Understanding the Energy Gap Law under Vibrational Strong Coupling Yong Rui Poh Sindhana Pannir-Sivajothi Joel Yuen-Zhou Department of Chemistry and Biochemistry University of California San Diego La Jolla California 92093 USA

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Understanding the Energy Gap Law under Vibrational Strong Coupling

Yong Rui Poh, Sindhana Pannir-Sivajothi, Joel Yuen-Zhou∗

Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA

(Dated: January 5, 2023)

The rate of non-radiative decay between two molecular electronic states is succinctly described

by the energy gap law, which suggests an approximately-exponential dependence of the rate on

the electronic energy gap. Here, we inquire whether this rate is modiﬁed under vibrational strong

coupling, a regime whereby the molecular vibrations are strongly coupled to an infrared cavity. We

show that, under most conditions, the collective light-matter coupling strength is not large enough to

counter the entropic penalty involved with using the polariton modes, so the energy gap law remains

unchanged. This eﬀect (or the lack thereof) may be reversed with deep strong light-matter couplings

or large detunings, both of which increase the upper polariton frequency. Finally, we demonstrate

how vibrational polariton condensates mitigate the entropy problem by providing large occupation

numbers in the polariton modes.

INTRODUCTION

When molecules are placed inside an optical cavity,

they may interact strongly with the quantised radia-

tion mode to form light-matter hybrid states called po-

laritons [1–5]. In particular, this phenomenon is most

signiﬁcant when energy cycles between the molecular

transitions and the photon mode at a rate faster than

the decay of each individual component. Molecular po-

laritons have diﬀerent frequencies and potential energy

surfaces than their pure-matter counterparts. As such,

they oﬀer an interesting avenue for controlling chemical

properties, often through parameters directly related

to light-matter interactions (like cavity frequencies and

light-matter coupling strengths) [6]. Of particular in-

terest to this paper are vibrational polaritons, formed

by strong coupling between the high-frequency vibra-

tional modes of molecules and an infrared optical cav-

ity that houses these molecules [2, 3]. This eﬀect, also

known as vibrational strong coupling (VSC), has been

experimentally shown to modify reaction pathways and

achieve chemoselectivity [7–11]. VSC is an ensemble ef-

fect: a large number of molecules, typically of the order

of 1010, must collectively couple to a cavity mode to

generate an appreciable collective light-matter coupling

strength. The result is the formation of two polari-

ton modes, which provide control over chemical proper-

ties and are desirable, alongside a large number of dark

modes that behave eﬀectively as uncoupled molecules in

the absence of disorder [12] (with disorder, dark modes

might behave diﬀerently from uncoupled molecular ex-

citations, although the extent to which this occurs is

still a subject of current exploration [13–15]). The lat-

ter is a source of concern; due to their sheer numbers,

these non-photonic dark modes have dominant control

over chemical properties of the system, potentially un-

doing any beneﬁt created by the polariton modes. In

principle, there exists a speciﬁc and reasonable param-

eter range that allows some unique properties of vibra-

tional polariton modes to outweigh the entropic cost

∗joelyuen@ucsd.edu

from using them [16]; yet, in practice, these molecu-

lar parameter requirements (such as ultralow reorgan-

isation energies) have not been experimentally found.

Meanwhile, more recently, polariton condensates have

been theoretically shown as an alternative avenue for

overcoming the entropic penalty associated with polari-

ton modes [17–19].

One chemical property that may beneﬁt from VSC

is the non-radiative decay between electronic states of

an excited molecule (Fig. 1a). The decay rate is con-

cisely and elegantly described by the energy gap law

[20, 21], which was ﬁrst derived by Englman and Jort-

ner in 1970 [20]. There, they applied Fermi’s golden

rule to approximate the transfer rate between two dis-

placed harmonic oscillators, representing the potential

energy surfaces of two electronic states (|Giand |Ei),

through a diabatic coupling term of amplitude |JGE|

that is treated perturbatively. More speciﬁcally, they

considered the Hamiltonian

H0=X

~ωvib,mˆ

b†

G,mˆ

bG,m|GihG|

+ X

~ωvib,mˆ

b†

E,mˆ

bE,m + ∆E!|EihE|,(1)

and a perturbative coupling term Vtrs =

JGE (|EihG|+|GihE|), where ∆Eis the electronic

energy gap, b†

x,m (bx,m) is the creation (annihilation)

operator of the m-th vibrational mode in electronic

state xwith frequency ωvib,m, and bE,m =bG,m −√Sm,

with Smbeing the Huang-Rhys factor representing

the displacement of the two potential energy surfaces

along mode m. In the low temperature limit, the

non-radiative decay rate is approximately

Wbare ≈|JGE|2

~s2π

~ωvib,M ∆E

×e−PmSmexp −γ∆E

~ωvib,M ,(2)

with γ= ln ∆E

SM~ωvib,M −1. Here, ωvib,M is the max-

imum vibrational frequency of the molecule, SM=

Pm∈{M} Smis the sum of the Huang-Rhys factors for

arXiv:2210.04986v2 [physics.chem-ph] 5 Jan 2023

FIG. 1. Non-radiative decay of a molecule from a higher electronic state |Eito a lower electronic state |Gi. (a) This process

may, in its simplest form, be modelled as a non-linear conversion of energy from electronic energy (∆E) in state |Eito

vibrational quanta (n~ωvib,M ) in state |Gialong the vibrational mode of maximum frequency ωvib,M . (b) Under collective

VSC in an infrared cavity, the same electronic energy (∆E) may be redistributed among polariton quanta (nP~ωP,P=

LP,UP) and dark mode quanta (nD~ωD), the former of which is useful because the UP mode has a higher frequency (ωUP)

than the dark modes (ωD=ωvib,M ) and can enhance the rate of non-radiative decay. However, under most circumstances,

this advantage is not realised because the large number of dark modes makes it entropically unfavourable to decay through

the polariton modes.

the set of vibrational modes {M} with frequencies near

ωvib,M , and the subscript “bare” indicates that the rate

expression is computed for a molecule outside the cavity.

There are theoretical foundations in grouping multiple

vibrational modes into a single, eﬀective vibrational

mode with contributions from the highest frequency

modes [20]. The inherent assumption in this expression

is that the electronic energy is most likely lost through a

number of high-frequency vibrations rather than many

more low-frequency ones, valid when the total reorgani-

sation energy PmSm~ωmof the high-frequency modes

is much larger than that of the low-frequency modes

[22]. An analytical generalisation of this model to ac-

count for low-frequency modes and ﬁnite temperatures

has been provided by Jang [22]. In that work, Jang de-

rives an improved rate expression that retains the same

qualitative phenomena as Eq. (2) but with improved

accuracy as compared to numerical results. Since Eq.

(2) is often ﬁtted into experimental data to obtain JGE,

one can potentially achieve more accurate estimates of

JGE with Jang’s improved model. Going back to Eq.

(2), in most cases, γmay be regarded as a constant,

giving an approximately-exponential dependence of the

decay rate on the energy gap ∆E(hence the name).

Overall, this model has found numerous applications in

molecular spectroscopy [23–26] and optoelectronics [27–

32], in particular for describing the quantum yields of

radiative processes.

In this paper, we demonstrate how, under VSC, a

large proportion of non-radiative decay occurs through

the dark modes, which was expected due to their large

numbers (Fig. 1b). At the same time, decay through

the higher-frequency polariton channel, if signiﬁcant,

can reduce the eﬀective energy gap for dark mode de-

cay, thereby increasing the overall decay rate. These

two eﬀects work against each other and the polaritonic

one dominates only under extreme conditions such as

deep strong couplings and large detunings; otherwise,

the rate of non-radiative decay, being dominated by the

dark modes, takes a value similar to the bare one outside

of the cavity. Finally, we investigate how this entropic

problem may be mitigated by the use of polariton con-

densates.

RESULTS AND DISCUSSIONS

The model system

We consider a model system of Nidentical molecules

i= 1,··· , N, each with two electronic states (|Eiiand

|Gii) and a set of molecular vibrational modes. The

collective eﬀects of these vibrational modes are repre-

sented by a single mode at the maximum frequency ωvib

with a collective Huang-Rhys factor S; this is consistent

with the approach taken by Englman and Jortner [20].

Along this eﬀective vibrational coordinate, the potential

energy curves for both electronic states are modelled as

a pair of displaced harmonic oscillators, each with the

same frequency. Finally, all Nmolecules are strongly

coupled to a lossless cavity mode of frequency ωph that

is in (or close to) resonance with the eﬀective vibra-

tional mode. To zeroth-order in the diabatic coupling

JGE, the Hamiltonian reads

H0=~ωphˆa†

phˆaph

i=1

~ωvibˆ

b†

G,iˆ

bG,i|GiihGi|

i=1 ~ωvibˆ

b†

E,iˆ

bE,i + ∆E|EiihEi|

i=1

~gˆ

b†

G,iˆaph + ˆa†

phˆ

bG,i|GiihGi|

i=1

~gˆ

b†

E,iˆaph + ˆa†

phˆ

bE,i|EiihEi|,(3)

where ˆa†

ph (ˆaph)is the creation (annihilation) operator

of the photon mode, ˆ

b†

x,i ˆ

bx,iis the creation (annihi-

lation) operator of the i-th molecule’s vibrational mode

(x=G,E) with ˆ

bE,i =ˆ

bG,i −√Sfor all i,∆Eis the

energy gap between the two electronic states of each

molecule and gis the single-molecule light-matter cou-

pling strength. Here, we have made the rotating wave

approximation (RWA) and assumed that gis the same

for both electronic states. Also, zero-point energies of

the vibrational states have been omitted since they only

contribute constants to the ﬁnal energies.

To model the non-radiative decay rate of a single

molecule cfrom the |Ecielectronic state to the |Gci

electronic state, we introduce a diabatic coupling term

into the Hamiltonian,

Vtrs,c =JGE (|EcihGc|+|GcihEc|),(4)

where JGE is the corresponding amplitude. Then, fol-

lowing the same procedure as Englman and Jortner

[20], we may compute the non-radiative decay rate of

molecule cby Fermi’s golden rule, which assumes ˆ

Vtrs,c

to be a perturbation with respect to ˆ

H0. Despite the

presence of light-matter couplings in ˆ

H0, this assump-

tion remains valid since non-radiative decay eﬀectively

couples a single electronic excitation to a large number

of vibrational excitations n≈∆E

~ωvib 1, all within the

same molecule. As such, the nonlinearity of this process

makes it slower than the molecule’s interactions with

the cavity mode, which is linear. More quantitatively,

non-radiative decay is characterised by the decay ampli-

tude multiplied by the Franck-Condon overlap between

the initial and ﬁnal vibrational states of the decaying

molecule; in the low-temperature limit, this takes the

form of

JGEre−SSn

n!≈10−9~ωvib,(5)

which is two orders of magnitude smaller than the

decaying molecule’s light-matter coupling strength

~g≈10−7~ωvibif we consider JGE ≈0.4~ωvib,

g√N≈0.01ωvib,S≈0.1,∆E≈10~ωvib and N≈

1010, conditions typical of S1→S0transitions of aro-

matic hydrocarbons [23, 33] under collective VSC [11].

To ﬁnd the initial and ﬁnal eigenstates of ˆ

H0, we may,

without loss of generality, focus on the decay of molecule

1 from |Eito |Giwhile keeping the remaining N−1

molecules in |Gi(i.e. pick c= 1). Then, the initial and

ﬁnal electronic eigenstates of ˆ

H0may be written col-

lectively as |E,G,··· ,Giand |G,G,··· ,Gi, where we

have listed the electronic state of each molecule in in-

creasing order of its index i. Each of the two electronic

states above comprises Nvibrational modes coupled to

a single cavity mode. Therefore, all that remains is ﬁnd-

ing the normal modes of ˆ

H0within these two electronic

subspaces. Working ﬁrst in the subspace of the initial

electronic state, we transform the Nvibrational modes

into a single bright (B) mode,

bB=1

√N ˆ

bE,1+

i=2

bG,i!,(6)

with the correct symmetry to interact with light and

N−1dark (D) modes,

bD,k =Ck,1ˆ

bE,1+

i=2

Ck,iˆ

bG,i,(7)

with 2≤k≤N, that do not couple to light. Note

that the constants {Ck,i}(1≤i≤N) are chosen

such that the dark modes are orthogonal to the bright

mode, i.e. PN

i=1 Ck,i = 0, and to each other, i.e.

i=1 C∗

j,iCk,i =δjk. In this basis, the dark modes are

already diagonal while the bright and photon modes

mix to give the upper polariton (UP) ˆ

bUP and lower

polariton (LP) ˆ

bLP modes,

bLP =−sin (θ) ˆaph + cos (θ)ˆ

bB,

bUP = cos (θ) ˆaph + sin (θ)ˆ

bB,(8)

with a mixing angle of

θ= tan−1Ω−∆

2√Ng ,(9)

where Ω = p∆2+ 4g2Nis the Rabi splitting, ∆ =

ωph −ωvib is the detuning and g√Nis the collective

light-matter coupling strength. Also, the mode frequen-

cies are

ωLP =ωvib +∆−Ω

ωUP =ωvib +∆+Ω

ωD,k =ωvib for all k. (10)

We may follow the same steps to deﬁne the bright,

dark and polariton modes ˆ

B,nˆ

D,ko,ˆ

LP and ˆ

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UnderstandingtheEnergyGapLawunderVibrationalStrongCouplingYongRuiPoh,SindhanaPannir-Sivajothi,JoelYuen-ZhouDepartmentofChemistryandBiochemistry,UniversityofCaliforniaSanDiego,LaJolla,California92093,USA(Dated:January5,2023)Therateofnon-radiativedecaybetweentwomolecularelectronicstatesissuccinctlyde...

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Understanding the Energy Gap Law under Vibrational Strong Coupling Yong Rui Poh Sindhana Pannir-Sivajothi Joel Yuen-Zhou Department of Chemistry and Biochemistry University of California San Diego La Jolla California 92093 USA

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