Stochastic Quantum Molecular Dynamics: a functional theory for electrons and nuclei dynamically coupled to an environment.

Transcription

1 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, / 38

2 Outline Open Quantum Systems Different approaches to deal with decoherence and dissipation Stochastic current density-functional theory Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

3 Outline Open Quantum Systems Different approaches to deal with decoherence and dissipation Stochastic current density-functional theory Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex Stochastic Quantum Molecular Dynamics: Theory and Applications Including nuclear motion: Stochastic Quantum Molecular Dynamics Application: Stochastic quantum MD of 4-(N,N-Dimethylamino)benzonitrile Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

4 Outline Open Quantum Systems Different approaches to deal with decoherence and dissipation Stochastic current density-functional theory Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex Stochastic Quantum Molecular Dynamics: Theory and Applications Including nuclear motion: Stochastic Quantum Molecular Dynamics 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, / 38

5 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

6 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Dissipation and Decoherence Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

7 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Dissipation and Decoherence 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. Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

8 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Dissipation and Decoherence 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: Quantum computing/quantum information theory Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

9 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Dissipation and Decoherence 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: Quantum computing/quantum information theory (time-resolved) transport and optics Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

10 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Dissipation and Decoherence 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: Quantum computing/quantum information theory (time-resolved) transport and optics Driven quantum phase transitions Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

11 Why Open Quantum Systems? General aspects: Cannot have perfectly isolated quantum systems Dissipation and Decoherence 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: Quantum computing/quantum information theory (time-resolved) transport and optics Driven quantum phase transitions Quantum measurement Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

12 Open Quantum System S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB Hamiltonian of combined system Ĥ = ĤS ÎB + ÎS ĤB + ĤSB Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

13 Open Quantum System S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB Hamiltonian of combined system Ĥ = ĤS ÎB + ÎS ĤB + ĤSB Unitary time evolution i tψ(t) = Ĥ(t)Ψ(t) d i hĥ(t), dt ˆρ(t) = i ˆρ(t) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

14 Reduced system dynamics S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB Tracing over bath degrees of freedom ˆρ S = Tr B ˆρ d i hĥ(t), dt ˆρS(t) = itrb ˆρ(t) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

15 Reduced system dynamics S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB Tracing over bath degrees of freedom ˆρ S = Tr B ˆρ d i hĥ(t), dt ˆρS(t) = itrb ˆρ(t) ρ S(t) represents in general no pure state. Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

16 Nakajima-Zwanzig projection operator technique Projection Operators Properties ˆP ρ = tr B{ρ} ρ B ρ S ρ B ˆQρ = ρ ˆP ρ ˆP + ˆQ = I ˆP 2 = ˆP ˆQ 2 = ˆQ ˆP ˆQ = ˆQ ˆP = 0 S. Nakajima, Progr. Theor. Phys., 20, (1958). R. Zwanzig, J. Chem. Phys., 33, (1960). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

17 Nakajima-Zwanzig projection operator technique Total Hamiltonian Ĥ = ĤS + ĤB + αĥsb Liouville von Neumann equation in interaction picture i hĥsb, t ˆρ(t) = iα ˆρ(t) α ˆL SB(t)ˆρ(t) Apply Projection Operators t ˆP ˆρ(t) = α ˆP ˆL SB(t) ˆP ˆρ(t) + α ˆP ˆL SB(t) ˆQˆρ(t) t ˆQˆρ(t) = α ˆQ ˆL SB(t) ˆP ˆρ(t) + α ˆQ ˆL SB(t) ˆQˆρ(t) S. Nakajima, Progr. Theor. Phys., 20, (1958). R. Zwanzig, J. Chem. Phys., 33, (1960). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

18 Formally solve for ˆQˆρ(t) where Nakajima-Zwanzig projection operator technique Nakajima-Zwanzig equation Z t ˆQˆρ(t) = G(t, t 0) ˆQˆρ(t 0) + α dsg(t, s) ˆQ ˆL SB(s) ˆQˆρ(s), t 0» Z t G(t, s) = T exp α ˆQL SB(s )ds, G(s, s) = I s Source Term z } { t ˆP ˆρ(t) =α ˆP L SB(t) ˆP ˆρ(t) + α ˆP L SB(t)G(t, t 0) ˆQˆρ(t 0) Z t + α 2 ˆP LSB(t)G(t, s) ˆQL SB(s) ˆP ˆρ(s)ds t 0 {z } Memory Term S. Nakajima, Progr. Theor. Phys., 20, (1958). R. Zwanzig, J. Chem. Phys., 33, (1960). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

19 Nakajima-Zwanzig projection operator technique Typical approximations for the Nakajima-Zwanzig equation Perturbation theory in α, e.g. Born approximation (up to α 2 ). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

20 Nakajima-Zwanzig projection operator technique Typical approximations for the Nakajima-Zwanzig equation Perturbation theory in α, e.g. Born approximation (up to α 2 ). Linked cluster/cumulant expansions. Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

21 Nakajima-Zwanzig projection operator technique Typical approximations for the Nakajima-Zwanzig equation Perturbation theory in α, e.g. Born approximation (up to α 2 ). Linked cluster/cumulant expansions. Markov approximation. Z t t 0... ˆρ(s)ds = Z t t 0... ˆρ(t)ds Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

22 Nakajima-Zwanzig projection operator technique Typical approximations for the Nakajima-Zwanzig equation Perturbation theory in α, e.g. Born approximation (up to α 2 ). Linked cluster/cumulant expansions. Markov approximation. Z t t 0... ˆρ(s)ds = Z t t 0... ˆρ(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, / 38

23 Lindblad theorem Lindblad equation d ˆρS(t) = LˆρS(t) dt Most general form for the generator of a quantum dynamical semigroup i Lˆρ S(t) = i hĥ, ˆρS(t) + X γ k ˆV k ˆρ S(t) ˆV k 1 2 ˆV ˆV k k ρ S(t) 1 «ˆρS(t) ˆV k 2 ˆV k k On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, (1976). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

24 Lindblad theorem Lindblad equation d ˆρS(t) = LˆρS(t) dt Most general form for the generator of a quantum dynamical semigroup i Lˆρ S(t) = i hĥ, ˆρS(t) + X γ k ˆV k ˆρ S(t) ˆV k 1 2 ˆV ˆV k k ρ S(t) 1 «ˆρS(t) ˆV k 2 ˆV k k On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, (1976). Semigroup properties: ˆQ(t) is completely positive ˆQ(t) ˆQ(s) = ˆQ(t + s) ˆQ(0) = Î Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

25 Lindblad theorem Lindblad equation d ˆρS(t) = LˆρS(t) dt Most general form for the generator of a quantum dynamical semigroup i Lˆρ S(t) = i hĥ, ˆρS(t) + X γ k ˆV k ˆρ S(t) ˆV k 1 2 ˆV ˆV k k ρ S(t) 1 «ˆρS(t) ˆV k 2 ˆV k k On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, (1976). Semigroup properties: ˆQ(t) is completely positive ˆQ(t) ˆQ(s) = ˆQ(t + s) ˆQ(0) = Î Semigroup preserves: Hermiticity (probabilities are real numbers) Trace (conservation of norm) Positivity (probabilities are positive) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

26 Lindblad theorem Lindblad equation d ˆρS(t) = LˆρS(t) dt Most general form for the generator of a quantum dynamical semigroup i Lˆρ S(t) = i hĥ, ˆρS(t) + X γ k ˆV k ˆρ S(t) ˆV k 1 2 ˆV ˆV k k ρ S(t) 1 «ˆρS(t) ˆV k 2 ˆV k k On the generator of quantum mechanical semigroups, G. Lindblad, Commun. Math. Phys., 48, (1976). Semigroup properties: ˆQ(t) is completely positive ˆQ(t) ˆQ(s) = ˆQ(t + s) ˆQ(0) = Î Semigroup preserves: Hermiticity (probabilities are real numbers) Trace (conservation of norm) 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, / 38

27 Reduced system dynamics S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB = Use density operator ˆρ S Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

28 Reduced system dynamics S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB = Use state vector Ψ S Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

29 Reduced system dynamics S + B : ĤS HB, Ψ, ˆρ System S : ĤS, ΨS, ˆρS Environment B : ĤB, ΨB, ˆρB = Use state vector Ψ S No need to work with composite objects like density matrices Can trace out bath directly on the level of state vectors Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

30 Feshbach Projection-Operator Method Ĥ = ĤS + ĤB + αĥsb, HBχn(xB) = εnχn(xb) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

31 Feshbach Projection-Operator Method Ĥ = ĤS + ĤB + αĥsb, HBχn(xB) = εnχn(xb) Expand total wavefunction in arbitrary complete and orthonormal basis of the bath Ψ(x S, x B; t) = X n φ n(x S; t)χ n(x B) Projection Operators ˆP := ÎS χ k χ k ˆQ := Î S X k n χ k χ k Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

32 Feshbach Projection-Operator Method Ĥ = ĤS + ĤB + αĥsb, HBχn(xB) = εnχn(xb) Expand total wavefunction in arbitrary complete and orthonormal basis of the bath Ψ(x S, x B; t) = X n φ n(x S; t)χ n(x B) Projection Operators ˆP := ÎS χ k χ k ˆQ := Î S X k n χ k χ k Apply to TDSE i t ˆP Ψ(t) = ˆP Ĥ ˆP Ψ(t) + ˆP Ĥ ˆQΨ(t) i t ˆQΨ(t) = ˆQĤ ˆQΨ(t) + ˆQĤ ˆP Ψ(t) P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

33 Feshbach Projection-Operator Method Effective equation for ˆP Ψ (still fully coherent) Source Term z } { i t ˆP Ψ(t) = ˆP Ĥ ˆP ˆP Ψ(t) + ˆP Ĥ ˆQe i ˆQĤ ˆQt ˆQΨ(0) Z t i dτ ˆP Ĥ ˆQe i ˆQĤ ˆQ(t τ) ˆQĤ ˆP ˆP Ψ(τ) 0 {z } Memory Term Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

34 Feshbach Projection-Operator Method Effective equation for ˆP Ψ (still fully coherent) Source Term z } { i t ˆP Ψ(t) = ˆP Ĥ ˆP ˆP Ψ(t) + ˆP Ĥ ˆQe i ˆQĤ ˆQt ˆQΨ(0) Z t i dτ ˆP Ĥ ˆQe i ˆQĤ ˆQ(t τ) ˆQĤ ˆP ˆP Ψ(τ) 0 {z } Memory Term = Formal similarity to quantum transport formulation of Kurth and Stefanucci et. al. Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

35 Feshbach Projection-Operator Method Effective equation for ˆP Ψ (still fully coherent) Source Term z } { i t ˆP Ψ(t) = ˆP Ĥ ˆP ˆP Ψ(t) + ˆP Ĥ ˆQe i ˆQĤ ˆQt ˆQΨ(0) Z t i dτ ˆP Ĥ ˆQe i ˆQĤ ˆQ(t τ) ˆQĤ ˆP ˆP Ψ(τ) 0 {z } Memory Term = Formal similarity to quantum transport formulation of Kurth and Stefanucci et. al. Non-Markovian Stochastic Schrödinger equation perturbative expansion to second order in α H SB random phase approximation, dense bath spectrum, bath in statistical equilibrium i tψ(t) =ĤSψ(t) + α X η α(t) ˆV αψ(t) Z α t iα 2 dτ X C αβ (t τ) ˆV α e iĥs (t τ) ˆVβ ψ(τ) + O(α 3 ) 0 {z } αβ Bath correlation functions P. Gaspard, M. Nagaoka, JCP, 111, 5675 (1999). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

36 δ-correlated bath Markovian Stochastic Schrödinger equation C αβ (t τ) = D αβ δ(t τ) Stochastic Schrödinger equation in Born-Markov approximation i tψ(t) =ĤSψ(t) + α X α η α(t) ˆV αψ(t) iα 2 X αβ D αβ ˆV α ˆVβ ψ(t) + O(α 3 ) Statistical average: ρ S(t) = ψ(t) ψ(t) ψ(t) ψ(t) Unravelling of Lindblad equation for static (linear) Hamiltonians Does not rely on semigroup property Valid for time-dependent Hamiltonians Gives always physical states Sound starting point to formulate stochastic TDDFT Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

37 TDDFT for Open Quantum Systems Approach in terms of density matrices K. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, (2005). Density matrix evolution may not obey positivity during time-evolution Scales as O(N 2 ) due to usage of density matrix Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

38 TDDFT for Open Quantum Systems Approach in terms of density matrices K. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, (2005). Density matrix evolution may not obey positivity during time-evolution Scales as O(N 2 ) due to usage of density matrix Approach in terms of stochastic Schrödinger equations M. Di Ventra and R. D Agosta, Phys. Rev. Lett. 98, (2007). Positivity is guaranteed by construction Scales as O(N) since only state vectors are used Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

39 TDDFT for Open Quantum Systems Approach in terms of density matrices K. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, (2005). Density matrix evolution may not obey positivity during time-evolution Scales as O(N 2 ) due to usage of density matrix Approach in terms of stochastic Schrödinger equations M. Di Ventra and R. D Agosta, Phys. Rev. Lett. 98, (2007). Positivity is guaranteed by construction Scales as O(N) since only state vectors are used Comparison to classical stochastic systems: Fokker-Planck equation Langevin equation Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

40 Stochastic Time-Dependent Current-Density-Functional Theory Can prove: For fixed bath operators ˆV and initial state j(r, t) 1:1 A(r, t) M. Di Ventra and R. D Agosta, Phys. Rev. Lett. 98, (2007). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

41 Stochastic Time-Dependent Current-Density-Functional Theory Can prove: For fixed bath operators ˆV and initial state j(r, t) 1:1 A(r, t) M. Di Ventra and R. D Agosta, Phys. Rev. Lett. 98, (2007). Mapping of fully interacting stochastic TDSE to stochastic TDKS equations 2 3 i tψ j(r, t) = 6 4ĤKS (t) 1 2 i ˆV ˆV + l(t) ˆV 7 ψj(r, t) {z } {z } 5 damping fluctuations Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

42 Stochastic Time-Dependent Current-Density-Functional Theory Can prove: For fixed bath operators ˆV and initial state j(r, t) 1:1 A(r, t) M. Di Ventra and R. D Agosta, Phys. Rev. Lett. 98, (2007). Mapping of fully interacting stochastic TDSE to stochastic TDKS equations 2 3 l(t) : stochastic process i tψ j(r, t) = 6 4ĤKS (t) 1 2 i ˆV ˆV + l(t) ˆV 7 ψj(r, t) {z } {z } 5 damping fluctuations l(t) = 0, l(t)l(t ) = δ(t t ) Assumes Factorization at initial time: ψ(t 0) = ψ S(t 0) ψ B(t 0) Markovian approximation: no bath memory Weak coupling to the bath (second order in H SB) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

43 Choice of bath operators: a simple model Order-N scheme and bath operators which obey Fermi statistics V j kk (r) = δ kj (1 δ kk ) p γ(r)f D(ɛ k ) ψ j(r) ψ k (r) Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev. B 78, (2008). Fermi-Dirac distribution Local relaxation rates f D(ε k ) =» «1 εk µ 1 + exp k BT γ kk (r) = ψ k (r) γ 0 ψ k (r) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

44 Choice of bath operators: a simple model Order-N scheme and bath operators which obey Fermi statistics V j kk (r) = δ kj (1 δ kk ) p γ(r)f D(ɛ k ) ψ j(r) ψ k (r) Yu. V. Pershin, Y. Dubi, and M. Di Ventra, Phys. Rev. B 78, (2008). Fermi-Dirac distribution Local relaxation rates f D(ε k ) =» «1 εk µ 1 + exp k BT γ kk (r) = ψ k (r) γ 0 ψ k (r) Operators ensure that Fermi statistics is obeyed. If a steady state is reached, it will be a thermal state. Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

45 Quantum jump algorithm 1) Draw uniform random number η j [0, 1] 2) Propagate auxilary wave function under non-hermitian Hamiltonian i tφ = Ĥ0Φ i ˆV ˆV Φ 3) Propagate system wave function under norm-conserving Hamiltonian i tψ = Ĥ0Ψ i ˆV ˆV Ψ + i ˆV Ψ 2 Ψ 4) If norm of auxilary wave function drops below η j, act with bath operator Φ(t j) η j, Ψ(t j) = ˆV Ψ(t j), Φ(t j) = Ψ(t j) 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 = Ψ j Ψ j guarantees always physical state! Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

46 Practical Implementation Full-potential, all-electron code with local orbitals Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

47 Practical Implementation Expansion of TD orbitals in non-orthorgonal atom centered basis functions φ k ψ j(t) = X k c jk (t)φ k Overlap matrix TDKS equations S jk = φ j φ k iŝ t c j(t) = ĤKS (t) c j(t) Exponential midpoint approximation for short-time propagator h i Û(t + t, t) = exp iŝ 1 Ĥ KS (t + t/2) t + O( t 3 ) h = Ŝ 1/2 exp iŝ 1/2 Ĥ KS (t + t/2) tŝ 1/2 + O( t 3 ) i Ŝ1/2 Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

48 Application (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

49 Application: zincbacteriochlorin-bacteriochlorin complex HOMO LUMO Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

50 Free Propagation of HOMO Free propagation of HOMO: (for real-valued orbitals) ψhomo(t) = ψ GS HOMO e iε HOMO t Re{ψHOMO(t)} = ψ GS HOMO cos(εhomot) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

51 Linear combination of HOMO and LUMO Initial state ψ TDKS HOMO(t = 0) = 1 h i ψ GS HOMO + e i π GS 2 ψ LUMO 2 Closed quantum system Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

52 Linear combination of HOMO and LUMO Initial state ψ TDKS HOMO(t = 0) = 1 h i ψ GS HOMO + e i π GS 2 ψ LUMO 2 Including coupling to environment Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

53 Outline Open Quantum Systems Different approaches to deal with Decoherence and Dissipation Stochastic current density-functional theory Application: Stochastic simulation of (1,4)-phenylene-linked zincbacteriochlorin-bacteriochlorin complex Stochastic Quantum Molecular Dynamics: Theory and Applications Including nuclear motion: Stochastic Quantum Molecular Dynamics 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, / 38

54 Molecular Dynamics for Open Systems Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD: Electronic degrees of freedom are treated with closed system approach Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

55 Molecular Dynamics for Open Systems Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD: Electronic degrees of freedom are treated with closed system approach Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces) Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

56 Molecular Dynamics for Open Systems Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD: Electronic degrees of freedom are treated with closed system approach Damping is added only to nuclear EOM (Langevin terms, velocity dep. forces) However: Electrons are the first to experience energy transfer to a bath Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

57 Molecular Dynamics for Open Systems Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD: Electronic degrees of freedom are treated with closed system approach 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 Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

58 Molecular Dynamics for Open Systems Standard approaches like Car-Parinello MD, Born-Oppenheimer MD, Ehrenfest MD: Electronic degrees of freedom are treated with closed system approach 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, / 38

59 Stochastic Quantum Molecular Dynamics Extension of stochastic TDCDFT to include nuclear degrees of freedom i tψ = Ĥ(t)Ψ 1 2 i ˆV ˆV Ψ + l(t) ˆV Ψ Ĥ(t) = ˆT e(r, t) + Ŵee(r) + Ûext,e(r, t)+ ˆT n(r, t) + Ŵnn(R) + Ûext,n(R, t)+ Ŵ en(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 V α(x, t) J(x, t) A(x, t) Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, (2009). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

60 Stochastic Quantum Molecular Dynamics: practical scheme Extension of stochastic TDCDFT to include nuclear degrees of freedom i tψ(x, t) = Ĥ(t)Ψ(x, t) 1 2 i ˆV ˆV Ψ(x, t) + l(t) ˆV Ψ(x, t) In practice: resort as approximation to classical nuclei Bath operators V j kk (r, R(t); t) = δ kj (1 δ kk ) p γ(r, R(t); t)f D(ɛ k ) ψ j(r, R(t); t) ψ k (r, R(t); t) Ehrenfest forces as approximation for classical nuclei Z M α Rα(t) = Ψ Rα Ĥ eψdr Note: Wavefunctions are stochastic = stochastic force on the nuclei Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, (2009). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

61 Vibronic excitation of Beryllium dimer, moving nuclei closed quantum system stretched initial condition Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

62 Vibronic excitation of Beryllium dimer, moving nuclei closed quantum system stretched initial condition open quantum system relaxation rate τ = 300 fs Maxwell-Boltzmann velocity distribution at jumps m «3/2 f( v) = exp m v2 2πkT 2kT Average over 15 stochastic realizations Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

63 Dual-Fluoresence in 4-(N,N-Dimethylamino)benzonitrile ɛ ( 45K) = 50.2 ɛ (+75K) = 30.3 polar solvent acetonitrile non-polar solvent n-hexane DMABN Experiment: J. Phys. Chem. A 2006, 110, Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

64 Stochastic Quantum MD simulation for 4-(N,N-Dimethylamino)benzonitrile Electron Localization Function Rotated dimethyl group as initial condition Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

65 Stochastic Quantum MD simulation for 4-(N,N-Dimethylamino)benzonitrile δ CH 3 N C N angle [deg] closed 0K open 0K open 300K CH time [fs] counts counts t=0 fs T=0 K t=46 fs T=300K angle [deg] t=137 fs T=0K t=137 fs T=300K angle [deg] t=273 fs T=0K t=273 fs T=300K angle [deg] t=550 fs T=0K t=550 fs T=300K angle [deg] Heiko Appel, Massimiliano Di Ventra, Phys. Rev. B 80, (2009). Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

66 Conclusions SQMD allows to describe ab-initio relaxation and dephasing of electronic and nuclear degrees of freedom Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

67 Conclusions SQMD allows to describe ab-initio relaxation and dephasing of electronic and nuclear degrees of freedom In constrast to standard QMD, stochastic QMD allows also coupling of electrons to a heat bath Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

68 Conclusions SQMD allows to describe ab-initio relaxation and dephasing of electronic and nuclear degrees of freedom In constrast to standard QMD, stochastic QMD allows also coupling of electrons to a heat bath Nuclei can feel the heat bath via electron-nuclear coupling Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

69 Conclusions SQMD allows to describe ab-initio relaxation and dephasing of electronic and nuclear degrees of freedom In constrast to standard QMD, stochastic QMD allows also coupling of electrons to a heat bath Nuclei can feel the heat bath via electron-nuclear coupling Within reach of Stochastic Quantum Molecular Dynamics Themopower/Thermoelectric effects Excited state relaxation in solution Decoherence and dephasing in pump-probe experiments Driven quantum phase transitions... Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

70 Future work Relaxation times from system-bath interaction Hamiltonian Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

71 Future work Relaxation times from system-bath interaction Hamiltonian Non-markovian dynamics: bath-correlation functions which are not delta correlated Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

72 Future work Relaxation times from system-bath interaction Hamiltonian Non-markovian dynamics: bath-correlation functions which are not delta correlated Functionals: Dependence of XC-functional on bath operators Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

73 Thanks! Massimiliano Di Ventra Yonatan Dubi Matt Krems Jim Wilson Roberto D Agosta Thank you for your attention! Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38

74 Positivity A density matrix which obeys positivity is a non-negative operator in Hilbert space Ψ ˆρ Ψ 0, Ψ This guarantees that the expectation value of a positive operator A Â (t) = X i p i Ψ(t) Â Ψ(t) = T r{ˆρ(t)â} 0 and the variance of any operator ˆB ˆB 2 (t) ˆB 2 (t) 0 Heiko Appel (UC San Diego) Stochastic Quantum Molecular Dynamics January 13, / 38