Resonance Theory for the Dynamics of OpenDimers

Marco Merkli

Memorial University, St. John’s, Canada

Seminaire Phys Math, Universite de Lille21 mars 2017

Based on 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 (∼ 1 nanosecond) [or theexcitation is lost in thermal environment]

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 the rates 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(qα − 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) but is heuristic

• 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 dynamics of weakly coupled dimer (V << ε)valid for all times and any reservoir coupling strength (λ)

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)? Happens in systems of interest!– Then cannot extend to second Riemann sheet to access poles.– How can we construct decay times and directions then?

• Task: develop method using only mild regularity condition:

z 7→ 〈ψ, (L− z)−1φ〉 bounded as Imz ↑ 0 (LAP)

Spec. defo. OK for spin-boson at weak coupling (λ small)

λ = 0: system and reservoir uncoupled, dynamics factorizes λ 6= 0: Uθ = spectral 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

Spec. defo. NOT OK for spin-boson at large λ

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 here !

Using Mourre theory instead of spectral deformation:

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 )

0XX X

Spec(L 0 )

0 |V| > 0

Two main technical tools in proof:

1. Positive commutator methods to show instability of eigenvaluesassuming an effective coupling “Fermi Golden Rule” condition

2. Limiting Absorption Principle (LAP) to show AC spectrum

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, 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 instable 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.

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 eigenvalue e of L0.

Rz = (L− z)−1 and RPez = (P⊥e LP⊥e − z)−1 RanP⊥e

Assumption: Limiting Absorption Principle

We assume

supy<0, x≈e

|⟨φ,RPe

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]

∃c > 0 s.t. if 0 < |V | < c then ∀t > 0,

eitL =∑

e partially stable

eitEe ΠEe +

me−1∑j=1

eit(e+V 2ae,j ) Π′e,j

+∑

e unstable

me−1∑j=0

eit(e+V 2ae,j ) Π′e,j + O(1/t)

(weakly on D). The exponents ae,j and the operators Π′e,j are closeto the spectral 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 fields

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 = explicit

reservoir 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− α1−α2

2

γ = relaxation rate ∝ V 2

(different values for local and collective cases)

α1,2 = renormalizations of energies ±ε (∝ λ21,2)

p∞ = equil. value w.r.t. renormalized dimer energies

Note: Remainder small on time-scale γt << 1, i.e., t << V−2

Properties of final populations

Final donor population (modulo O(V ) correction)

p∞ ≈1

2− ε

4T, for T >> |ε|.

where ε := ε− αD−αA2 is effective energy gap. If donor strongly

coupled (λ2D >> maxλ2

A, ε) then ε ∝ −λ2D , 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 a 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 ∝ λ21 − λ2

2, y ∝ (λ1 − λ2)2 Local: εj ∝ λ2j − λ1λ2

Surfaces = γc,l Red curve = symmetric coupling

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

X

Spec(L )

0XX X

Spec(L 0 )

0 |V| > 0

Want:

eitL =∑

e part. stable

eitEe ΠEe +

me−1∑j=1

eit(e+V 2ae,j ) Π′e,j

+∑

e unstable

me−1∑j=0

eit(e+V 2ae,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

– Subdivide integration into regions close to e and away from 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

-iw

∫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 projection 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 dynamics of adimer strongly coupled to reservoirs.

• Technical novelty: Analytic spectral deformation theory doesnot apply, so we build singular Mourre theory (very ‘irregular’operators) and extract decay times and directions from it.

• Establish generalized Marcus formula for donor-acceptorreaction rate, uncovering physical properties (e.g. populationvalues) different from the previously known usual formula.

de votre attention!

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 λ1, λ2 arbitrary.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