e ective evolution of open dimersmerkli/talks/oberwolfach2016.pdf · when a molecule is excited...
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Effective evolution of open dimers
Marco Merkli
Deptartment of Mathematics and StatisticsMemorial University, St. John’s, Canada
Obwerwolfach, September 2016
Collaborations with
M. Konenberg (2016)G.P. Berman, R.T. Sayre, S. Gnanakaran,
M. Konenberg, A.I. Nesterov and H. Song (2016)
The plan
I. Physical motivation
II. Resonance expansion
III. Application: dynamics of a dimer
IV. Outline: proof of resonance expansion
I. Physical motivation
Excitation transfer processWhen a molecule is excited electronically by absorbing a photon, itluminesces by emitting another photon or excitation is lost inthermal enviornment (∼ 1 nanosecond).
Fluorescence
However, when another molecule with similar excitation energy ispresent within ∼ 1− 10 nanometers, the excitation can beswapped between the molecules (∼ 1 picosecond).
Excita'on transfer process: D*+ A D + A*
D*
A D
A*
Excitation transfer happens in biological systems (in chlorophyllmolecules during photosynthesis)
Similar charge transfer (electron, proton) happens in chemicalreactions: D + A → D− + A+ (reactant and product)
Processes take place in noisy environments (molecular vibrations,protein and solvent degrees of freedom)
Environment D
Collec/ve Environment
Donor D
Acceptor A
Environment A V
Local model (red) and collective model (blue)
V : exchange or dipole-dipole interaction
Local (uncorrelated) model: D, A have individual environments
Collective (correlated) model: D, A have common environment
Electronic excitation energy transfer theory = Forster theoryCharge transfer theory = Marcus theory
Goal: Derive transfer rate constants in these transport processes
I Forster formula (1948)
γF =9000 (ln 10)κ2
128π5NA τD n4r R
6
∫ ∞0
fD(ν)εA(ν)
ν4dν
κ2 = orientation factor, NA = Avogadro’s number, τD = spontaneous decay life-time of excited donor, nr= refractive index of medium, R = donor-acceptor distance, fD (ν) = normalized donor emission spectrum,εA(ν) = acceptor molar extinction coefficient
I Marcus formula (1956)
γM =2π
~|V |2 1√
4πλkBTexp
[−(∆G + λ)2
4λkBT
]V = electronic coupling, λ = reorganization energy, ∆G = Gibbs free energy change in reaction
Marcus approach
HMarcus = |R〉ER〈R|+ |P〉EP〈P|+ |R〉V 〈P|+ |P〉V 〈R|
=
(ER VV EP
)Marcus Hamiltonian
R = reactant (donor), P = product (acceptor)• Collective reactant/product energies:
ER =∑α
(p2α
2mα+ 1
2mαω2αq
2α
),
EP =∑α
(p2α
2mα+ 1
2mαω2α(q2
α − q0,α)2 − ε0,α
)• In quantum mechanical treatment, ER and EP become operatorsHR and HP
Marcus Hamiltonian ↔ spin-boson model
• Xu-Schulten ‘94: Marcus Hamiltonian equivalent to spin-bosonHamiltonian
HSB = Vσx + ε σz + HR + λσz ⊗ ϕ(h)
with λ2 ∝ εrec reconstruction energy,
HR =∑α
ωα(a†αaα + 1/2)
ϕ(h) = 1√2
∑α
hαa†α + h.c., hα = form factor
• For spin-boson model can use heuristic procedure ‘time-de-pendent perturbation theory’ of Leggett et al. ‘87 to getrelaxation rate γ for initially populated donor
“ pdonor = e−γt ”
Towards a structure-based exciton Hamiltonian for the CP29 antenna of photosystem II
Frank Műh, Dominik Lindorfer, Marcel Schmidt am Busch and Thomas Renger, Phys. Chem. Chem. Phys., 16, 11848 (2014)
Our chlorophyll dimer:
604: Chla, Eaexc= 14 827cm-1
= 1.8385eV 606: Chlb, Eb
exc= 15 626cm-1
= 1.9376eV ε = Eb
exc- Eaexc = 99.1meV
V = 8.3meV
Our chlorophyll dimer is weakly coupled: Vε≈ 0.08 ≪ 1.
Donor
Acceptor
For the weakly coupled dimer V << ε and at high temperaturekBT >> ~ωc , rate γ is given by Marcus formula
γMarcus =V 2
4
√π
T εrece−
(ε−εrec)2
4Tεrec
εrec = reconstruction energy ∝ λ2 is ≈ ε (giving max of γMarcus)
• Marcus theory works for large (any) interaction strength withenvironment (εrec)
• The ‘usual’ theory of open quantum systems is Bloch-Redfieldtheory, designed for small interactions with environment
• Rudolph A. Marcus received the 1992 Nobel Prize in Chemistry“for his contributions to the theory of electron transfer reactionsin chemical systems”
Our main contributions:
1. Develop dynamical resonance theory, a controlled pertur-bation theory for the dynamics of the weakly coupled dimer(V << ε), valid for all times and reservoir coupling strengths
2. Extract from it validity of exponential decay law and rates ofrelaxation and decoherence
3. Consider individual coupling strenghts of donor and acceptor toenvironment and/or independent environments
II. Resonance expansion
Spectral deformation ...... or not !
I Propagator eitL, L self-adj. Liouville operator (∼ Hamiltonian)
I Map z 7→ f (z) ≡ 〈ψ, (L− z)−1φ〉 is analytic in z ∈ C−I Spec. defo. technique: f (z) extends meromorphically
into C+, has poles on 2nd Riemann sheet
meromorphic con+nua+on of f(z)
Γ
>
cut
X
X
X X
X X
z
poles in 2nd Riemann sheet X
1st Riemann sheet
Γ’
>
I Contour deformation reveals dynamics
〈ψ, eitLφ〉 =−1
2πi
∫Γ
eitz f (z)dz =∑
poles a
eitaf (a) + O(e−αt
)
I Get continuation of f (z): Uθ, θ ∈ R suitable unitary group
f (z) = 〈ψ, (L− z)−1φ〉 = 〈ψθ, (Lθ − z)−1φθ〉
(Lθ = UθLU∗θ , ψθ = Uθψ)
I Some regularity ⇒ above f (z) extends to θ ∈ CI θ ∈ C\R: Lθ is non-self-ajdoint, has eigenvalues in C+ =
poles of f (z)
I z 7→ f (z) for Imθ 6= 0 fixed is desired meromorphic extension
• What if Lθ does not have a meromorphic extension (lack ofregularity)? This happens in systems of interest.– Then cannot extend to second Riemann sheet to access poles.– How can we recover decay times and directions?
• Task: develop method using only regularity condition:
z 7→ 〈ψ(L− z)−1φ〉 bounded as Imz ↑ 0 (LAP)
Example where spectral deformation works:Spin-boson at weak coupling (λ small)
λ = 0: system and reservoir uncoupled, dynamics factorizes λ 6= 0: Uθ = translation 4, spec(Lθ):
X
X
X X
continuous spectrum
a− a0 a+ γ
λ2
0
Eigenvalues of Lθ disjoint from cont. spec., Resonance expansion
eitL(θ) =∑j
eitaj Πj + O(e−γt)
Regularity requirement: I ∝ field operator ‘translation analytic’ 4
Spin-boson at weak tunneling (V small )
Syst. & res. already interacting for unperturbed dynamics V = 0.‘Undo interaction’ by unitary polaron transformat. U = U(λ)
L0 ≡ U L0 U∗ = LS + LR (V = 0)
Perturbation V 6= 0:L = L0 + V II = I(λ) ∼ σ+ ⊗Wβ(λh) + adj.
Wβ(λh) = eiϕβ(λh) thermal Weyl operator
Perturbation I = bounded operator, but behaves badly underspectral deformation:
spectral defo. of Wβ(λh) ∼ eλ√N N = number op.
Too unbounded to do perturbation theory in V ! Don’t knowhow to implement spectral deformation technique !
Using Mourre theory instead of spectral deformation can prove:
Theorem (Konenberg-Merkli-Song 2014). ∀λ ∈ R, if V 6= 0 issmall enough, then L has absolutely continuous spectrum coveringR and a single simple eigenvalue at the origin. The eigenvector isthe coupled equilibrium state Ω.
X
Spec(L )
0
XX X
Spec(L )0
0 |Δ| > 0
Two main technical tools in proof:
1. Positive commutator methods to show instability of eigenvaluesassuming an effective coupling “Fermi Golden Rule” condition
2. Show limiting Absorption Principle (LAP) to then show AC spec.
supx∈(a,b);y>0
∣∣〈ψ, (L− x + iy)−1ψ〉∣∣ ≤ C (ψ)
for ψ in a dense set =⇒ spec of L in (a, b) purely AC.
Dynamical consequences of the Theorem:
L = 0 · PΩ ⊕ LP⊥Ω & spec(L) purely AC, so
eitL = PΩ ⊕ eitLP⊥Ω −→ PΩ ⊕ 0, weakly, as t →∞
This is rather incomplete information compared to spectral defo.case, where decay rates and directions are obtained as resonanceenergies and projections
How can we recover full dynamical information
eitL ∼∑j
eitaj Πj + remainder
using Mourre theory ?
Result: Resonance Expansion via Mourre Theory
Setup
Family of self-adjoint operators on Hilbert space H
L = L0 + V I , V ∈ R perturbation parameter
Call eigenvalue e of L0
I unstable if for V 6= 0 small, L does not have eigenvalues in aneighbourhood of e
I partially stable if for V 6= 0 small, L has eigenvalues in aneighbourhood of e with summed multiplicity < mult(e)
We suppose all eigenvalues are either unstable or partially stablewith a reduction to dimension one.
Level shift operators
I If e was isolated eigenvalue of L0, with spectral projection Pe ,then by analytic pert. theory, eigenvalues of L near e would bethose of
ePe + VPe IPe − V 2Pe IP⊥e (L0 − e)−1IPe + O(V 3)
I We assume Pe IPe = 0
I Since e is actually embedded eigenvalue, the resolventP⊥e (L0 − e)−1 does not exist, but we can expect the 2nd ordercorrections to be linked to the level shift operator
Λe = −Pe IP⊥e (L0 − e + i0+)−1IPe
Assumption: Fermi Golden Rule Condition(Instability of eigenvalues visible at order V 2)
(1) The eigenvalues of all the level shift operators Λe are simple.
(2) e unstable =⇒ all the eigenvalues λe,0, . . . , λe,me−1 of Λe
have strictly positive imaginary part.
(3) e partially stable =⇒ Λe has a single real eigenvalue λe,0. Allother eigenvalues λe,1, . . . , λe,me−1 have imaginary part > 0.
Then Λe is diagonalizable:
Λe =me−1∑j=0
λe,jPe,j
where Pe,j are the (rank one) spectral projections.
Let Pe = eigenprojection associated to eigenval. e of L0.
Rz = (L− z)−1 and Rez = (P⊥e LP⊥e − z)−1 RanP⊥e
Assumption: Limiting Absorption Principle
We assume
supy<0, x≈e
|⟨φ, Re
x+iyψ⟩| ≤ C (φ, ψ) <∞
andsup
y<0, x away fromall e| 〈φ,Rx+iyψ〉 | ≤ C (φ, ψ) <∞
for vectors φ, ψ in a dense set D.
ΠEe := spectral projection of L associated to eigenvalue Ee (near e)
Theorem (Resonance expansion) [Konenberg-Merkli, 2016]
∃V0 > 0 s.t. if 0 < |V | < V0 then ∀t > 0, (weakly on D)
eitL =∑
e partially stable
eitEe ΠEe +
me−1∑j=1
eit(e+ε2ae,j ) Π′e,j
+∑
e unstable
me−1∑j=0
eit(e+ε2ae,j ) Π′e,j + O(1/t).
The exponents ae,j and the projections Π′e,j are close to thespectral data of the level shift operator Λe ,
ae,j = λe,j + O(V ), Π′e,j = Pe,j + O(V ).
III. Application: dynamics of a dimer
We present results for collective (blue) environment model.
Environment D
Collec/ve Environment
Donor D
Acceptor A
Environment A V
H =1
2
(ε VV −ε
)+ HR +
(λD 00 λA
)⊗ φ(g)
HR =
∫R3
ω(k) a∗(k)a(k)d3k
φ(g) =1√2
∫R3
(g(k)a∗(k) + adj.
)d3k
Free bosonic quantum field.
Initial states, reduced dimer stateIntital states unentangled,
ρin = ρS ⊗ ρR
ρS = arbitrary, ρR reservoir equil. state at temp. T = 1/β > 0
Reduced dimer density matrix
ρS(t) = TrReservoir
(e−itHρine
itH)
Dimer site basis ϕ1 =
(10
)and ϕ2 =
(01
).
Donor population
p(t) = 〈ϕ1, ρS(t)ϕ1〉 = [ρS(t)]11, p(0) ∈ [0, 1]
Evolution t 7→ p(t) called relaxation while decoherence isevolution of off-diagonal matrix element
t 7→ [ρS(t)]12 = 〈ϕ1, ρS(t)ϕ2〉
Dynamics for V = 0
I Populations are constant in time (HS commutes with I )
I Total system has 2-dimensional manifold of stationary statesand one equilibrium (KMS) state
ρβ,λ =e−
β2
(ε−αD)
Zren|ϕ1〉〈ϕ1|⊗ρR,1 +
e−β2
(−ε−αA)
Zren|ϕ2〉〈ϕ2|⊗ρR,2
effective energy shifts αD,A ∝ λ2D,A and ρR,j = res. states
I Reduced dimer equilibrium state
ρren = TrR ρβ,λ =e−βHren
Zren, Hren = 1
2
(ε− αD 0
0 −ε− αA
)I Will show: as V 6= 0 (small), total system has a unique stat.
state (=KMS), all initial states converge to it as t →∞.
Reservoir spectral function & dynamics for V 6= 0
• Effect of reservoir on dimer encoded in spectral density
J(ω) =√
2π tanh(βω/2) C(ω), ω ≥ 0,
C(ω) = Fourier transform of reservoir correlation function
• Our resonance expansion requires regularity condition
J(ω) ∼ ωs with s ≥ 3 as ω → 0
J(ω) ∼ ω−σ with σ > 3/2 as ω →∞
(Minimal a ‘priori condition’: s > 1 super-ohmic; range 1 < s < 3 not
treatable up to now.)
Theorem (Population dynamics, relaxation) [M. et al, 2016]Consider the local/collective reservoirs model. Let λD , λA bearbitrary. There is a V0 > 0 s.t. for 0 < |V | < V0:
p(t) = p∞ + e−γt (p(0)− p∞) + O( t1+t2 ),
where
p∞ =1
1 + e−βε+ O(V ) with ε = ε− αD−αA
2
γ = relaxation rate ∝ V 2
αD,A = renormalizations of energies ±ε (∝ λ2D,A)
p∞ = equil. value w.r.t. renormalized dimer energies
Note: Remainder small on time-scale γt << 1, i.e., t << V−2; butcan further expand remainder
Properties of final populations
Final donor population (modulo O(V ) correction)
p∞ ≈1
2− ε
4T, for T >> |ε|.
If donor strongly coupled then ε ∝ −λ2D < 0, so
Increased donor-reservoir coupling increases final donorpopulation
Effect intensifies at lower temperatures
p∞ ≈
1, if λ2D >> maxλ2
A, ε0, if λ2
A >> maxλ2D , ε
for T << |ε|
Acceptor gets entirely populated if it is strongly coupled toreservoir
Expression for relaxation rate
γc = V 2 limr→0+
∫ ∞0
e−rt cos(εt) cos
[(λD − λA)2
πQ1(t)
]× exp
[−(λD − λA)2
πQ2(t)
]dt
where
Q1(t) =
∫ ∞0
J(ω)
ω2sin(ωt) dω,
Q2(t) =
∫ ∞0
J(ω)(1− cos(ωt))
ω2coth(βω/2) dω
This is the Generalized Marcus Formula – in the symmetric caseλD = −λA and at high temperatures, kBT >> ~ωc , it reduces tothe usual Marcus Formula
γMarcus =
(V
2
)2√π
T εrece−
(ε−εrec)2
4Tεrec (0 < εrec ∝ λ2)
Some numerical results
• Accuracy of generalized Marcus formula:– ωc/T . 0.1 rates given by the gen. Marcus formula
coincide extremely well (∼ ±1%) with true values γc,l– ωc/T & 1 get serious deviations (& 30%)
• Asymmetric coupling can significantly increase transfer rate:
Collective: x ∝ λ2D − λ2
A, y ∝ (λD − λA)2 Local: εj ∝ λ2j − λ1λ2
Surfaces = γc,l Red curve = symmetric coupling
Non-interacting dimer V = 0
Populations constant in time and
[ρS(t)]12 = e−it εD(t) [ρS(0)]12
– limt→∞[ρS(t)]12 = 0 called full phase decoherence– limit non-zero called partial phase decoherence
Full decoherence ⇐⇒ low frequency modes well coupled to dimer:
Lemma. Full phase deco. ⇐⇒ J(ω) ∼ ωs with s ≤ 2 (ω → 0)
Graph: D(t) = e−Γ(t)
s = 3
Red: βωc = 0.1Green: βωc = 1
Blue: βωc = 5
Decoherence of the interacting dimer
For s > 2: residual asymptotic coherence limt→∞
D(t) = e−Γ∞ > 0
Theorem (Decoherence) [M. et al, 2016]Consider the local/collective reservoirs model with λD , λAarbitrary. There is a V0 > 0 such that if 0 < |V | < V0, then
[ρS(t)]12 = e−Γ∞ e−γt/2 e−it(ε+xLS) [ρS(0)]12 + O(V ) + O( 11+t ),
where γ is the relaxation rate, xLS ∈ R is the Lamb shift.
• Well-known relation from weak coupling theory (Bloch-Redfield)holds for all coupling strengths:
γdecoherence = γrelaxation/2
• Theorem holds for s ≥ 3: regime of partial deco., Γ∞ <∞• We expect to get rigorous result for s > 1. But if s ≤ 2: Γ∞ =∞,
above expansion not useful, analysis needs modification
IV. Outline: proof of resonance expansion
Situation: L0 is perturbed into L0 + V I s.t.
I all eigenvalues of L0 are embedded
I all eigenvalues of L0 are either unstable or reduce todimension one under perturbation
I the Limiting Absorption Principle holds
Want:
eitL =∑
e part. stable
eitEe ΠEe +
me−1∑j=1
eit(e+ε2ae,j ) Π′e,j
+∑
e unstable
me−1∑j=0
eit(e+ε2ae,j ) Π′e,j + O(1/t)
Resonance data ae,j , Π′e,j obtained by perturbation theory in V
Decomposing resolvent using Feshbach map
• Pe := spectral projection assoc. to eigenvalue e of L0
• Feshbach map: resolvent (L− z)−1 has decomposition
Rz ≡ (L− z)−1 = F−1z + Rz + Bz
where
Fz := Pe
(e − z − V 2I Rz I
)Pe
Rz := (P⊥e LP⊥e − z)−1 RanP⊥e
Bz = −VF−1z I Rz − V Rz IF
−1z + V 2Rz IF
−1z I Rz
F−1z finite-dimensional (acts on RanPe)
Rz dispersive (LAP, purely AC spectrum, time-decay) Bz higher order terms in V plus dispersive
– Standard resolvent representation (any w > 0)
eitLψ =−1
2πi
∫R−iw
eitz (L− z)−1ψ dz
– Divide into regions close to each e and away from all e
• Focus on partially stable eigenvalue e of L0. Assume e = 0 and
that L has a single, simple eigenvalue E = 0 (no shift) for small V .
– Let J be interval around 0 (containing no eigenvalue of L0 but 0)
– Contribution to⟨ϕ, eitLψ
⟩from integral over J∫
J−iweitz 〈ϕ,Rzψ〉 dz
=
∫J−iw
eitz[⟨ϕ,F−1
z ψ⟩
+⟨ϕ, Rzψ
⟩+ 〈ϕ,Bzψ〉
]dz
Contribution of F−1z
Set P ≡ P0 (e = 0). Feshbach map equals
Fz = P(−z − V 2I Rz I )P ≡ −z + V 2Az , z ∈ C−
• Diagonalize operator Az :
Az =d−1∑j=0
aj(z)Qj(z) =⇒ F−1z =
d−1∑j=0
Qj(z)
−z + V 2aj(z)
=⇒∫J−iw
eitz⟨ϕ,F−1
z ψ⟩dz
=d−1∑j=0
∫J−iw
eitz
−z + V 2aj(z)〈ϕ,Qj(z)ψ〉 dz
Az ≈ Level Shift Operator Λ0
Az = −PI (L− z)−1IP = −PI (L0 + i0+)−1IP︸ ︷︷ ︸Level Shift Operator Λ0
+ O(V ) + O(z)
Eigenvalues aj of Az ≈ eigenvalues λj of Λ0
aj(z) = λj + O(V ) + O(z), j = 1, . . . , d − 1
a0(z) = O(z)
(L has simple eigenvalue 0 ⇒ a0(0) = 0 for all V by “isospectrality of
Feshbach map”)
X
X
Ja0
ajX XX
X
∫J−iw
eitz⟨ϕ,F−1
z ψ⟩dz =
d−1∑j=0
∫J−iw
eitz
−z + V 2aj(z)〈ϕ,Qj(z)ψ〉 dz
≈d−1∑j=0
〈ϕ,Qj(0)ψ〉∫J−iw
eitz
−z + V 2aj(0)dz
X
J
a0
Xja
S
T∫S
eitz
−z + V 2aj(0)dz ∼
∫ ∞0
e−ytdy = O(1/t)∫T
eitz
−z + V 2aj(0)dz ∼ 0∮
eitz
−z + V 2aj(0)dz ∼ eitV
2aj (0)
=⇒ 1
2πi
∫J−iw
eitz⟨ϕ,F−1
z ψ⟩dz
= 〈ϕ,Q0(0)ψ〉+d−1∑j=1
eitV2aj (0) 〈ϕ,Qj(0)ψ〉+ O(1/t)
• Part constant in time: spectral proj. of L for eigenvalue 0 is
Π0 = limV→0+
(iV )(L− 0 + iV )−1
By Feshbach decomposition of resolvent get
〈ϕ,Q0(0)ψ〉 = 〈ϕ,Π0ψ〉+ O(V )
Contributions from Bz will add up precisely to give this O(V )
• Decaying parts: rate given by “Fermi Golden Rule”
eitV2aj (0) = eitV
2[λj+O(V )]
Contributions of Bz and Rz
• Obtain ∫J−iw
eitz⟨ϕ, Rzψ
⟩dz = O(1/t)
by integrating by parts w.r.t. z and using LAP for Rz to show thatsupz∈C−
∣∣ ddz
⟨ϕ, Rzψ
⟩ ∣∣ ≤ C .
• To treat ∫J−iw
eitz 〈ϕ,Bzψ〉 dz
Bz = −VF−1z I Rz − V Rz IF
−1z + V 2Rz IF
−1z I Rz use again spectral
representation of F−1z =⇒ get corrections (to all orders in V ) of
contributions coming from F−1z
Summary
• We develop a resonance expansion for the reduced dynamicsof a dimer strongly coupled to reservoirs.
• Technical novelty: Analytic spectral deformation theory doesnot apply, so we extract decay times and directions fromMourre theory, and for ‘singular perturbations’.
( 6←→ Cattaneo-Graf-Hunziker ‘06Bach-Frohlich-Sigal, Bach-Frohlich-Sigal-Soffer,Derezinski-Jaksic, DeRoeck-Kupiainen,
Faupin-Møller-Skibsted, Jaksic-Pillet)
• Establish generalized Marcus formula for donor-acceptorreaction rate which reduces to known formula in usual settingbut also describes new physical regime (faster reaction rate,
different final population values for asymmetrically coupled dimer).