Optomechanical Eﬀects in Nanocavity-enhanced Resonant Raman Scattering of a Single Molecule Xuan-Ming Shen1Yuan Zhang1Shunping Zhang2Yao Zhang3Qiu-Shi Meng3Guangchao

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Optomechanical Eﬀects in Nanocavity-enhanced Resonant Raman Scattering of a

Single Molecule

Xuan-Ming Shen,1Yuan Zhang,1, ∗Shunping Zhang,2Yao Zhang,3Qiu-Shi Meng,3Guangchao

Zheng,4Siyuan Lv,5Luxia Wang,5Roberto A. Boto,6, 7 Chongxin Shan,1, †and Javier Aizpurua6, 7, ‡

1Henan Key Laboratory of Diamond Optoelectronic Materials and Devices,

Key Laboratory of Material Physics, Ministry of Education, School of Physics and Microelectronics,

Zhengzhou University, Daxue Road 75, Zhengzhou 450052, China

2School of Physics and Technology, Center for Nanoscience and Nanotechnology,

and Key Laboratory of Artiﬁcial Micro- and Nano-structures of

Ministry of Education, Wuhan University, Wuhan 430072, China

3Hefei National Research Center for Physical Sciences at the Microscale and

Synergetic Innovation Centre of Quantum Information and Quantum Physics,

University of Science and Technology of China, Hefei, Anhui 230026, China

4School of Physics and Microelectronics, Zhengzhou University, Daxue Road 75, Zhengzhou 450052

5Department of Physics, University of Science and Technology Beijing, 100083 Beijing, China

6Center for Material Physics (CSIC - UPV/EHU and DIPC) Paseo

Manuel de Lardizabal 5, Donostia-San Sebastian Gipuzkoa 20018, Spain

7Donostia International Physics Center, Paseo Manuel de Lardizabal 4, Donostia-San Sebastian 20018, Spain

In this article, we address the optomechanical eﬀects in surface-enhanced resonant Raman scat-

tering (SERRS) from a single molecule in a nano-particle on mirror (NPoM) nanocavity by develop-

ing a quantum master equation theory, which combines macroscopic quantum electrodynamics and

electron-vibration interaction within the framework of open quantum system theory. We supplement

the theory with electromagnetic simulations and time-dependent density functional theory calcula-

tions in order to study the SERRS of a methylene blue molecule in a realistic NPoM nanocavity.

The simulations allow us not only to identify the conditions to achieve conventional optomechanical

eﬀects, such as vibrational pumping, non-linear scaling of Stokes and anti-Stokes scattering, but

also to discovery distinct behaviors, such as the saturation of exciton population, the emergence of

Mollow triplet side-bands, and higher-order Raman scattering. All in all, our study might guide

further investigations of optomechanical eﬀects in resonant Raman scattering.

I. INTRODUCTION

Surface-enhanced Raman scattering (SERS) refers to

the enhacement of the Raman signal of molecules located

near metallic nanostructures [1]. This eﬀect is partially

due to the charge transfer between the molecule and the

metal (chemical enhancement [2]) but the main con-

tribution is due to the enhanced electromagnetic ﬁeld

by the metallic nanostructure (electromagnetic ﬁeld en-

hancement [3]). Moreover, since Raman enhancement

can reach tens of orders of magnitude for molecules near

electromagnetic hot-spots [4], vibrational pumping asso-

ciated with the enhanced Stokes scattering can poten-

tially compete with thermal vibrational excitation, and

is thus able to introduce non-linear scaling of anti-Stokes

scattering with increasing laser intensity [5–8].

Earlier studies on vibrational pumping were hindered

by the diﬃculty of quantitatively estimating the vibra-

tional pumping rate. To overcome this problem, in

recent years, M. K. Schmidt et al. [9] and P. Roelli,

et al. [10] were inspired by cavity optomechanics [11],

and developed a molecular optomechanics theory [12] for

∗yzhuaudipc@zzu.edu.cn

†cxshan@zzu.edu.cn

‡aizpurua@ehu.es

oﬀ-resonant Raman scattering. According to this the-

ory, molecular vibrations interact with the plasmonic re-

sponse of metallic nanostructures through optomechan-

ical coupling, and the vibrational pumping rate can be

quantitatively determined by this coupling together with

the molecular vibrational energy and the plasmonic re-

sponse (such as mode energy, damping rate and laser

excitation). Besides vibrational pumping, molecular op-

tomechanics also allows us to investigate many novel

eﬀects, such as non-linear divergent Stokes scattering

(known as parametric instability in cavity optomechan-

ics [11]), collective optomechanical eﬀects [13], higher-

order Raman scattering [14], and optical spring eﬀect [15]

among others.

Metallic nanocavities, formed by metallic nanoparticle

dimers [16], metal nano-particle on mirror (NPoM) [17]

or STM tip-on-metallic substrate [18], are ideal conﬁgu-

ration for observation of novel optomechanical eﬀects in

experiments, because they provide hundred folds of local

ﬁeld enhancement inside the nano-gaps, and thus provide

easily SERS enhancement over 1012. Using biphenyl-4-

thiol molecules inside NPoM gold nanocavities, F. Benz,

et al. [19] observed non-linear scaling of anti-Stokes SERS

with increasing continuous-wave laser excitation (thus

justifying vibrational pumping). Later on, using the

same system, non-linear scaling of Stokes SERS was ob-

served for pulsed laser illumination of much stronger in-

arXiv:2210.02639v1 [physics.optics] 6 Oct 2022

tensity at room temperature [20] (proving the precursor

of parametric instability and the collective optomechan-

ical eﬀect). Recently, Y. Xu, et al. [21] observed also the

similar non-linear Stokes SERS with a MoS2 monolayer

within metallic nanocube-on-mirror nanocavities.

Most of studies on molecular optomechanics so-far fo-

cus on the oﬀ-resonant Raman scattering. Because oﬀ-

resonant Raman scattering is usually weak, the observa-

tion of its optomechanical eﬀects requires normally very

strong laser excitation. To reduce the required laser in-

tensity, in this article, we propose to combine the metal-

lic nanocavities with the molecular resonant eﬀect to en-

hance the Raman scattering of molecules. The molecular

resonant eﬀect refers to the enhancement of the Raman

scattering when the laser is resonant with molecular elec-

tronic excited states. Indeed, the earliest studies on vi-

brational pumping focused on surface enhanced resonant

Raman scattering (SERRS) of dye molecules [5–8] (such

as crystal violet, or rhodamine 6G).

In our previous works [22, 23], we extended the molec-

ular optomechanics approach based on single plasmon

mode to SERRS, and showed that the electron-vibration

coupling, leading to resonant Raman scattering, resem-

bles the plasmon-vibration optomechanical coupling, and

thus we show that optomechanical eﬀects can also occur

in resonant Raman scattering. Moreover, since the for-

mer coupling is usually much larger, and the electronic

excitation is usually much narrower than the plasmon

excitation, optomechanical eﬀects in SERRS can poten-

tially occur for much smaller laser intensities. Further-

more, in our other works [15, 24], we showed that the

plasmonic response of metallic nanocavities is far more

complex than that of a single mode, and the molecu-

lar optomechanics is strongly aﬀected by the plasmonic

pseudo-mode, formed by the overlapping higher-order

plasmonic modes [25].

To address SERRS from molecules in realistic metallic

nanocavities, here, we develop a theory that combines the

macroscopic quantum electrodynamics description [26,

27] with the electron-vibration interaction, and derive

a quantum master equation for the molecular electronic

and vibrational dynamics. As an example, we apply our

theory to a single methylene blue molecule inside a gold

NPoM nanocavity, as shown in Fig. 1. To maximize

the methylene blue molecule-nanocavity interaction, we

assume that the methylene blue molecule is encapsulated

by a cucurbit[n] cage [28] so that the molecule stands

vertically.

Our study shows that most of the optomechanical ef-

fects, such as vibrational pumping, parametric instabil-

ity, vibrational saturation, Raman line shift and narrow-

ing and so on, can occur in SERRS at lower laser inten-

sity threshold. However, the molecular excitation satu-

rates for strong laser excitation because of its two-level

fermionic nature (in contrast to the inﬁnite-levels of a

bosonic plasmon), and the SERRS signal saturates and

even vanishes for strong laser excitation. In addition,

we also ﬁnd that the resonant ﬂuorescence is red-shifted

ωl±ωv

ωl

ωv

Figure 1. A vertically-orientated methylene blue molecule

(with black, white, blue, yellow spheres for carbon, hydro-

gen, nitrogen, sulfur atoms, respectively) in the middle of a

nano-particle on mirror (NPoM) nanocavity of 0.9nm thick,

formed by a truncated gold sphere with 40 nm diameter and

bottom facet of 10 nm diameter on top of a ﬂat gold sub-

strate. The laser excitation of frequency ωlis enhanced in the

nanocavity, and the enhanced local ﬁeld excites the molecule

vibrating with frequency ων. The emitted ﬁeld at frequency

ωl(Rayleigh scattering) and at frequencies ωl−ων,ωl+ων

(Stokes and anti-Stokes scattering), as well as frequencies in-

dependent of ωl(ﬂuorescence), is enhanced and propagated

to the far-ﬁeld.

by about 40 meV (plasmonic Lamb shift [29, 30]), and

broadened by about 22 meV (due to the Purcell eﬀect),

and also shows three broad peaks for strong laser excita-

tion [31] (corresponding to the Mollow triplet similar to

the situation in quantum optics [32]).

Our article is organized as follows. We present ﬁrst

our theory for SERRS of single molecule in plasmonic

nanocavities in Section II, which is followed by the time-

dependent density functional theory (TDDFT) calcula-

tion of the methylene blue molecule in Section III and

the electromagnetic simulation of the NPoM nanocavity

in Section IV. In Section V, we study the evolution of

the SERRS and ﬂuorescence with increasing laser illumi-

nation, which is blue-, zero- or red-detuned with respect

to the molecular excitation, respectively. In the end, we

conclude our work and comment on the extensions in fu-

ture.

II. QUANTUM MASTER EQUATION

To address the processes shown in Fig. 1, we have

developed a theory that combines macroscopic quantum

electrodynamics and electron-vibration interaction. In

Appendix A, we detail the treatment of the interaction

between a single molecule and the plasmonic (electromag-

netic) ﬁeld of the metallic nanocavity. To reduce the de-

grees of freedom, we apply the open quantum system the-

ory [33] where we consider the plasmonic ﬁeld as a reser-

voir and treat the molecule-plasmonic ﬁeld interaction as

a perturbation in second-order, to ﬁnally achieve an eﬀec-

tive master equation for the single molecule. Here, this

treatment is valid since the single molecule couples rela-

tively weakly with the nanocavity. However, further con-

sideration is required for the system with more molecules,

which might enter into the strong coupling regime [28].

To account for other mechanisms, like the molecular

vibrations and the molecular excitation, we generalize

the eﬀective master equation to obtain

∂

∂t ˆρ=−i

~hˆ

Hele +ˆ

Hlas +ˆ

Hvib +ˆ

Hele−vib +ˆ

Hpla,ˆρi

+ (Γ + γe)Dˆσ[ˆρ] + χe

2Dˆσz[ˆρ]

γν{nth

ν+ 1Dˆ

bν[ˆρ] + nth

νDˆ

b†

ν[ˆρ]}.(1)

In the above equation, we treat the molecular electronic

ground and excited state as a two-level system, and

specify its Hamiltonian as ˆ

Hele =~(ωe/2)ˆσzwith tran-

sition frequency ωeand Pauli operator ˆσz. We treat

the molecular excitation semi-classically with the Hamil-

tonian ˆ

Hlas =~ˆσ†ve−iωlas t+v∗eiωlastˆσ, where the

molecule-near ﬁeld coupling ~v=−dm·E(rm, ωlas)is de-

termined by the enhanced local electric ﬁeld E(rm, ωlas)

at the molecular position rm, activated by a laser with

frequency ωlas. Here, ˆσ†,ˆσare the raising and lowering

operator of the molecular excitation. We approximate

the molecular vibrations as quantized harmonic oscilla-

tors and specify their Hamiltonian ˆ

Hvib =~Pνωνˆ

b†

νˆ

bν

with the frequency ων, the creation ˆ

b†

νand annihila-

tion ˆ

bνoperators of the ν-th vibrational mode. The

electron-vibration interaction takes the form ˆ

Hele−vib =

~Pνωνdνˆσ†ˆσˆ

b†

ν+ˆ

bνwith the dimensionless displace-

ment dν=√Sν(Sνis known as the Huang-Rhys fac-

tor [34]) of the parabolic potential energy surface for the

electronic ground and excited state. Here, the electron

transition frequency ωehas already accounted for the

shift Pνdνωνdue to the electron-vibration coupling.

The elimination of the plasmonic ﬁeld leads to one

Hamiltonian ˆ

Hpla =~(Ω/2)ˆσzand one Lindblad term

ΓDˆσ[ˆρ]with the superoperator Dˆo[ˆρ] = ˆoˆρˆo†−1

2ˆo†ˆoˆρ−

2ˆρˆo†ˆo(for any operator ˆo), which describe the shift

Ω = −1

~0

ω2

c2dm·Re←→

G(rm,rm;ωe)·d∗

m(2)

of the transition frequency (plasmonic Lamb shift [29,

30]), and the Purcell-enhanced decay rate

Γ = 2

~0

ω2

c2dm·Im←→

G(rm,rm;ωe)·d∗

m(3)

of the molecule. Here, 0, c are the vacuum permittivity

and the speed of light, and dmis the molecular transi-

tion dipole. ←→

G(r,r0;ω)is the dyadic Green’s function,

which connects usually the electric ﬁeld with frequency ω

at the position rwith the dipole point source at another

position r0in classical electrodynamics. The remaining

terms of Eq. (1) describe the dissipation of the elec-

tronic states, including the non-radiative decay γeDˆσ[ˆρ]

with rate γe, and the dephasing χe

2Dˆσz[ˆρ]with rate χe,

and the thermal decay and pumping of the vibrational

modes Pνγνnth

ν+ 1Dˆ

b[ˆρ]+γνnth

νDˆ

b†[ˆρ]with rate γν,

where nth

ν= [exp {~ων/kBT} − 1]−1is the thermal vi-

brational population at temperature T(kBis Boltzmann

constant).

In Appendix B, we have derived the formula to com-

pute the spectrum dW (ω)/dΩmeasured by a detector in

the far-ﬁeld:

dΩ(ω)≈K(ω)Re Z∞

dτeiωτ tr {ˆσˆ%(τ)}.(4)

In this equation, the propagation factor is deﬁned as

K(ω) = ~2cr2

4π20

ω2

c2←→

G(rd,rm;ω)·d∗

m

,(5)

where the dyadic Green’s function ←→

G(rd,rm;ω)con-

nects the electric ﬁeld at frequency ωat position rdof

the detector with the molecule at the position rm.rde-

notes the distance between the molecule and the detector.

In addition, the operator ˆ%(τ)satisﬁes the same master

equation as ˆρ, but with initial condition ˆ%(0) = ˆρˆσ†.

To solve the master equation [Eq.(1)], we introduce the

density matrix ρan,bm (with a, b =g, e) in the basis of the

product states {|ei|mi,|gi|ni}, where |ei,|gidenote the

electronic excited and ground state of the molecule, and

|mi=Qν|mνi,|ni=Qν|nνiare the occupation num-

ber states of the vibrational modes (mν, nνare positive

integers). From Eq. (1), we can easily derive the equa-

tion for the density matrix. In the current study, we uti-

lize the QuTip toolkit [35, 36] to solve the density matrix

equation. By solving this equation, we can determine the

electronic excited state population ˆσ†ˆσ= tr{ˆσ†ˆσˆρ},

the mean vibrational population Dˆ

b†

νˆ

bνE= tr{ˆ

b†

νˆ

bνˆρ}, the

two-time correlation C(τ)≡tr {ˆσˆ%(τ)}, and the spec-

trum dW (ω)/dΩ.

In the weak-excitation limit ˆσ†ˆσ1, we can ap-

ply the Holstein-Primakoﬀ approximation [37] to the

molecule by introducing the bosonic creation ˆaand an-

nihilation operator ˆa†such that: ˆσz≡2ˆa†ˆa−1,ˆσ†≡

ˆa†√1−ˆa†ˆa≈ˆa†,ˆσ≡√1−ˆa†ˆaˆa≈ˆa. In this case,

we can carry out the replacement ˆσz→ˆa†ˆa,ˆσ†→ˆa†,

ˆσ→ˆain the master equation given above, and then

the resulted equation is equivalent to the one used in

the molecular optomechanics theory [9, 10]. This indi-

cates that the molecular excitation works eﬀectively as

the plasmon mode, and the electron-vibration coupling

as the optomechanical coupling i.e. gν=ωνdν. Thus, we

expect to ﬁnd many interesting optomechanical eﬀects

in the current system, namely vibrational pumping, non-

linear Raman scattering, Raman line-shift and broaden-

ing [13] and so on. For typical value of ~ων= 200 meV

and d= 0.1, we obtain ~gν= 20 meV, which is about

two orders of magnitude larger than in typical molecular

optomechanics [9, 13]. Thus, we expect to observe a va-

riety of optomechanical eﬀects in SERRS with however

lower laser intensity threshold.

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OptomechanicalEectsinNanocavity-enhancedResonantRamanScatteringofaSingleMoleculeXuan-MingShen,1YuanZhang,1,ShunpingZhang,2YaoZhang,3Qiu-ShiMeng,3GuangchaoZheng,4SiyuanLv,5LuxiaWang,5RobertoA.Boto,6,7ChongxinShan,1,yandJavierAizpurua6,7,z1HenanKeyLaboratoryofDiamondOptoelectronicMaterialsandDevices...

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Optomechanical Eﬀects in Nanocavity-enhanced Resonant Raman Scattering of a Single Molecule Xuan-Ming Shen1Yuan Zhang1Shunping Zhang2Yao Zhang3Qiu-Shi Meng3Guangchao

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