Multidimensional Coherent Spectroscopy of Molecular Polaritons Langevin Approach Zhedong Zhang1 2Xiaoyu Nie3Dangyuan Lei4and Shaul Mukamel5 6 1Department of Physics City University of Hong Kong Kowloon Hong Kong SAR

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Multidimensional Coherent Spectroscopy of Molecular Polaritons: Langevin Approach

Zhedong Zhang,1, 2, ∗Xiaoyu Nie,3Dangyuan Lei,4and Shaul Mukamel5, 6, †

1Department of Physics, City University of Hong Kong, Kowloon, Hong Kong SAR

2City University of Hong Kong, Shenzhen Research Institute, Shenzhen 518057, Guangdong, China

3Centre for Quantum Technologies, National University of Singapore, Singapore 117543

4Department of Materials Science and Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR

5Department of Chemistry, University of California Irvine, Irvine, California 92697, United States

6Department of Physics and Astronomy, University of California Irvine, Irvine, California 92697, United States

(Dated: October 25, 2022)

We present a microscopic theory for nonlinear optical spectroscopy of Nmolecules in an optical cavity. A

quantum Langevin analytical expression is derived for the time- and frequency-resolved signals accounting for

arbitrary numbers of vibrational excitations. We identify clear signatures of the polariton-polaron interaction

from multidimensional projections of the signal, e.g., pathways and timescales. Cooperative dynamics of cavity

polaritons against intramolecular vibrations is revealed, along with a cross talk between long-range coherence

and vibronic coupling that may lead to localization eﬀects. Our results further characterize the polaritonic

coherence and the population transfer that is slower.

Introduction.–Strong molecule-photon interaction has

drawn considerable attention in recent study of molecular

spectroscopy. New relaxation channels have been demon-

strated to control fast electron dynamics and reaction activity

[1–9]. Optical cavities create hybrid states between molecules

and conﬁned photons, known as polaritons [10–14]. Theoret-

ically, this requires a substantial generalization of quantum

electrodynamics (QED) into molecules containing many

more degrees of freedom than atoms and qubits.

It has been demonstrated that light in a conﬁned geome-

try can signiﬁcantly alter the molecular absorption and emis-

sion signals [15–17]. The collective interaction between ex-

citations of many molecules and photons is of fundamental

importance, leading to interesting phenomena, e.g., superra-

diance and cooperative dynamics of polaritons [18–23]. In

contrast to atoms whereby superradiance and cavity polaritons

are well understood, molecular polaritons are more complex

in theory and experiments. This arises from the complicated

couplings between electronic and nuclear degrees of freedom,

which possess new challenges for optical spectroscopy. Re-

cently the absorption and ﬂuorescence spectra are described

by Holstein-Tavis-Cummings model [24, 25]. Exact diago-

nalization of the full Hamiltonian was used to calculate the

optical responses, by only taking a few vibrational excita-

tions into account [11, 26]. Here we focus on the polaritonic

relaxation pathways involving the population and coherence

dynamics, which are however open issues. Ultrafast spec-

troscopic technique has been used to monitor the dynamics

of vibrational polaritons [22, 27, 28]. Time- and- frequency-

gated photon-coincidence counting was employed to monitor

the many-body dynamics of cavity polaritons, making use of

nonlinear interferometry [28, 29]. Polaritons reveal the eﬀects

of strongly modifying the energy harvesting and migration in

chromophore aggregates, through novel control knobs not ac-

cessible by classical light [7, 13, 30–32]. Elaborate nonlinear

optical measurements of molecular polaritons have demon-

strated unusual correlation properties [33–35]. That calls for

an extensive understanding of dark states with a high mode

density [36–42], nonlinearities and multiexciton correlations

[43–47].

Previous spectroscopic studies of cavity polaritons were

mostly based on wave function methods including nonadia-

batic nuclear dynamics [48–50], Redﬁeld theory and quantum

chemistry simulations of low excitations [51–56]. Absorption

and emission associated with multiple phonons and optically

dark states depend on a strong polariton-polaron interaction,

which raises a fundamental issue in cavity polaritons and how-

ever complicates the simulation of ultrafast spectroscopy.

In this Letter, we employ a quantum Langevin theory for

time-frequency-resolved coherent spectroscopy of molecular

polaritons. Analytical solution for multidimensional third-

order spectroscopic signals is developed. The results reveal

multiple channels and timescales of the cooperative relaxation

of polaritons, and also the trade-oﬀwith dark states.

Langevin model for polaritons.–Given Nidentical

molecules in an optical cavity, each has two energy surfaces

corresponding to electronically ground and excited states,

i.e., |gjiand |eji(j=1,2,··· ,N), respectively. Electronic

excitations forming excitons couple to intramolecular vibra-

tions and to cavity photons, as depicted in Fig.1(b), and are

described by the Holstein-Tavis-Cummings Hamiltonian

n=1h∆nσ+

nσ−

n+ωvb†

nbn+gnσ+

na+σ−

na†+δca†ai(1)

where ∆n=δ−λωv(bn+b†

n) and δdenotes the detuning be-

tween excitons and external pulse ﬁeld. [σ−

n, σ+

m]=σz

nδnm.

σ+

n=|enihgn|and σ−

n=|gnihen|are the respective raising

and lowering operators for the excitons in the nth molecule.

bndenotes the bosonic annihilation operator of the vibra-

tional mode with a high frequency ωv, in the nth molecule.

aannihilates cavity photons. Each molecule has one high-

frequency vibrational mode. In addition to the strong cou-

pling to the single-longitudinal cavity mode, the molecules

are subject to three temporally separated laser pulses whose

electric ﬁelds Ej(t−Tj)e−iv j(t−Tj);j=1,2,3 described by

arXiv:2210.13366v1 [quant-ph] 24 Oct 2022

FIG. 1: Schematic of time-resolved spectroscopy for molecular polaritons. (a) Emission signal is collected along a certain

direction, once the molecules are excited by laser pulses. (b) Exciton-photon interaction in molecules in the presence of vibronic

coupling attached to individual molecule. This results in the dark states and emitter dark states (EDSs) weakly interacting

with cavity, apart from the upper and lower polariton modes; Rich timescales and channels of excited-state relaxation are thus

expected. (c) Linear absorption of molecular polaritons with 10 organic molecules in an optical cavity. The parameters are taken

to be ωD=16113cm−1,˜

δ=δc=0, Γ = 20cm−1,γ=1cm−1,γc=0.9cm−1,ωv=1200cm−1, typically from cyanine dyes [60].

V(t)=P3

j=1PN

n=1Vj,n(t)+h.c. with Vj,n(t)=−σ+

nΩj(t−

Tj)e−i(vj−v3)teivjTjwhere Ωj(t−Tj)=µeg Ej(t−Tj) is the Rabi

frequency with the jth pulse ﬁeld and µeg is molecular dipole

moment [57]. The full Hamiltionian is H(t)=H+V(t), which

yields the quantum Langevin equations (QLEs) for σ−

n,a,bn.

We incorporate the polaron transform via the displacement

operator Dn=eλ(bn−b†

n)into the QLE for the dressed operator

˜σ−

n=σ−

nD†

n. This is to involve the exciton-vibration coupling

to all orders, as it is normally moderate or strong. The QLEs

for operators read a matrix form

V=−ˆ

MV +Vin(t)+i

j=1

Ωj(t−Tj)eivjTje−i(vj−v3)tWx(2)

after a lengthy algebra, where the term ∝(bn−b†

n)=

i√2pn(pnis the dimensionless momentum of nuclear) has

been dropped due to the nuclear velocity much lower than

electrons [59]. The vector V =[ ˜σ−

1,˜σ−

2,··· ,˜σ−

N,a]T

involves N+1 components, and Wx=[(2n1−

1)D†

1,··· ,(2n1−1)D†

N,0]T,nl=˜σ+

l˜σ−

l. Vin(t)=

[p2γ˜σ−,in

1(t),··· ,p2γ˜σ−,in

N(t),p2γcain(t)]Tgroups the noise

operators originated from exciton decay and cavity leakage.

The matrix ˆ

M in Eq.(2) reads







i˜

δ+γ0··· 0igσz

1D†

0i˜

δ+γ··· 0igσz

2D†

0 0 ··· i˜

δ+γigσz

ND†

igD1igD2··· igDNiδc+γc







.(3)

We solve for the vibration dynamics: bn(t)≈e−(iωv+Γ)tbn(0) +

√2ΓRt

0e−(iωv+Γ)(t−t0)bin

n(t0)dt0, neglecting back inﬂuence from

excitons, along the line of the stochastic Liouville equation

[22, 55]. Eq.(2) represents the dynamics of molecular po-

laritons. Perturbation theory of the molecule-ﬁeld interaction

V(t) will be used and we will calculate two-dimensional pho-

ton emission signals by placing the detectors oﬀthe cavity

axis, shown in Fig.1(a). These signals are governed by mul-

tipoint correlation functions of the dipole operators and the

corresponding Green’s functions, which are determined by the

exact solution to the QLEs in Eq.(2).

The polariton emission.–We ﬁrst present a general result

for the emission spectrum of cavity polaritons. Subject to

a probe pulse, Eq.(2) solves for the far-ﬁeld dipolar radia-

tion from molecules governed by the macroscopic polariza-

tion P(t)=µ∗

eg PN

i=1hσ−

i;1(t)i. We ﬁnd the emission signal

PE(ω, T)=2i|µeg|2

i=1

l=1Z∞

−∞

dtZt

dτeiωtE(τ−T)

×e−iv(τ−T)hGil(t−τ)nl(τ)D†

l(τ)Di(t)i

(4)

where G(t)=Te−Rt

0ˆ

Mdt0is the free propagator without pulse

actions. We note that, from the dressed excited-state popula-

tions nl(τ)D†

l(τ), the cavity polaritons of molecules undergo

a dynamics against the local ﬂuctuations from polaron eﬀect.

The polaron-induced localization as a result of dark states will

compete with the cooperative dynamics of polaritons. These

can be visualized from the emission signal, which will thus

be a real-time monitoring of polariton dynamics through pulse

shaping and grating. More advanced information will be elab-

orated by the multidimensional projections of the signals.

Linear absorption.–Assuming ωv/Tb1 that applies

for organic molecules at room temperature, the vibra-

tional correlation functions can be evaluated within vacuum

state. Using Eq.(2), the absorption spectra reads SA(ω)=

i,l=1P∞

m=0Sλ

mδm

il Re Gil(ω−ξ∗

m)and Sλ

m=e−λ2λ2m/m!

is the Franck-Condon factor. ξm=m(ωv+iΓ) and

Gmn(Ω)=R∞

0Gmn(t)eiΩtdt is the Fourier component of

the propagator G(t). SA(ω) resolves the lower polariton

(LP) and upper polariton (UP) ωLP/UP, EDSs ωD+mωv

decoupled from cavity photons while the dark states ωD

are not visible. To see these closely, we assume γ=

γc,˜

δ=δc=0. The peak intensities can be thus found

SA(ωD+mωv)/SAωLP/UP≈2(λ2m/m!)(γLP/UP/mΓ) and

SAωLP/UP +mωv/SAωLP/UP≈0 when N1. The

modes at ωLP/UP +mωvare hard to observe. Yet the EDSs

at ωD+mωvmay be of comparable intensities with polari-

ton modes. Such spectral-line properties will be shown to be

generally true in the time-resolved spectroscopic signals.

Fig.1(c,up) illustrates the absorption spectra where the LP

and UP are prominent from the peaks at 14300cm−1and

17900cm−1seperated by 2g√N. In between, we can observe

an extra peak at ωD+ωvsupporting an EDS decoupled from

cavity photons and the large oscillator strength owing to the

density of states ∼N. Fig.1(c,down) shows that the EDSs are

masked by the Rabi splitting for weaker vibronic coupling.

This, as a benchmark to the strong-coupling case, elaborate

the eﬀect of vibronic coupling against the collective coupling

to cavity photons. The localization nature of the EDSs is thus

indicated from eroding the cooperativity between molecules,

which will be elaborated in time-resolved spectroscopy.

2D polariton spectroscopy.–To have multidimensional pro-

jections of the emission signal, a sequential laser pulses

have to interact with the molecular polaritons. As the ﬁrst

two pulses create excited-state populations and coherences

nl;2(t)=σ+

l;1(t)σ−

l;1(t) where the 1st-order correction σ±

l;1 is

calculated from Eq.(2), we ﬁnd

nl;2(t)D†

l(t)=

j=1

j0=1"t

dt00dt0E∗

1(t0−T1)E2(t00 −T2)

×G∗

l j0(t−t0)Gl j(t−t00)Dj0(t0)D†

j(t00)D†

l(t).

(5)

The 3rd-order correction to the polarization follows Eq.(4)

when the third pulse serves as probe. Given the time-ordered

pulses, the transition pathways are selective resulting from the

term cancellation. Inserting Eq.(5) into Eq.(4), we therefore

proceed to the far-ﬁeld polarization for the emission along the

direction kI=−k1+k2+k3, i.e., P(t)=µ∗

eg PN

i=1hσ−

i;3(t)i

which yields

P(ω)=2i

i,l=1

j,j0=1&∞

dtdτdt00dt0eiωtE3(τ−T3)

× E2(t00 −T2)E∗

1(t0−T1)h0|Gil(t−τ)G∗

l j0(τ−t0)

×Gl j(τ−t00)Dj0(t0)D†

j(t00)D†

l(τ)Di(t)|0i

(6)

where the four-point correlation function of vibrations

h0|Dj0(t0)D†

j(t00)D†

l(τ)Di(t)|0ihas to be evaluated explicitly.

The 2D signal is usually detected via a reference beam

as a local oscillator interfering with the emission. This

leads to the heterodyne-detected signal S2D(Ω3,T,Ω1)=

Im R∞

0E∗

LO(Ω3)P(Ω3)eiΩ1τdτwith the Fourier transform

against the 1st delay τ=T2−T1, where ELO(Ω3) is the Fourier

component of the local oscillator ﬁeld. In general, calculating

the signal with Eq.(6) is hard due to the integrals over pulse

shapes. The procedures can be simpliﬁed by invoking the im-

pulsive approximation such that the pulse is shorter than the

dephasing and solvent time scales. We further consider the

few-photon cavity that draws much attention in recent exper-

iments, and notice the vibronic coupling predominately ac-

counted by the polarons. The most signiﬁcant terms may be

remained, allowing the approximation gσz

lD†

l≈gD†

l≈gin

Eq.(3). The higher-order corrections will be presented else-

where. We obtain an analytical solution to the 2D polariton

signal (2DPS), up to a real constant

S(Ω3,T,Ω1)=ieiφ

i,l=1

j,j0=1

N+1

p=1

∞

{m}=0

Sλ

{m}δm1

j0jδm2

il δm3

jl δm4

j0l

×δm5

i j δm6

i j0(−1)m3+m6Gil Ω3+ξm2+m5+m6G∗

lp(T)Gl j(T)

×eiξm3+m4+m5+m6TG∗

p j0−Ω1−ξm1+m4+m6

(7)

subsequently from Eq.(6), where Sλ

{m}=Q6

s=1Sλ

msand φen-

codes the global phase from the four classical pulses. Details

of the derivation of the signals via QLEs are given in Supple-

mental Material [59].

Simulations.–We have simulated the 2DPS to study polari-

ton, exciton and polaron dynamics from the analytical solu-

tions. We set g√N/ωv=1.5 for strong coupling.

The lower and upper rows in Fig.2 illustrate the 2DPS re-

spectively for N=1 and 10 molecules with ﬁxed Rabi fre-

quency 2g√N. For 10 molecules in cavity, the signal reveals

the real-time population transfer and coherence dynamics be-

tween polaritons and EDSs. The EDSs, however, cannot be

resolved when one molecule coupled to cavity only. This is

evident by the absence of the peaks at Ω1,3=ωD±nωv(n=

0,1,2, ...) in the lower row, compared with the upper row. The

2DPS for N=1 can monitor the states at ωUP −integer ×ωv

and their population transfer as well as coherence with the po-

lariton states, as seen from the variation of the cross peaks,

for example, at (Ω1=ωUP −nωv,Ω3=ωUP −mωv),m,n

with the time delay T. We will present more details about the

polariton dynamics against the EDSs next.

Fig.2(a) shows the 2D signal at T=0, from which the LP

and UP states can be observed, evident by the two diagonal

peaks at ω±g√N. The cross peaks may result from the co-

herence and the polariton-polaron coupling, since energy and

dephasing are not allowed at T=0. The former is due to

the broadband nature of pump pulses while the latter is re-

sponsible for the change of phonon numbers associated with

optical transitions. To have a closer look, we notice the states

at Ω1=14300,17300 and 17900cm−1, after the absorbing en-

ergy from the pump pulse. These agree with the absorbance in

Fig.1(c,up). The cross peaks imposing Ω1−Ω3=integer ×ωv

indicates the population of the EDSs which decouple from

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MultidimensionalCoherentSpectroscopyofMolecularPolaritons:LangevinApproachZhedongZhang,1,2,XiaoyuNie,3DangyuanLei,4andShaulMukamel5,6,y1DepartmentofPhysics,CityUniversityofHongKong,Kowloon,HongKongSAR2CityUniversityofHongKong,ShenzhenResearchInstitute,Shenzhen518057,Guangdong,China3CentreforQuantum...

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Multidimensional Coherent Spectroscopy of Molecular Polaritons Langevin Approach Zhedong Zhang1 2Xiaoyu Nie3Dangyuan Lei4and Shaul Mukamel5 6 1Department of Physics City University of Hong Kong Kowloon Hong Kong SAR

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