Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Local Coherent-State Approximation to
System-Bath quantum dynamics
Rocco Martinazzo
Department of Physical Chemistry and Electrochemistry
University of Milan, Milan, Italy
Multidimensional Quantum Mechanics with Trajectories
Leeds, Sept, 2nd 2008
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dcfe@university of milan
LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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dcfe@university of milan
LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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dcfe@university of milan
LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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dcfe@university of milan
LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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dcfe@university of milan
LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
System-Bath dynamics
System: relevant part,
experimentally probed
⇒ Few, important DOFs
System
x 1 , x2 ,..xn
Bath: irrelevant part, but
responsible for energy
transfer
⇒ Large number of DOFs
of non-direct relevance
Bath
q1 , q2 ,..qk ,..qF
Quantum description is mandatory for inherently quantum
systems and/or low-temperature baths..
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
System-Bath dynamics
..e.g. in surface science
sticking of hydrogen atoms on cold surfaces
vibrational relaxation of light adsorbates at surfaces
surface diffusion of hydrogen atoms at low temperatures
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
System-Bath dynamics
Reduced equations of motion
Unitary (approximate) evolution
First project, then evolve
First evolve, then project
Density operator,
ρsys
ρsb
→
(Approximate) equation of
motion for ρsys
∼ (t)
..also REOM for system
observables if Hisenberg
picture is preferred
Appromixate
time-evolution of the whole
system ρsb
∼ (t)
Project whole state onto
system space, ρsb → ρsys
..also exact evolution if the bath
is not too large
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Reduced Equations of Motion
Reduced density operator:
ρS = trB [U(t, 0)ρ0 U(t, 0)† ]
Initial factorizing conditions
ρ(0) = ρS (0) ⊗ ρB (0)
Dynamical map:
ρS (t) = V (t)ρS (0)
Markov Approximation, τB τR
V (t1 )V (t2 ) = V (t1 + t2 ), t1 , t2 ≥ 0
Lindblad’s theorem
L = Lsys + Ldiss
Lsys ρS = −i[H S , ρS ]
L
Pdiss ρS =
†
1 †
k γk (Ak ρS Ak − 2 [Ak Ak , ρS ]+ )
V (t) = exp(Lt) or equiv
dρs (t)
= Lρs
dt
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Approximate Unitary Evolution
Dissipative dynamics for t < trec ∝ ∆ω −1
H = H sys + H coup + H bath
H bath =
pk2
k =1 ( 2mk
PF
+ 12 mk ωk2 qk2 ) and H coupl =
PF
k =1 fk qk
where fk = fk (x) and eventually ωk = ωk (x)
System-bath interactions are local in system coordinates
Bath oscillators are (at least locally) harmonic
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Approximate Unitary Evolution
Dissipative dynamics for t < trec ∝ ∆ω −1
H = H sys + H env
H sys = H sys (x, p)
H env = H env (q1 , p1 , ..qF , pF ; x)
H env is local in system coordinates
H env is approximately harmonic in bath coordinates
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Basic idea
Main ingredients
Local, approximately harmonic environment hamiltonian
H = H sys + H env
H sys = H sys (x, p)
H env = H env (q1 , p1 , ..qF , pF ; x)
System locality ⇒ Discrete Variable Representation
Bath harmonicity ⇒ Coherent-States
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Basic idea
DVR and CSs
Main ingredients
Discrete Variable
Representation (DVR)
P = P†, P = P2
E = PH interesting subspace
H = L2 (M), M ⊃ {xα }
e.g. for M = RN
1
|ξα i = √
P |xα i
Nα
|∆α i = P |xα i
˛
˙
¸
∆α ˛ ∆β = ∆β (xα ) = Nα δαβ
˙ ˛
¸
V = V (x) ⇒ ξα ˛ V |ξβ ∼ δαβ V (xα )
Coherent-States (CSs)
H[ψ] = ∆pψ ∆xψ
δH[ψ] = 0 ⇒ (p − p0 ) |ψi = α(x − x0 ) |ψi
n
o
n
o
x0
p0
p
x
|ψi = 2∆x
|ψi
+ i 2∆p
+ i 2∆p
2∆x
(a − z) |ψi = 0 → |zi := |ψi
∆x =
p
~/2mω → H HO = ~ω(a† a + 21 )
e−
H HO
~
t
i
|z0 i = e− 2 ωt |zt i
zt =
e−iωt z0
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Basic idea
LCSA ansatz for T = 0 K case
DVR approximation in system space
Z
X
|Ψi = dx |xi hx | Ψi ≈
|ξα i hξα | Ψi
α
hξα | Ψisys = Cα |Φα ibath
Hartree product approximation to local bath states
|Φα i ≈ |φ1α i |φ2α i .. |φFα i
Coherent-State approximation to spfs
|φkα i ≈ |zαk i
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Basic idea
LCSA ansatz for T = 0 K case
DVR approximation in system space
Z
X
|Ψi = dx |xi hx | Ψi ≈
|ξα i hξα | Ψi
α
hξα | Ψisys = Cα |Φα ibath
Hartree product approximation to local bath states
|Φα i ≈ |φ1α i |φ2α i .. |φFα i
Coherent-State approximation to spfs
|φkα i ≈ |zαk i
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Basic idea
LCSA ansatz for T = 0 K case
DVR approximation in system space
Z
X
|Ψi = dx |xi hx | Ψi ≈
|ξα i hξα | Ψi
α
hξα | Ψisys = Cα |Φα ibath
Hartree product approximation to local bath states
|Φα i ≈ |φ1α i |φ2α i .. |φFα i
Coherent-State approximation to spfs
|φkα i ≈ |zαk i
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Basic idea
LCSA ansatz for T = 0 K case
Dynamical variables
System: Cα
Bath: zαk
|Ψi =
P
α Cα |ξα i |Zα i
|Zα i = |zα1 i |zα2 i .. |zαF i
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
Working equations
Dirac-Frenkel variational principle
hδΨ | i~∂t − H|Ψi = 0
⇓
i~Ċα =
i~Cα żαk =
P
β
damp
Hαβ
Cβ + vαeff Cα
damp k
k
β Hαβ (zβ − zα )Cβ + Cα
P
env
∂Hord
∂ak†
(..zαk ∗ , zαk ..; xα )
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
Working equations: subsystem equations
Damped hamiltonian:
damp
Hαβ
˛ ¸
¸˙ ˛ ¸
˙ ˛
sys ˙
= ξα ˛ H sys |ξβ Zα ˛ Zβ = Hαβ Zα ˛ Zβ
«
„
F
X
˙ ˛ ¸
Γββ
Γαα
k∗ k
zα
zβ
−
with Γαβ =
Zα ˛ Zβ = exp Γαβ −
2
2
k =1
Effective potential:
gauge
eff
lmf
vα
= vα
+ vα
˙ ˛
¸
lmf
env
k∗ k
vα
= Zα ˛ H env |Zα = Hord
(..zα
, zα ..; xα )
gauge
vα
= −i~
F D
X
˛ E
k ˛ ˙k
zα
˛ zα
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k =1
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
Working equations: bath equations
˙k
k = z˙k
z˙α
αclass + zαquant
Classical ‘force’:
env
i ∂Hord
k∗ k
k
z˙αclass
(..zα
, zα ..; xα )
=−
~ ∂a†
k
env =
e.g. for Hord
PF
k =1
” P
“
”
“
~ωk ak† ak + 21 − Fk=1 λk ak† + λ†k ak
one gets
env
∂Hord
†
∂ak
k − λ (x )
= ~ωk zα
k α
Quantum ‘force’:
i X damp k
k
k
z˙αquant
=−
H
(zβ − zα
)Cβ
~Cα β αβ
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
Working equations: properties
By using the Dirac-Frenkel variational principle, one gets for
free
Norm conservation
Energy conservation
..and more generally a symplectic structure
computational cost ∝ F α
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
Working equations: properties
Power-law scaling
Hours on Intel Xeon [email protected]
4000
3000
100 DVR points
2000
50 DVR points
1000
0
0
4
1×10
4
2×10
4
3×10
Nbath
4
4×10
4
5×10
(No inter-mode coupling)
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
T > 0 case
ρsb =
X
ED
LCSA pi ΨLCSA
Ψ
i
i
i
⇒ Independent wavepacket propagation
⊗ k
E.g. ρsb (0) = ρsys (0) ⊗ ρbath
β (0), harmonic bath in thermal equilibrium, ρβ = Πk ρβ
ρkβ =
R
d 2 z wβk (|z|2 ) |zβ/2 i hzβ/2 |
⇒ Monte Carlo sampling of
R
wβk (s) =
1−e−β~ωk
π
e−s(1−e
−β~ωk
)
d 2 z 1 d 2 z 2 ..d 2 z F for initial conditions
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
LCSA equations
Comparison with Gaussian-MCTDH
|Ψi =
P
α
1 i .. |z F i
Cα |ξα i |zα
α
|Ψi =
P
i0 ,I
Ci0 ,I |φi0 i |zi1 i .. |ziF i
1
F
System-states |ξα i are
time-independent
System-states |φi0 i are
time-dependent
For each α there is one CS product,
1 i .. |z F i
|zα
α
For each i0 there is a number of CS
products, {|zi1 i .. |ziF i}I = {ZI }I
Different α → different Zα
Different i0 → same {ZI }I
Orthogonal configurations
1 i .. |z F i,
|Ψα i = |ξα i |zα
α
hΨα |Ψβ i = δαβ
Non fully orthogonal configurations
|Ψi0 ,I i = |φi0 i |zi1 i .. |ziF i,
1
F
hΨi0 ,I |Ψj0 ,J i = δi0 ,j0 hZI |ZJ i
1
F
LCSA is a selected-configuration G-MCTDH variant, with simpler eq.s of motion
because it is tailored to system-bath problems
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Model systems
Subsystem: H atom
Vibrational
dynamics
Tunneling
dynamics
Sticking
dynamics
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Model systems
Bath
Truncated, discretized Ohmic bath
H bc = H HO + H coup
‘Shape function’ f (x)
H coup = −
†
†
k (λk ak + λk ak )
P
J(ω) = π
P
k
λk = f (x)Λk
Relaxation time γ −1
Λ2k δ(ω − ωk ) = mωγ(ω)
f (x) =
Ohm ⇒ J(ω) = mωγ
ωk = k ∆ω =
k ω cutoff /F
Λk =
q
ω cutoff
mγωk ∆ω
π
ωsys
8
<
1−eαx
α
vibrational dyn
:
x
tunneling dyn
γ −1 = 10 − 1000 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Model systems
Subsystem density matrix
Vibrational
dynamics
Tunneling
dynamics
Sticking
dynamics
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Model systems
Bath occupation numbers
Vibrational dynamics
Sticking dynamics
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Small baths
Vibrational dynamics
1D bath
Harmonic oscillator
Morse oscillator
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Small baths
Vibrational dynamics
Morse + 2D bath, γ −1 = 8 fs
Morse + 3D bath, γ −1 = 32 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Small baths
Tunneling dynamics
1D bath
Double well, Eb ∼ 6 Kcal mol −1 , ωsys = 1530 cm−1
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Vibrational relaxation
Standard LCSA
Morse + 50D bath, γ −1 = 50 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Vibrational relaxation
Damped LCSA
Morse + 50D bath, γ −1 = 50 fs, η −1 = 12.5 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Vibrational relaxation
Damped LCSA
Morse + 50D bath, η −1 = 12.5 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
“Damped” LCSA
LCS approximation to system + p-bath + s-bath
s ∼ zs
Classical approximation to s-bath, zα
β
System
x 1 , x2 ,..xn
Continuum limit, Ohmic s-bath, T=0 k
Primary bath
q1 , q2 ,..qk ,..qF
ς (t)
Xαk =
Secondary bath
r 1, r 2,..rs ,..rS
1
~Cα
η
k = iX k − k
i żα
+ ωk Re(Xαk )
α
2∆pK
k
env
P
damp k
k )+ 1 ∂Hord (..z k ∗ , z k ..; x )
(z
−z
H
α
α
α
α
†
β αβ
β
~
∂ak
P
A2
s
ηk (t) = m1
s m ω 2 cos(ωs t) → ηk (t) = 2ηk δ(t)
k
k k
P
ps (0)
s
s
ςk (t) =
s As [(q (0) − qeq (0))cos(ωs t) + ms ωs sin(ωs t)] → 0
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
“Damped” LCSA
η Optimization: Morse + 50D bath, γ −1 = 50 fs, t = 150.0 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
“Damped” LCSA
True dissipative dynamics
Recurrence problem removed
Morse + nD bath, ω cutoff = 2ωsys , γ −1 = 200 fs, η −1 = 12.5 fs group-logo
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Tunneling dynamics
Double well + 50D bath
Eb ∼ 2 Kcal mol −1 , ωsys = 816 cm−1
ω cutoff = 940 cm−1 , γ −1 = 50 fs
Standard LCSA, η −1 = ∞ fs
Eb ∼ 6 Kcal mol −1 , ωsys = 1530 cm−1
ω cutoff = 4390 cm−1 , γ −1 = 50 fs
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Damped LCSA, η −1 = 40 fs
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Sticking dynamics
Ecoll = 200 meV
Morse + 50D bath, γ −1 = 1.0 ps
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Sticking dynamics
Overview
Morse + 50D bath, γ −1 = 1.0 ps
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Large baths
Sticking dynamics
Damped LCSA
Morse + 50D bath, Ecoll = 350 meV, γ −1 = 1.0 ps
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Outline
1
Introduction
2
Local Coherent-State Approximation
Basic idea
Equations of motion
3
Application to model systems
Small dimensional baths
Large dimensional baths
4
Symplectic structure
General considerations
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Generalities
Hamiltonian flows
Symplectic manifold (M, ω)
Manifold equipped
non-degenerate
P withi a closed,
2-form ω, ω =
ωij dx dx j
Non-degeneracy (|ωij | 6= 0):
P
α=
αi dx i ↔Xα , ω(Xα , v) = α(v)
α has an associated flow, ẋ = Xα
In particular, H→dH↔XH
⇒ Hamiltonian flow, ẋ = XH
dH(XH ) = dH/dt = ω(XH , XH ) = 0
Closedness (dω = 0):
LXH ω = 0
T∗M
x
α
T M x
Ιω
x v
X
α
M
e.g. classical Phase-Space is a
cotangent bundle M = T ∗ (C)
⇒ natural, built-in
form,
P symplectic
ω = dα, α =
yi dx i
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Generalities
Hamiltonian flows in variational QD
Time-Dependent Variational Principle δS = 0
L=
i~ hΨ|Ψ̇i − hΨ̇|Ψi
hΨ | H|Ψi
−
2
hΨ | Ψi
hΨ | Ψi
Sample space ⇒ M with coordinates x i (variational parameters) → |Ψ(x)i,
X
L=
ẋ i Zi (x) − H(x)
P
α=
Zi dx i ⇒ ω = dα
If ω = {ωij }, ωij = ∂Zi /∂x j − ∂Zj /∂x i is non-singular ⇒ (M, ω) is symplectic
Variational flow:
X
∂H
ωij ẋ i =
∂x j
⇒ the variational flow is the hamiltonian flow of the hamiltonian H(x)
X ∂f ∂g
{f , g} := ω(Xf , Xg ) =
ξ ij ξ = ω −1
∂x i ∂x i
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Summary
The Local Coherent-State Approximation is reasonably
accurate and computationally cheap to sudy
vibrational relaxation
tunneling
sticking
..dynamics in the quantum regime.
It experiences numerical problems but they might be solved in
the near future with robust, symplectic propagation schemes
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LCSA for system-bath dynamics
Introduction
Local Coherent-State Approximation
Application to model systems
Symplectic structure
Summary
Acknowledgements
University of Milan
Gian Franco Tantardini
Fausto Martelli
Ecole Normal Superieure
Irene Burghardt
Universitat Potsdam
Mathias Nest
Peter Saalfrank
+-x:
C.I.L.E.A. Supercomputing
Center
I.S.T.M.
Chemical Dynamics Theory Group
http://users.unimi.it/cdtg
group-logo
dcfe@university of milan
LCSA for system-bath dynamics