Resonance Theory for the Dynamics of OpenDimers

Marco Merkli

Memorial University, St. John’s, Canada

BCAM, BilbaoApril 11, 2017

Based on collaborations with

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

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)

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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

Principles of quantum theory

– The state ψ of a system is given by a normalized vector in aHilbert space H (which one?); ψ = wave function.

– The observables of a system are operators A acting on H(which ones?).

– When a measurement of an observable A is made, the onlypossible measurement outcomes are its eigenvalues {a1, . . . , an}. Ifthe system is in the state ψ, then a measurement of A yields theoutcome ak with probability pk = ‖χA=akψ‖2.

– The evolution of a quantum system is given by a path t 7→ ψt inHilbert space. It obeys the Schrodinger equation:

i~d

dtψt = Hψt

H is the energy observable, a distinguished observable called theHamiltonian.

Examples

• A spin

H = C2, ψ = c1|↑〉+ c2|↓〉, H =

(1 00 −1

)• A particle moving in 3-dimensional space

H = L2(R3, d3x), ψ = ψ(x), H = −∆ + V (x)

• A quantum oscillator

H = `2(N), ψ = {ψn}n∈N, H = a∗a

Eigenvalues of H are {0, 1, 2, . . .} with associated eigenvectors|n〉. Creation operator a∗|n〉 ∝ |n + 1〉 creates excitation.

• A quantum field is a system of (uncountably many) quantumoscillators, one at each space location x ∈ R3 (with associatedcreation operator a∗(x)).

Irreversibility (?)

– Any system having discrete energy spectrum

spec(H) = {Ej}j∈N

exhibits quasi-periodic motion: Schrodinger equation ⇒

ψt = e−itHψ0 =∞∑j=1

e−itEjPjψ0

oscillates (Ej ∈ R).

– In nature, irreversible effects occurall the time. For instance, the excitationtransfer in photosynthesis is irreversible!

– To describe irreversibility need to take a limit of continuousenergies.

• Generally: the larger a system, the smaller its energy gaps

E.g. A particle in a box of side length L has energy gaps ∝ 1/L2

irreversibility ⇔ possibility to escape to spatial ∞⇒ continuous energies come from thermodynamic limit (L→∞)

• This limit often causes all kinds ofmathematical trouble...

...because

as L→∞, Hilbert space is lost !

→ Need to formulate theory purely in terms of observables (algebraicquantum field theory)

→ Need to reconstruct Hilbert space after (GNS representation ofoperator algebras)

→ Need to link dynamics to spectrum of some (new) Hamiltonian acting

on (new) Hilbert space

Open quantum systems

– A system S is called open if itis connected with a heat bath /environment / reservoir R.

– The total complex, S + R is closed.

– The Hilbert space of the closed system S + R is HS ⊗HR, itsdynamics is given by the Schrodinger equation with Hamiltonian

H = HS + HR + HSR

HS, HR = Hamiltonians of the isolated S, RHSR = interaction Hamiltonian (interaction energy).

– One is interested in properties (observables, state) of S alone,and those properties depend on the interaction with R.

– Suppose initial state of S + R of form ψS ⊗ ψR.

– Principles of quantum mechanics ⇒ average of systemobservable A at time t is

〈A〉t =⟨ψS ⊗ ψR,

{eitH(A⊗ 1R)e−itH

}ψS ⊗ ψR

⟩.

– It defines the reduced system density matrix ρS(t) by

〈A〉t = TrS

(ρS(t)A

).

– ρS(t) is operator acting on HS alone, but encodes allenvironment effects on S.

– Not surprisingly: The map t 7→ ρS(t) complicated !

– Main task of theory of open quantum syst.: Examine this map.

Back to Donor-Acceptor systems: Marcus approach

R = reactant (donor)P = product (acceptor)

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HMarcus = |R〉ER〈R|+ |P〉EP〈P|+ |R〉V 〈P|+ |P〉V 〈R|

=

(ER VV EP

)Marcus Hamiltonian

• Collective reactant/product energies (harmonic oscillators)

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

operators HR 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 and at high temperature

V << ε, kBT >> ~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 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 acceptorto environment and/or independent environments

II. Resonance expansion

Spectral deformation.....or not!

∗ Take a Hamiltonian L (e.g. the Spin-Boson Hamiltonian)∗ Time evolution of an initial state ψ0:

ψt = eitLψ0

∗ Want to find suitable representation of the propagator eitL

∗ If z 7→ f (z) ≡ 〈ψ, (L− z)−1φ〉 has meromorphic continuation,

〈ψ, eitLφ〉 =−1

2πi

∫Γ

eitz f (z)dz =∑

poles a

eitaf (a) + O(e−αt

)

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’  

>  

∗ This is called spectral deformation method and works well forSpin-Boson system when system-reservoir coupling λ is small.

∗ Do not know how to obtain meromorphic continuation in regimeof large λ ! (Lack of regularity.)

∗ How then can we construct decay times and directions forsystems with less regularity?

∗ Task: develop method using only mild regularity condition

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

called Limiting Absorption Principle (LAP).

(Which does hold for Spin-Boson with large coupling!)

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 − V 2Pe IP⊥e (L0 − e)−1IPe + O(V 3)

I Since e is actually embedded eigenvalue, the resolvent

P⊥e (L0 − e)−1 does not exist.

But we can expect the 2nd order corrections (∝ V 2) to belinked 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)

We assume that Λe is diagonalizable:

Λe =me−1∑j=0

λe,jPe,j

λe,j = eigenvalues, Pe,j = (rank one) spectral projections

Moreover, we assume that:

e unstable eigenvalue ⇒ Imλe,j > 0 for all je partially stable ⇒ Imλe,0 = 0 and Imλe,j > 0 for all j 6= 0

These hypotheses are readily verified in many applications.

Assumption: Limiting Absorption Principle(Dispersiveness due to continuous spectrum)

Let Pe = eigenprojection associated to eigenvalue e of L0.

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

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.

Gure emaitza nagusiak:

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.

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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 state

Intital 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)

Donor population = (1, 1) density matrix element

p(t) = 〈ϕ1, ρS(t)ϕ1〉 = [ρS(t)]11, p(0) ∈ [0, 1]

Goal: Find t 7→ p(t) ( & derive Marcus formula!)

Reservoir spectral function & dynamics

• General mechanism (that we show)

For V = 0, S+R has two-dimensional manifold of stationary states

For V 6= 0 this collapses to single stationary (equilibrium) state- equilibrium state is dynamically attractive- detailed approach to equilibrium given by resonance data

• 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 →∞

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 >> max{λ2

A, ε}) then ε ∝ −λ2D , so

Increased donor-reservoir coupling increases final donorpopulation

Effect intensifies at lower temperatures

p∞ ≈{

1, if λ2D >> max{λ2

A, ε}0, if λ2

A >> max{λ2D , 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.

Eskerrik zure arreta mota asko!