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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
E. Coccia (CNR) Oct 5 2017, Modena 2 / 29
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.
E. Coccia (CNR) Oct 5 2017, Modena 3 / 29
Part I:Plasmonic control of molecular
absorption
E. Coccia (CNR) Oct 5 2017, Modena 4 / 29
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⇐
E. Coccia (CNR) Oct 5 2017, Modena 5 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 6 / 29
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⇐
E. Coccia (CNR) Oct 5 2017, Modena 7 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 9 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 10 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 11 / 29
Quantum jump algorithm
∆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
E. Coccia (CNR) Oct 5 2017, Modena 12 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 13 / 29
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))
E. Coccia (CNR) Oct 5 2017, Modena 14 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 15 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 16 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 17 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 17 / 29
T2 vs Tplas
Plasmon lifetime Tplas ∼ 50− 100 fsInterference with dephasing time T2 (5 and 50 fs, 1 ps)
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 18 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 19 / 29
Part II:fs pulse-shaping spectroscopy
E. Coccia (CNR) Oct 5 2017, Modena 20 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 21 / 29
fs pulse-shaping spectroscopy
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)
E. Coccia (CNR) Oct 5 2017, Modena 22 / 29
fs pulse-shaping spectroscopy
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)
E. Coccia (CNR) Oct 5 2017, Modena 23 / 29
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)
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 24 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 24 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 25 / 29
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?
E. Coccia (CNR) Oct 5 2017, Modena 26 / 29
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)
E. Coccia (CNR) Oct 5 2017, Modena 27 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 28 / 29
Acknowledgements
Prof. Stefano Corni (University of Padova & CNR)Jacopo Fregoni (Unimore & CNR)
Fundings:
http://www.tame-plasmons.eu/
E. Coccia (CNR) Oct 5 2017, Modena 29 / 29
Part I:Plasmonic control of molecular
absorption
E. Coccia (CNR) Oct 5 2017, Modena 30 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 31 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 33 / 29
Quantum jump algorithm
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)〉
E. Coccia (CNR) Oct 5 2017, Modena 34 / 29
Quantum jump algorithm
∆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
E. Coccia (CNR) Oct 5 2017, Modena 35 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 36 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 37 / 29
LiCN+NP
0 1 2 3 4 5 6 7 8 9 10D (nm)
Gro
und-
stat
e en
ergy
(Har
tree)
Isolated LiCN
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 38 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 39 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 40 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 41 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 42 / 29
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)
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 43 / 29
Part II:fs pulse-shaping spectroscopy
E. Coccia (CNR) Oct 5 2017, Modena 44 / 29
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
E. Coccia (CNR) Oct 5 2017, Modena 45 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 46 / 29
Fs pulse-shaping spectroscopy (T2 = 60 fs)
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|
∆t = 0 fs∆t = 10 fs∆t = 50 fs∆t = 100 fs∆t = 200 fs∆t = 400 fs∆t = 600 fs
∆φ = 0
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 47 / 29
Fs pulse-shaping spectroscopy (T2 = 60 fs)
0 100 200 300 400 500 600 700 800 900 1000Time (fs)
0
0.0005
0.001
0.0015
0.002
|1><
0|
∆t = 0 fs∆t = 10 fs∆t = 50 fs∆t = 100 fs∆t = 200 fs∆t = 400 fs∆t = 600 fs
∆φ = π
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 48 / 29
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
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 49 / 29
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 = 30 fs
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 50 / 29
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 = 120 fs
EC and S. Corni, in preparation
E. Coccia (CNR) Oct 5 2017, Modena 51 / 29