t) + αXα
ηα(t)Vαψ(t)
− iα2
Z t
0
dτXαβ
Cαβ(t− τ)| {z }Bath correlation functions
V †αe−iHS(t−τ)Vβψ(τ) +O(α3)
P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 14 / 38
Markovian Stochastic Schrodinger equation
δ-correlated bath
Cαβ(t− τ) = Dαβδ(t− τ)
Stochastic Schrodinger equation in Born-Markov approximation
i∂tψ(t) =HSψ(t) + αXα
ηα(t)Vαψ(t)
− iα2Xαβ
DαβV†α Vβψ(t) +O(α3)
Statistical average:
ρS(t) =|ψ(t) 〉〈ψ(t) |〈ψ(t) |ψ(t) 〉
I Unravelling of Lindblad equation for static (linear) HamiltoniansI Does not rely on semigroup propertyI Valid for time-dependent HamiltoniansI Gives always physical statesI Sound starting point to formulate stochastic TDDFT
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 15 / 38
TDDFT for Open Quantum Systems
I Approach in terms of density matricesK. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, 146803 (2005).
I Density matrix evolution may not obey positivity during time-evolution
I Scales as O(N2) due to usage of density matrix
I Approach in terms of stochastic Schrodinger equationsM. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).
I Positivity is guaranteed by construction
I Scales as O(N) since only state vectors are used
I Comparison to classical stochastic systems:
Fokker-Planck equation ⇐⇒ Langevin equation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 16 / 38
TDDFT for Open Quantum Systems
I Approach in terms of density matricesK. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, 146803 (2005).
I Density matrix evolution may not obey positivity during time-evolution
I Scales as O(N2) due to usage of density matrix
I Approach in terms of stochastic Schrodinger equationsM. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).
I Positivity is guaranteed by construction
I Scales as O(N) since only state vectors are used
I Comparison to classical stochastic systems:
Fokker-Planck equation ⇐⇒ Langevin equation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 16 / 38
TDDFT for Open Quantum Systems
I Approach in terms of density matricesK. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, 146803 (2005).
I Density matrix evolution may not obey positivity during time-evolution
I Scales as O(N2) due to usage of density matrix
I Approach in terms of stochastic Schrodinger equationsM. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).
I Positivity is guaranteed by construction
I Scales as O(N) since only state vectors are used
I Comparison to classical stochastic systems:
Fokker-Planck equation ⇐⇒ Langevin equation
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 16 / 38
Stochastic Time-Dependent Current-Density-Functional Theory
Can prove: For fixed bath operators V and initial state
j(r, t)1:1←→ A(r, t)
M. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).
Mapping of fully interacting stochastic TDSE to stochastic TDKS equations
i∂tψj(r, t) =
26664HKS(t)− 1
2iV †V| {z }
damping
+ l(t)V| {z }fluctuations
37775ψj(r, t)l(t) : stochastic process
l(t) = 0, l(t)l(t′) = δ(t− t′)
Assumes
I Factorization at initial time: ψ(t0) = ψS(t0)× ψB(t0)
I Markovian approximation: no bath memory
I Weak coupling to the bath (second order in HSB)
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 17 / 38
Stochastic Time-Dependent Current-Density-Functional Theory
Can prove: For fixed bath operators V and initial state
j(r, t)1:1←→ A(r, t)
M. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).
Mapping of fully interacting stochastic TDSE to stochastic TDKS equations
i∂tψj(r, t) =
26664HKS(t)− 1
2iV †V| {z }
damping
+ l(t)V| {z }fluctuations
37775ψj(r, t)
l(t) : stochastic process
l(t) = 0, l(t)l(t′) = δ(t− t′)
Assumes
I Factorization at initial time: ψ(t0) = ψS(t0)× ψB(t0)
I Markovian approximation: no bath memory
I Weak coupling to the bath (second order in HSB)
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 17 / 38
Stochastic Time-Dependent Current-Density-Functional Theory
Can prove: For fixed bath operators V and initial state
j(r, t)1:1←→ A(r, t)
M. Di Ventra and R. D’Agosta, Phys. Rev. Lett. 98, 226403 (2007).
Mapping of fully interacting stochastic TDSE to stochastic TDKS equations
i∂tψj(r, t) =
26664HKS(t)− 1
2iV †V| {z }
damping
+ l(t)V| {z }fluctuations
37775ψj(r, t)l(t) : stochastic process
l(t) = 0, l(t)l(t′) = δ(t− t′)
Assumes
I Factorization at initial time: ψ(t0) = ψS(t0)× ψB(t0)
I Markovian approximation: no bath memory
I Weak coupling to the bath (second order in HSB)
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 17 / 38
Choice of bath operators: a simple model
Order-N scheme and bath operators which obey Fermi statistics
V jkk′(r) = δkj(1− δkk′)pγ(r)fD(εk) |ψj(r) 〉〈ψk′(r) |
Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev. B 78, 054302 (2008).
Fermi-Dirac distribution
fD(εk) =
»1 + exp
„εk − µkBT
«–−1
Local relaxation ratesγkk′(r) = |ψk(r) 〉γ0〈ψk′(r) |
I Operators ensure that Fermi statistics is obeyed.
I If a steady state is reached, it will be a thermal state.
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 18 / 38
Choice of bath operators: a simple model
Order-N scheme and bath operators which obey Fermi statistics
V jkk′(r) = δkj(1− δkk′)pγ(r)fD(εk) |ψj(r) 〉〈ψk′(r) |
Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev. B 78, 054302 (2008).
Fermi-Dirac distribution
fD(εk) =
»1 + exp
„εk − µkBT
«–−1
Local relaxation ratesγkk′(r) = |ψk(r) 〉γ0〈ψk′(r) |
I Operators ensure that Fermi statistics is obeyed.
I If a steady state is reached, it will be a thermal state.
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 18 / 38
Quantum jump algorithm
1) Draw uniform random number ηj ∈ [0, 1]
2) Propagate auxilary wave function under non-Hermitian Hamiltonian
i∂tΦ = H0Φ− iV †V Φ
3) Propagate system wave function under norm-conserving Hamiltonian
i∂tΨ = H0Ψ− iV †VΨ + i||VΨ||2Ψ
4) If norm of auxilary wave function drops below ηj , act with bath operator
||Φ(tj)|| ≤ ηj , Ψ(tj) = VΨ(tj), Φ(tj) = Ψ(tj)
5) Go to step 1)
cf. H.P. Breuer and F. Petruccione, Theory of Open Quantum Systems, Oxford University Press
=⇒ Leads to piecewise deterministic evolution
Average over stochastic realizations:
ρ = |Ψj 〉〈Ψj | guarantees always physical state!
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 19 / 38
Practical Implementation
Full-potential, all-electron code with local orbitals
http://www.fhi-berlin.mpg.de/aims/
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 20 / 38
Practical Implementation
Expansion of TD orbitals in non-orthorgonal atom centered basis functions φk
ψj(t) =Xk
cjk(t)φk
Overlap matrixSjk = 〈φj |φk 〉
TDKS equations
iS∂
∂tcj(t) = HKS(t) cj(t)
Exponential midpoint approximation for short-time propagator
U(t+ ∆t, t) = exph−iS−1HKS(t+ ∆t/2)∆t+O(∆t3)
i= S−1/2 exp
h−iS−1/2HKS(t+ ∆t/2)∆tS−1/2 +O(∆t3)
iS1/2
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 21 / 38
Application
(1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 22 / 38
Application: zincbacteriochlorin-bacteriochlorin complex
HOMO
LUMO
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 23 / 38
Free Propagation of HOMO
Free propagation of HOMO:
ψHOMO(t) = ψGSHOMO × e−iεHOMO t
Re{ψHOMO(t)} = ψGSHOMO × cos(εHOMOt)
(for real-valued orbitals)
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 24 / 38
Linear combination of HOMO and LUMO
Initial state
ψTDKSHOMO(t = 0) =
1√2
hψ
GSHOMO + e−i
π2 ψ
GSLUMO
iClosed quantum system
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 25 / 38
Linear combination of HOMO and LUMO
Initial state
ψTDKSHOMO(t = 0) =
1√2
hψ
GSHOMO + e−i
π2 ψ
GSLUMO
iIncluding coupling to environment
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 26 / 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-linkedzincbacteriochlorin-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 27 / 38
Molecular Dynamics for Open Systems
Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD:
I Electronic degrees of freedom are treated with closed system approach
I Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces)
However:
Electrons are the first to experience energy transfer to a bath
Nuclei feel bath directly but also through electron-ion interaction=⇒ different forces on nuclei
Need open quantum theory for both electrons and nuclei
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38
Molecular Dynamics for Open Systems
Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD:
I Electronic degrees of freedom are treated with closed system approach
I Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces)
However:
Electrons are the first to experience energy transfer to a bath
Nuclei feel bath directly but also through electron-ion interaction=⇒ different forces on nuclei
Need open quantum theory for both electrons and nuclei
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38
Molecular Dynamics for Open Systems
Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD:
I Electronic degrees of freedom are treated with closed system approach
I Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces)
However:
Electrons are the first to experience energy transfer to a bath
Nuclei feel bath directly but also through electron-ion interaction=⇒ different forces on nuclei
Need open quantum theory for both electrons and nuclei
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38
Molecular Dynamics for Open Systems
Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD:
I Electronic degrees of freedom are treated with closed system approach
I Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces)
However:
Electrons are the first to experience energy transfer to a bath
Nuclei feel bath directly but also through electron-ion interaction=⇒ different forces on nuclei
Need open quantum theory for both electrons and nuclei
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38
Molecular Dynamics for Open Systems
Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD:
I Electronic degrees of freedom are treated with closed system approach
I Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces)
However:
Electrons are the first to experience energy transfer to a bath
Nuclei feel bath directly but also through electron-ion interaction=⇒ different forces on nuclei
Need open quantum theory for both electrons and nuclei
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 28 / 38
Stochastic Quantum Molecular Dynamics
Extension of stochastic TDCDFT to include nuclear degrees of freedom
i∂tΨ = H(t)Ψ− 1
2iV †VΨ + l(t)VΨ
H(t) =Te(r, t) + Wee(r) + Uext,e(r, t)+
Tn(R, t) + Wnn(R) + Uext,n(R, t)+
Wen(r,R)
Total current〈 J(x, t) 〉 = 〈 j(r, t) 〉+ 〈 J(R, t) 〉, x = (r,R)
For given initial state Ψ(x, t = 0) and bath operators Vα(x, t)
〈 J(x, t) 〉 ←→ A(x, t)
Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, 212303 (2009).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 29 / 38
Stochastic Quantum Molecular Dynamics: practical scheme
Extension of stochastic TDCDFT to include nuclear degrees of freedom
i∂tΨ(x, t) = H(t)Ψ(x, t)− 1
2iV †VΨ(x, t) + l(t)VΨ(x, t)
In practice: resort as approximation to classical nuclei
Bath operators
V jkk′(r,R(t); t) = δkj(1− δkk′)pγ(r,R(t); t)fD(εk) |ψj(r,R(t); t) 〉〈ψk′(r,R(t); t) |
Ehrenfest forces as approximation for classical nuclei
MαRα(t) = −Z
Ψ∗∇RαHeΨdr
Note:
Wavefunctions are stochastic =⇒ stochastic force on the nuclei
Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, 212303 (2009).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 30 / 38
Vibronic excitation of Beryllium dimer, moving nuclei
closed quantum systemstretched initial condition
open quantum systemrelaxation rate τ = 300 fs
Maxwell-Boltzmann velocity distribution atjumps
f(~v) =“ m
2πkT
”3/2exp
„−m~v2
2kT
«
Average over 15 stochastic realizations
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 31 / 38
Vibronic excitation of Beryllium dimer, moving nuclei
closed quantum systemstretched initial condition
open quantum systemrelaxation rate τ = 300 fs
Maxwell-Boltzmann velocity distribution atjumps
f(~v) =“ m
2πkT
”3/2exp
„−m~v2
2kT
«
Average over 15 stochastic realizations
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 31 / 38
Dual-Fluoresence in 4-(N,N-Dimethylamino)benzonitrile
polar solventacetonitrile
non-polar solventn-hexane
ε(−45K) = 50.2ε(+75K) = 30.3
DMABN
Experiment: J. Phys. Chem. A 2006, 110, 2955-2969
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 32 / 38
Stochastic Quantum MD simulation for4-(N,N-Dimethylamino)benzonitrile
Electron Localization Function
Rotated dimethyl group asinitial condition
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 33 / 38
Stochastic Quantum MD simulation for4-(N,N-Dimethylamino)benzonitrile
CH3CH
CH3CH
C NN
δ
0
20
40
coun
ts
t=0 fsT=0 K
t=137 fsT=0K
t=273 fsT=0K
0102030
t=550 fsT=0K
-25 0 25angle [deg]
0
20
40
coun
ts
t=46 fsT=300K
-25 0 25angle [deg]
t=137 fsT=300K
-25 0 25angle [deg]
t=273 fsT=300K
-25 0 25angle [deg]
0
10
20t=550 fsT=300K
0 100 200 300 400 500 600 time [fs]
-30-20-10
0102030
angl
e [d
eg]
closed system @ 0Kopen system @ 0Kopen system @ 300K
Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, 212303 (2009).
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 34 / 38
Conclusions
I SQMD allows to describe ab-initio relaxation and dephasing of electronic andnuclear degrees of freedom
I In constrast to standard QMD, stochastic QMD allows also coupling of electrons toa heat bath
I Nuclei can feel the heat bath via electron-nuclear coupling
Within reach of Stochastic Quantum Molecular Dynamics
I Themopower/Thermoelectric effects
I Excited state relaxation in solution
I Decoherence and dephasing in pump-probe experiments
I Driven quantum phase transitions
I ....
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38
Conclusions
I SQMD allows to describe ab-initio relaxation and dephasing of electronic andnuclear degrees of freedom
I In constrast to standard QMD, stochastic QMD allows also coupling of electrons toa heat bath
I Nuclei can feel the heat bath via electron-nuclear coupling
Within reach of Stochastic Quantum Molecular Dynamics
I Themopower/Thermoelectric effects
I Excited state relaxation in solution
I Decoherence and dephasing in pump-probe experiments
I Driven quantum phase transitions
I ....
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38
Conclusions
I SQMD allows to describe ab-initio relaxation and dephasing of electronic andnuclear degrees of freedom
I In constrast to standard QMD, stochastic QMD allows also coupling of electrons toa heat bath
I Nuclei can feel the heat bath via electron-nuclear coupling
Within reach of Stochastic Quantum Molecular Dynamics
I Themopower/Thermoelectric effects
I Excited state relaxation in solution
I Decoherence and dephasing in pump-probe experiments
I Driven quantum phase transitions
I ....
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38
Conclusions
I SQMD allows to describe ab-initio relaxation and dephasing of electronic andnuclear degrees of freedom
I In constrast to standard QMD, stochastic QMD allows also coupling of electrons toa heat bath
I Nuclei can feel the heat bath via electron-nuclear coupling
Within reach of Stochastic Quantum Molecular Dynamics
I Themopower/Thermoelectric effects
I Excited state relaxation in solution
I Decoherence and dephasing in pump-probe experiments
I Driven quantum phase transitions
I ....
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 35 / 38
Future work
I Relaxation times from system-bath interaction Hamiltonian
I Non-markovian dynamics: bath-correlation functions which are not delta correlated
I Functionals: Dependence of XC-functional on bath operators
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 36 / 38
Future work
I Relaxation times from system-bath interaction Hamiltonian
I Non-markovian dynamics: bath-correlation functions which are not delta correlated
I Functionals: Dependence of XC-functional on bath operators
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 36 / 38
Future work
I Relaxation times from system-bath interaction Hamiltonian
I Non-markovian dynamics: bath-correlation functions which are not delta correlated
I Functionals: Dependence of XC-functional on bath operators
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 36 / 38
Thanks!
Massimiliano Di VentraYonatan DubiMatt KremsJim Wilson
Roberto D’Agosta
Thank you for your attention!
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 37 / 38
Positivity
A density matrix which obeys positivity is a non-negative operator in Hilbert space
〈Ψ | ρ |Ψ 〉 ≥ 0, ∀ |Ψ 〉
This guarantees that the expectation value of a positive operator A
〈 A 〉(t) =Xi
pi〈Ψ(t) | A |Ψ(t) 〉 = Tr{ρ(t)A} ≥ 0
and the variance of any operator B
〈 B2 〉(t)− 〈 B 〉2(t) ≥ 0
Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, 2010 38 / 38