How the environment affects the optical response of a ... · Time-domain theoretical model Real-time approach: System in the ground state and subject to a time-dependent potential

How the environment affects the opticalresponse of a molecule? Application to

nanoparticle-chromophore systems and tofluorophores

Emanuele Coccia

Centro S3, CNR Istituto di Nanoscienze, Modena

Nano Colloquia

E. Coccia (CNR) Oct 5 2017, Modena 1 / 29

Outline

1 Part I: Plasmonic control of molecular absorption

2 Part II: fs pulse-shaping spectroscopy


Time-domain theoretical model

Real-time approach:

System in the ground state and subject to a time-dependentpotential (switched on at time t0)

Time-evolution of populations, energies, dipoles etc.


Part I:Plasmonic control of molecular

absorption


Molecular plasmonics

Molecule-nanoparticle (NP) interaction (SERS, metalenhanced fluorescence or quenching...)Molecular response:

Applied electromagnetic fieldNP shape, nature and sizeMutual orientation and⇒ distance⇐⇒ Surrounding environment⇐


NP-induced enhancement vs quenching

Molecular absorption and emission affected by NP

~µtotq0 = ~µmol

q0 + ~µNPq0

⇒Absorption⇐:A ∝ |~µtot

q0 |2

Emission:

Γrad ∝ ω3q0|~µtot

q0 |2

Nonradiative relaxation channel:energy transfer molecule→ NP

Γnonrad = Γnonradmet + Γnonrad

0

Exp: E. Dulkeit et al., Phys. Rev. Lett., 89 203002 (2002)Exp: E. Dulkeit et al., Nano Lett., 5 585 (2005)Theory: M. Caricato, O. Andreussi and S. Corni J. Phys. Chem. B, 110 16652 (2006)Theory: S. Vukovic, S. Corni and B. Mennucci J. Phys. Chem. C, 113 121 (2008)


Molecule and NP: multiscale model

System: molecule+NPS. Pipolo and S. Corni, J. Phys. Chem. C, 120 28774 (2016)

Effects of the environment (containing molecule+NP):⇒ relaxation and dephasing⇐


Open quantum systems

Total Hamiltonian H by considering system (S) and bath (B):

H(t) = HS(t)⊗ IB + IS ⊗ HB + HSB

HSB =∑

q

Sq ⊗ Bq

HS(t) = H0 − ~µ · ~E(t)

Effective treatment of the bath→ reduced density matrix ρS = TrB ρE. Coccia (CNR) Oct 5 2017, Modena 8 / 29

Lindblad master equation for ρS

Weak interaction between system and bathMarkovian limitLindblad master equation:

d

dtρS = −i[HS , ρS ] + L

L = −12

∑q

S†q Sq, ρS+∑

q

SqρSS†q

From ρS to |ΨS〉 → stochastic Schrodinger equation

H.-P. Breuer and F. Petruccione, The theory of open quantum systems, OUP Oxford (2007)


Stochastic Schrodinger equation (SSE)

SSE: approach equivalent to the master equation for ρS

idt|ΨS(t)〉 = HS(t)|ΨS(t)〉+∑

q

lq(t)Sq|ΨS(t)〉 − i

2

∑q

S†q Sq|ΨS(t)〉

SSE in Markovian limitDissipation: − i

2

∑q S†q Sq

(nonHermitian)Fluctuation:

∑q lq(t)Sq (stochastic)

t

lq(t)

Computational features:Cost proportional to Nstates (instead of N2

states for ρS)Average over M independent trajs→ trivial parallelization

Implementation: Dissipative dynamics + random jumpsK. Mølmer, Y. Castin and J. Dalibard, J. Opt. Soc. Am. B, 10 524 (1993)R. Biele and R. D’Agosta, J. Phys.: Condens. Matter, 24 273201(2012)


Quantum jump algorithm

“Dissipative” Hamiltonian

Hdis(t) = HS(t)− i

2

∑q

S†q Sq

Time-step δt discretisationWave function norm at first order in δt

〈ΨS(t + δt)|ΨS(t+ δt)〉 = 1−∆p

∆p = δt∑

q

〈ΨS(t)|S†q Sq|ΨS(t)〉 =∑

q

∆pq

∆pq = ∆t〈ΨS(t)|S†q Sq|ΨS(t)〉

K. Mølmer, Y. Castin and J. Dalibard, J. Opt. Soc. Am. B, 10 524 (1993)EC and S. Corni, in preparation



∆p compared at each step with an uniform random numberε ∈ [0,1]:

if ∆p < ε no quantum jump occurs, and the wave function isnormalizedif ∆p ≥ ε, a quantum jump occurs, and the new function isdefined as

|ΨS(t+ δt)〉 =Sq|ΨS(t)〉√

∆pq/δt

with probability given by ∆pq

∆p

K. Mølmer, Y. Castin and J. Dalibard, J. Opt. Soc. Am. B, 10 524 (1993)EC and S. Corni, in preparation


Interaction channels

|ΨS(t)〉 =Nstates∑

q

Cq(t)|φq〉, (ρS)kl = C∗kCl

Relaxation: Srelaxq =

√Γq|φ0〉〈φq|

Spontaneous emission and nonradiative relaxationPopulation |Cq(t)|2 exponentially decays with Γq

Dephasing:

|ΨS(t)〉 = C0(t)|φ0〉+ C1(t)|φ1〉+ ...

Sdeph0 |ΨS(t)〉 ∝

√γ0/2 (−C0(t)|φ0〉+ C1(t)|φ1〉+ ...

Sdeph1 |ΨS(t)〉 ∝

√γ1/2 (C0(t)|φ0〉−C1(t)|φ1〉+ ...

...

Decay rate of coherence 〈C∗k(t)Cl(t)〉 equal to γk + γlEC and S. Corni, in preparation


Testing the model: LiCN close to a silver NP

LiCN as model for our workδ pulse linearly polarised along y

Spherical silver NP of radius 5 nm

LiCN: CIS expansion, 6-31G(d) basis set (same setup as in J. Chem. Phys., 129 084302 (2008))


How to describe NP

Real-time description:Simulation of short pulsesTime evolution of properties of interestNonlinear optical properties of the molecule

Drude dielectric function

ε(ω) = 1−Ω2

p

ω2 + iγω

But: generic ε(ω) are also possible

Molecule+NP with time-dependent BEMS. Pipolo and S. Corni, J. Phys. Chem. C, 120 28774 (2016)


LiCN+NP

11 CI states considered (beyond two-state model)

Focus on states |2 > and |3 > (degenerate)

Nonradiative relaxation time 1 ps

Pure dephasing time T2 50 fs

LiCN No relax and dephasing

LiCN+NP(D=4 nm) No relax and dephasing

0 100 200 300 400Time (as)

Elec

tric

field

(a.u

.)

I = 0.1 W/cm2

LiCN+NP(D=4 nm) + relax and dephasing

EC and S. Corni, in preparation


Population for LiCN+NP+envPopulation |2〉+ |3〉 with δ pulseTotal transition dipole moment

~µtotq0 = ~µmol

q0 + ~µNPq0

Absorption ∝ |~µtotq0 |2

0 1 2 3 4 5 6 7 8 9 10D (nm)

0

5e-19

1e-18

1.5e-18

2e-18

2.5e-18

3e-18

3.5e-18

LiCN+NP+env

no NP

500 fs



Population for LiCN+NP+envPopulation |2〉+ |3〉 with δ pulseTotal transition dipole moment

~µtotq0 = ~µmol

q0 + ~µNPq0

Absorption ∝ |~µtotq0 |2

0 1 2 3 4 5 6 7 8 9 10D (nm)

0

5e-19

1e-18

1.5e-18

2e-18

2.5e-18

3e-18

3.5e-18

LiCN+NP+envLiCN+NP

no NP

no NP500 fs



T2 vs Tplas

Plasmon lifetime Tplas ∼ 50− 100 fsInterference with dephasing time T2 (5 and 50 fs, 1 ps)



Part I: conclusions

Developed a multiscale approach that also includesrelaxation and dephasing:Molecule: ab initioNP: classical electrodynamicsRelaxation and dephasing: SSE

Perspectives:

Continuous SSE: quantum state diffusion modelBeyond Markov limit→ relaxation and dephasing directlyfrom NP modelNP shape effects on electronic properties


Part II:fs pulse-shaping spectroscopy


fs pulse-shaping spectroscopy

Control of electronic coherences in single molecules

Variations of electronic coherences↔ changes of the emission

Phase memory broken by dephasing (threshold delay time)

Single molecule

Excita0on

Δt, ΔΦ = 0

NO ENV

ENV (dephasing T2)

Emission indipendent

of Δt

Emission decreases with Δt

Δt=const,ΔΦ=π

Emission suppressed

Emission (par0ally) suppressed

Δt, ΔΦ = 0 Δt<T2,ΔΦ=π

D. Brinks et al., Chem. Soc. Rev., 43 2476 (2014)



Single terrylenediimide (TDI) molecule: pure electronic transition

Distinct emission responses with ∆t and ∆φ

Theory: SSE

TDI$

Environment Dephasing

Absorp3on/emission

Calc.: B3LYP structure, CIS(D) energies, CIS transition dipoles, 11 statesExp.: R. Hildner, D. Brinks and N. F. van Hulst, Nature Phys., 7 172 (2011)



0 100 200 300 400 500 600Delay time (fs)

0

5e-06

1e-05

1.5e-05

2e-05

Exci

ted

stat

e po

pula

tion

at 1

ps

∆φ = 0∆φ = π

I = 5 kW/cm2

FWHM = 70 fsλ = 501 nmT1 = 3.5 ns

T2 = 30 fs

Calc.CIS(D)

0 100 200 300 400 500 600Delay time (fs)

0

5e-06

1e-05

1.5e-05

2e-05

Exci

ted

stat

e po

pula

tion

at 1

ps ∆φ = 0

∆φ = π

T2 = 120 fs

Calc.CIS(D)

0 50 100 150 200 250 300 350 400 450 500 550 600Delay time (fs)

0

5e-06

1e-05

1.5e-05

2e-05

Exci

ted

stat

e po

pula

tion

at 1

ps ∆φ = 0

∆φ = π ∆φ = 0 no dephasing∆φ = π no dephasing

T2 = 60 fs

Calc.CIS(D)

Calc.: EC and S. Corni, in preparationExp.: R. Hildner, D. Brinks and N. F. van Hulst, Nature Phys., 7 172 (2011)


Effect of the detuning

Detuning δ = |ωpulse − ωTDI|

Experimental crossing: ∼ 110− 125 fs

0 50 100 150 200 250 300 350 400 450 500 550 600Delay time (fs)

0

5e-06

1e-05

1.5e-05

2e-05

Exci

ted

stat

e po

pula

tion

at 1

ps ∆φ = 0

∆φ = π

T2 = 60 fs

Calc.CIS(D)



Effect of the detuning

Detuning δ = |ωpulse − ωTDI|

Experimental crossing: ∼ 110− 125 fs

0 50 100 150 200 250 300 350 400 450 500 550 600Delay time (fs)

0

5e-06

1e-05

1.5e-05

2e-05

Exci

ted

stat

e po

pula

tion

at 1

ps ∆φ = 0

∆φ = π ∆φ = 0 δ = 80 cm-1

∆φ = π δ = 80 cm-1

T2 = 60 fs

Calc.CIS(D)

δ



Vibrational signatures: the DN-QDI fluorophore

Single-molecule exampleDominant frequency at around 1000 cm−1

Ultrafast response: positions, widths and strengths of the vibrationallines

D. Brinks al., Nature, 465, 905 (2010)


Ongoing: including vibrational levels

k normal modes, Nvib levels

|v〉 = |v1〉 ⊗ |v2〉 ⊗ |v3〉 ⊗ ...|vk〉

Harmonic approximation(Nvib)k levels for each electronic state (SSE efficient choice!)Correct transition dipole moments with Franck-CondonfactorsJ.-L. Chang, J. Mol. Spectr., 232, 102 (2005)

Add nonadiabatic coupling for decay rates?


Two-color experiment in LH2

Oscillations due to quantuminterference between twopathways that populate theB850 levelCoherence induced by thecoupling J

Open questionR. Hildner et al., Science, 340, 1448 (2013)


Part II: conclusions

Dephasing introduced using SSEDephasing induced by the environment essential toreproduce experimentsRole of detuning δ

Perspectives:Molecular dynamics for sampling configurations?Including vibrational levels

DN-QDI, D. Brinks al., Nature, 465, 7300 (2010)

Vibronic correction to relaxation and dephasing


Acknowledgements

Prof. Stefano Corni (University of Padova & CNR)Jacopo Fregoni (Unimore & CNR)

Fundings:

http://www.tame-plasmons.eu/


Part I:Plasmonic control of molecular

absorption


Lindblad master equation

Effective treatment of the bath

ρS = TrB ρ ρS → reduced density matrix

Weak interaction between system and bathFrom ρS to |ΨS〉 → stochastic Schrodinger equationMarkovian limit→ thermalization time scales of S (fast) and B(slow) decoupledWeak coupling between system and bathLindblad master equation:

d

dtρS = −i[HS , ρS ] + L

L = −12

∑q

S†q Sq, ρS+∑

q

SqρSS†q

Sq defines the q-th channelρS proportional to N2

states


CI expansion of the wave function

|ΨS(t)〉 =N∑m

Cm(t)|φm〉

Cm(t) time-dependent coefficients|φm〉 m-th CI eigenstate

For sake of clarity

HSSE(t) ≡ HS(t) + α∑

q

lq(t)Sq − iα2

2

∑q

S†q Sq

Matrix form of the SSE

i∂C(t)∂t

= HSSEC(t)

Propagation via second-order Euler:

C(t+ ∆t) = C(t−∆t)− 2i∆tHSSE(t)C(t)

(ρS)kl = C∗k(t)Cl(t)E. Coccia (CNR) Oct 5 2017, Modena 32 / 29

Pure dephasing

Sdephq =

√γq/2

N∑p

M(p, q)|φp〉〈φp|

M(p, q) = 1 if p 6= q

M(p, q) = −1 if p = q



Deterministic evolution + random jumps

Hqjump(t) = HS(t)− i

2

∑q

S†q Sq

At first order in ∆t

〈ΨS(t + ∆t)|ΨS(t+ ∆t)〉 = 1−∆p

∆p = ∆t∑

q

〈ΨS(t)|S†q Sq|ΨS(t)〉 =∑

q

∆pq

∆pq = ∆t〈ΨS(t)|S†q Sq|ΨS(t)〉



∆p compared at each step with an uniform random numberε ∈ [0,1]:

if ∆p < ε no quantum jump occurs, and the wave function isnormalized;if ∆p ≥ ε, a quantum jump occurs, and the new function isdefined as the following

|ΨS(t+ ∆t)〉 =Sq|ΨS(t)〉√

∆pq/∆t

with probability given by ∆pq

∆p


Absorption

0 0.1 0.2 0.3 0.4 0.5Intensity (W/cm2)

0

1e-14

2e-14

3e-14

4e-14

5e-14

pop |2> SSE at 25 fsLinear regressionpop |2> no SSE at 25 fsLinear regression


Absorption

0.2 0.22 0.24 0.26 0.28 0.3ω

0

2e-15

4e-15

6e-15

8e-15

popula

tion |2

>

no SSESSE

0.25 0.275ω

0


LiCN+NP

0 1 2 3 4 5 6 7 8 9 10D (nm)

Gro

und-

stat

e en

ergy

(Har

tree)

Isolated LiCN



LiCN+NP

0 1 2 3 4 5 6 7 8 9 10D (nm)

0.2415

0.2416

0.2417

0.2418

0.2419

0.242

0.2421

0.2422

0.2423

0.2424

Exci

tatio

n en

ergy

0->

2 (H

artre

e)

Isolated LiCN



Population for LiCN+NP

0 200 400 600 800 1000Time (fs)

0

5e-18

1e-17

Popu

latio

n (|2

>+|3

>)

D=0.3 nmD=0.6 nmD=1 nmD=2 nmD=4 nmD=6 nmD=10 nmisolated LiCN

No environmentδ pulse



Coherence for LiCN+NP+env

0 50 100 150 200Time (fs)

0

1e-09

2e-09

3e-09

4e-09

5e-09

|2><

0| +

|3><

0|

D=0.3 nmD=0.6 nmD=1 nmD=2 nmD=4 nmD=6 nmD=10 nmisolated LiCN



Population for LiCN+NP+env

0 100 200 300 400 500 600 700 800 900Time (fs)

0

2e-15

4e-15

6e-15

8e-15

popu

latio

n (|2

>+|3

>)

D=0.3 nmD=0.6 nmD=1 nmD=2 nmD=4 nmD=6 nmisolated LiCN

ω pulse0->2 transition of isolated LiCN



Population for LiCN+NP+env

0 1 2 3 4 5 6 7 8D (nm)

0

2e-15

4e-15 LiCN+NP+env

Population (|2> + |3>) at 500 fsω pulse

LiCN+env (no NP)



Part II:fs pulse-shaping spectroscopy


Bloch equations

Two levels |0〉 and |1〉

µ = −(1T2− iω10)µ+ iΩ∗eiωtv

v = −(ω − ω0)T1

+ 2iΩe−iωtµ− 2iΩ∗eiωtµ

v = ρ11 − ρ00

ω10 = E1 − E0

µ = 〈1|µ|0〉ρ10 + cc


Fs pulse-shaping spectroscopy (T2 = 60 fs)

0 100 200 300 400 500 600 700 800 900 1000Time (fs)

0

5e-06

1e-05

1.5e-05

2e-05

Popu

latio

n

∆t = 0 fs∆t = 10 fs∆t = 50 fs∆t = 100 fs∆t = 200 fs∆t = 400 fs∆t = 600 fs

∆φ = 0




0 100 200 300 400 500 600 700 800 900 1000Time (fs)

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

|1><

0|


∆φ = 0




0 100 200 300 400 500 600 700 800 900 1000Time (fs)

0

0.0005

0.001

0.0015

0.002

|1><

0|


∆φ = π



Fs pulse-shaping spectroscopy

0 100 200 300 400 500 600Delay time (fs)

0

0.0001

0.0002

0.0003

0.0004

0.0005

|1><

0| c

oher

ence

at 1

ps

∆φ = 0∆φ = π

I = 5 kW/cm2

FWHM = 70 fs

λ = 501 nm

T1 = 3.5 nsT2 = 60 fs




0 100 200 300 400 500 600Delay time (fs)

0

0.0001

0.0002

0.0003

0.0004

0.0005

|1><

0| c

oher

ence

at 1

ps

∆φ = 0∆φ = π

I = 5 kW/cm2

FWHM = 70 fs

λ = 501 nm

T1 = 3.5 nsT2 = 30 fs




0 100 200 300 400 500 600Delay time (fs)

0

0.0001

0.0002

0.0003

0.0004

0.0005

|1><

0| c

oher

ence

at 1

ps

∆φ = 0∆φ = π

I = 5 kW/cm2

FWHM = 70 fs

λ = 501 nm

T1 = 3.5 nsT2 = 120 fs



Documents

How the environment affects the optical response of a ... · Time-domain theoretical model Real-time approach: System in the ground state and subject to a time-dependent potential