Content uploaded by Felipe Herrera

Author content

All content in this area was uploaded by Felipe Herrera on Sep 24, 2021

Content may be subject to copyright.

Multi-level quantum Rabi model for anharmonic vibrational polaritons

Federico Hern´andez1and Felipe Herrera1, 2 , ∗

1Department of Physics, Universidad de Santiago de Chile, Av. Ecuador 3493, Santiago, Chile.

2Millennium Institute for Research in Optics MIRO, Chile.

(Dated: June 12, 2019)

We propose a cavity QED approach to describe light-matter interaction between an individual

anharmonic molecular vibration and an infrared cavity ﬁeld. Starting from a generic Morse oscilla-

tor with quantized nuclear motion, we derive a multi-level quantum Rabi model to study vibrational

polaritons beyond the rotating-wave approximation. We analyze the spectrum of vibrational polari-

tons in detail and compare with available experiments. For high excitation energies, the spectrum

exhibits a dense manifold of true and avoided level crossings as the light-matter coupling strength

and cavity frequency are tuned. These crossings are governed by a pseudo parity selection rule

imposed by the cavity ﬁeld. We also analyze polariton eigenstates in nuclear coordinate space. We

show that the bond length of a vibrational polariton at a given energy is never greater than the bond

length of a bare Morse oscillator with the same energy. This type of bond hardening of vibrational

polaritons occurs at the expense of the creation of virtual infrared cavity photons, and may have

implications in chemical reactivity.

Cavity quantum electrodynamics (QED) has been in-

tensely studied for the development of quantum technol-

ogy over the last decade [1,2]. Precision experiments

under carefully controlled conditions have been imple-

mented to reach the regime where quantum optical ef-

fects become relevant for applications [3–5]. Chemical

systems and molecular materials at ambient conditions

for long have been considered to be unnecessarily com-

plex and uncontrollable to enable useful quantum optical

eﬀects. In recent years, the demonstration of reversible

modiﬁcations of chemical properties in molecular mate-

rials via strong coupling to conﬁned light has stimulated

the study of cavity QED as an emerging research direc-

tion in chemical physics [6]. Light-matter interaction in

the strong coupling (SC) and ultrastrong coupling (USC)

regimes opens the possibility of creating novel hybrid

photon-molecule states whose unique properties may en-

able novel applications in chemistry and material science.

In the infrared regime, the coupling of an intramolecu-

lar vibration to the quantized electromagnetic vacuum of

a Fabry-P´erot cavity can lead to the formation of vibra-

tional polaritons [7–21]. These hybrid light-matter states

exhibit fundamentally novel properties in comparison

with free-space vibrations. For instance, vibrational po-

laritons may enable the selective control of chemical reac-

tions [21–23], a long-standing goal in physical chemistry

[24]. Strong light-matter coupling provides a reversible

way of modifying reactive processes without changing the

chemical composition of materials, and also modify the

radiative and non-radiative dynamics of molecular vibra-

tions [25–31]. Several recent studies on vibrational strong

coupling (VSC) within the ground electronic state have

shown that chemical reactions can proceed through novel

pathways in comparison with free space. Under VSC, re-

actions may be inhibited or catalyzed, or the product

∗Electronic address: felipe.herrera.u@usach.cl

branching ratios tilted [7,16,22,23]. For instance, by

strongly coupling the carbon-silyl (Si-C) bond stretch-

ing vibration of 1-phenyl-2-trimethylsilylacetylene to the

electromagnetic vacuum of a resonant infrared microﬂu-

idic cavity, the rate of Si-C bond breakage has been shown

to decrease by a moderate factor of order unity [16]. In

another recent study [22], the catalytic eﬀect of VSC

on the hydrolysis of cyanate ions and ammonia borane,

under coupling of a Fabry-P´erot cavity witht the broad

OH infrared absorption band of water was demonstrated.

The measured increase on the reaction rate constant by

two orders of magnitude relative to free space for cyanate

and by four orders of magnitude for ammonia borane,

is one of the ﬁrst reports of cavity-enhanced reactivity,

a possibility earlier predicted for electron transfer reac-

tions in microcavities [32]. In another study, very precise

chemical control was carried out on a compound hav-

ing two available silyl bond cleavage sites, SiC and SiO.

For this system, simultaneous VSC with three spectrally

distinct vibrational modes was shown to modify the re-

activity landscape such that the branching ratio of SiC

and SiO cleavage was altered, by simply tuning the cavity

resonance conditions [23].

The diverse experimental evidence on VSC represents

a challenge for theoretical modeling, mainly due to the

inherent complexity of potential energy landscapes of re-

active species, as well as collective eﬀects that are rele-

vant in molecular ensembles. Despite recent theoretical

progress [14,28,31,33–40], it remains unclear whether

there is a universal mechanism for the modiﬁcation of

ground state chemical reactivity under VSC, or the prob-

lem is system-speciﬁc. Common explanations for exper-

imental observations are based on a traditional chemical

concepts such as changes in the potential energy surface,

modiﬁcations of an activation energy, or changes in the

relative energy of reactants and products. However, as

we discuss throughout, under VSC it is diﬃcult to jus-

tify the conventional physical meaning assigned to these

traditional concepts.

arXiv:1906.04374v1 [physics.chem-ph] 11 Jun 2019

2

In this work, we introduce a cavity QED approach to

study anharmonic vibrational polaritons in the single-

molecule limit. The method describes the molecular

sub-system as an individual anharmonic polar bond

with quantized nuclear motion. The vibration inter-

acts with a vacuum ﬁeld via electric dipole coupling.

Discrete-variable representation (DVR) is used to de-

scribe the cavity-free anharmonic vibration, and the elec-

tric dipole interaction is treated with a cavity QED ap-

proach that includes counter-rotating terms. This total

system Hamiltonian corresponds to a multi-level quan-

tum Rabi (MLQR) model. By construction, vibrational

polariton states can be analyzed in Hilbert space and

also in coordinate space. The method can be scaled to

the many-molecule regime.

The rest of the article is organized as follows: In Sec-

tion I, we review the properties of Morse oscillators.

The construction of the multi-level quantum Rabi model

is discussed in Section II. In Section III, we describe

the spectrum of vibrational polaritons arising from the

model. In Section IV, we analyze the representation of

vibrational polaritons in nuclear coordinate space. In

Section V, we analyze the level crossings that occur in

the excited polariton manifold. Section VI discusses the

eﬀect of the dipole self-energy term in the multipolar

Hamiltonian. We conclude and discuss future develop-

ments in Section VII.

I. MORSE OSCILLATOR

We model the nuclear motion of an anharmonic polar

vibration (e.g. carbonyl) with a Morse potential [41]

V(q) = De(1 −exp[−a(q−qe)])2,(1)

where Deis the classical dissociation energy (without

zero point motion), qeis the equilibrium bond length,

and ais a parameter. The vibrational Schr¨odinger equa-

tion with a Morse potential can be solved analytically

in terms of associated Laguerre polynomials [41,42], or

numerically using grid-based methods [43]. We use DVR

on a uniform grid with Fourier basis functions [44] to

obtain the vibrational wavefunctions and eigenvalues of

the Morse potential. For the dimensionless parameters

De= 12.0, qe= 4.0, and a= 0.2041, the correspond-

ing potential is shown in Fig. 1a. For the dimensionless

mass µ= 1, this potential has 24 bound states and is used

throughout. In the Supplemental Material, we show that

our conclusions do not vary qualitatively for other Morse

potential parameters.

As we explain below, the degree of anharmonicity in

the potential has a profound eﬀect in the behaviour of

vibrational polaritons. For the Morse potential, the an-

harmonicity can be easily tuned by changing the param-

eters aand µ, for ﬁxed binding energy De. The relation

between these parameters and the degree of anharmonic-

ity can be understood from the exact eigenvalues of the

FIG. 1: (a) Morse potential V(q), in units of the classical

dissociation energy De. We use De= 12, qe= 4, a= 0.204

(dimensionless) to generate 24 bound states. (b) Anharmonic-

ity of the energy spacing between adjacent Morse eigenstates

∆E≡ων,ν−1−ω10 . We use De= 12.0, qe= 4.0 for all points,

with µ= 3 and a= 0.204 (circles), µ= 1 and a= 0.175 (dia-

monds), and µ= 1 and a= 0.233 (squares). The dashed line

is the harmonic oscillator limit (∆E= 0).

Morse Hamiltonian [41]

Eν=−De+a~s2De

µ(ν+ 1/2) −a2~2

2Deµ(ν+ 1/2)2(2)

where νis the vibrational quantum number. By compar-

ing this expression with the Dunham expansion [45]

Eν=Y00 +ω0(ν+ 1/2) −ω0χe(ν+ 1/2)2+. . . (3)

where ω0is the vibrational frequency, the anharmonic

coeﬃcient χecan be written as

χe=π~2a

(2µ)1/2D3/2

e

.(4)

Vibrations with lower µand higher atherefore have

stronger spectral anharmonicity, for ﬁxed dissociation en-

ergy. We illustrate this dependence in Fig. 1b, where the

change in the vibrational energy level spacing relative to

the fundamental frequency ω10 for ν= 1 ←ν= 0 is

shown for diﬀerent values of µand a. The level spacing

between adjacent vibrational states can be signiﬁcantly

smaller than the harmonic oscillator value ω10, even for

relatively low values of ν.

The electronic wavefunction determines the contribu-

tion to the molecular dipole moment of the electron

charge distribution, which in the Born-Oppenheimer ap-

proximation is a parametric function d({q}) of all nu-

clear coordinates {q}. In general, the dipole function

d({q}) can be obtained using ab-initio quantum chem-

istry for simple molecular species. Since we are inter-

ested in understanding universal features of anharmonic

vibrational polaritons, we adopt a model functional form

d(q) that captures the correct physical behaviour of a

one-dimensional polar bond. The function must be: (i)

continuous over the entire range of q; (ii) have a maxi-

mum at some value of q, not necessarily the equilibrium

distance; (iii) asymptotically vanish as the neutral bond

dissociates into neutral species. These requirements are

satisﬁed by a Rayleigh distribution function of the form

d(q)=(q+c0) exp [−q2/2σ2],(5)

3

FIG. 2: (a) Rayleigh distribution model for the electric dipole function of a polar bond d(q) normalized to its maximum value.

(b) Permanent dipole moment matrix elements hν|de(q)|νias a function of the vibrational quantum number ν. (c) Dipole

matrix elements hν|de(q)|ν0ias a function of ν, for ν0= 0 (circles) and ν0= 5 (squares). Dipole matrix elements are normalized

to the maximum of d(q). The Morse parameters used are De= 12.0, qe= 4.0, a= 0.204 and µ= 1.

which we show in Fig. 2a for c0= 0 and σ/qe= 1.25.

These parameters are used throughout. σis the coor-

dinate at which the electronic dipole moment is greater.

The parameter c0is the dipole moment for q/qe1. In

the Supplemental Material, we show that our results and

conclusions do not qualitatively vary for diﬀerent choices

of σand c0.

In order to describe light-matter coupling properly,

we not only need a reasonable description of the elec-

tric dipole moment near the equilibrium distance qe, but

also in the long range up to the dissociation threshold.

This is because strong light-matter coupling in a cavity

can strongly admix several vibrational eigenstates with

high ν. Furthermore, we are interested in studying how

highly exited polaritons behave near the energy disso-

ciation threshold of the free-space molecular sub-system.

Therefore, dipole matrix elements between all bound and

unbound states of the Morse potential must be accurately

estimated. In Figs. 2b and 2c we show the scaling with ν

of the diagonal and oﬀ-diagonal vibrational dipole matrix

elements. The permanent dipole moments hν|ˆ

d(q)|νide-

crease with ν(panel 2b), as expected from the behaviour

of d(q) on a neutral molecular system. This trend also

holds for Morse oscillators with diﬀerent aand µparame-

ters. The higher the oscillator’s mass, the lower the rate

of decrease. Fig. 2c shows that for a ﬁxed vibrational

eigenstate |ν0i, the transition dipole moments with neigh-

bouring states |νi(ν6=ν0) are not negligible, and must

be taken into account in the light-matter coupling.

In linear infrared absorption of high-frequency modes

(e.g., ω10 ≈200 meV for carbonyl), only the ground vi-

brational level ν= 0 is populated at room temperature

(kT /~ω10 1). The oscillator strength of the funda-

mental absorption peak (ν= 0 →1) and its overtones

(|∆ν| ≥ 2) are thus proportional to |hν|ˆ

d(q)|0i|2with

ν≥1. Figure 2c (circles) captures the typical IR absorp-

tion pattern of decreasing overtone strength for higher

∆ν[46]. This qualitatively correct behaviour validates

the dipole model function d(q) in Eq. (5).

Using strong infrared laser pulses, it is possible to

prepare vibrational modes with high quantum numbers

ν1, even when kT/~ω10 1. This oﬀ-resonant driv-

ing is known as vibrational ladder climbing, and has been

used in nonlinear spectroscopic measurements [47]. Vi-

brational ladder climbing is determined by the matrix el-

ements hν|ˆ

d|ν0iwith ν06=ν≥1, corresponding to dipole

transitions between overtones. Fig. 2c (squares) shows

that these high-νmatrix elements can be as strong as

the ﬁrst overtones of the fundamental transition (circles),

over a range of neighbouring levels with |ν−ν0| ≤ 4, for

our choice of d(q). We show below that ignoring dipole

couplings between high-νovertones fails to describe the

rich and complex physics of the excited polariton mani-

fold up to the dissociation threshold. Excited polariton

levels can be expected to be relevant in the description of

nonlinear cavity transmission signals, chemical reactions,

and heat transport.

II. MULTI-LEVEL QUANTUM RABI MODEL

We derive the total Hamiltonian for the molecule-

cavity system starting using the Power-Zineau-Wolley

(PZW) multipolar formulation of light-matter interaction

[48]. The PZW frame is equivalent to minimal-coupling

by a unitary transformation that eliminates the vector

potential A(x) from the Hamiltonian [48]. We divide

the total Hamiltonian ˆ

Hin the three terms of the form

ˆ

H=ˆ

HM+ˆ

HC+ˆ

HLM. The molecular part is given by

ˆ

HM=ˆ

Hel +ˆ

Hvib +ˆ

Hrot +Zdx|P(x)|2,(6)

where the ﬁrst three terms represent the electronic, vi-

brational and rotational contributions, respectively. The

last term corresponds to the dipole self-energy, with P(x)

being the macroscopic polarization density.

The free cavity Hamiltonian ˆ

HCis given by

ˆ

HC=1

2Zdx|D(x)|2+1

µ0|H(x)|2

=X

ξ

~ωξˆa†

ξˆaξ+ 1/2,(7)

4

where D(x) and H(x) are the macroscopic displacement

and magnetic ﬁelds, respectively. µ0is the magnetic per-

meability. In the second line, we imposed canonical ﬁeld

quantization into a set of normal modes with contin-

uum label ξ, frequencies ωξ, and annihilation operators

ˆaξ. Light-matter interaction in the PZW frame, ignoring

magnetic moments, is given by [48]

ˆ

HLM =Zdx P(x)·D(x).(8)

We consider a non-rotating polar bond and therefore

set ˆ

Hrot = 0. The solutions of the electronic Hamil-

tonian ˆ

Hel are assumed to be known within the Born-

Oppenheimer approximation, such that they give the

dipole function d(q). The self-energy term in Eq. (6) will

be shown to produce a state-dependent vibrational shift

that does not qualitatively aﬀect the polariton spectrum

and eigenstates, and can be ignored to simplify the analy-

sis. In Section VI we put the dipole self-energy back into

the Hamiltonian and discuss its eﬀect on the polariton

spectrum. We adopt a point-dipole approximation for

the polarization density, i.e. P(x) = dδ(x−x0), where

dis the electric dipole vector and x0is the location of

the molecule.

We use a single-mode approximation for the cavity

Hamiltonian in Eq. (7) by setting ωξ≡ωcfor all ξ

and deﬁning the eﬀective ﬁeld operators ˆa=Pξˆaξ(up

to a normalization constant). This simpliﬁcation is jus-

tiﬁed in Fabry-P´erot cavities with a large free-spectral

range (FSR ∼300 −500 cm−1[9]), and low transmission

linewidths (FWHM ∼10 −40 cm−1[9]). In this approx-

imation, the intracavity displacement ﬁeld operator can

be approximated by ˆ

D≈ E0(ˆa+ ˆa†), where E0can be

considered as the amplitude of the vacuum ﬁeld ﬂuctua-

tions, or the electric ﬁeld per photon (ignoring vectorial

character). E0scales as 1/√Vmwith the eﬀective cavity

mode volume Vm[49]. We thus write the light-matter

interaction term as

ˆ

HLM =E0(ˆ

d++ˆ

d−)⊗(ˆa+ ˆa†),(9)

where the up-transition operator ˆ

d+projected into the

vibrational energy basis |νiis given by

ˆ

d+=X

ν,ν0>ν hν0|d(q)|νi |ν0i hν|,(10)

with ˆ

d−= ( ˆ

d+)†. By combining Eqs. (6) without self-

energy, Eq. (7) in the single-mode approximation, and

Eq. (9), we can arrive at the total system Hamiltonian

ˆ

H=ωcˆa†ˆa+X

ν

ων|νihν|(11)

+X

νX

ν0>ν

gν0ν(|ν0ihν|+|νihν0|)(ˆa+ ˆa†)

where ωνis the energy of the vibrational eigenstate |νi,

and gν0ν=E0hν0|d(q)|νifor ν0> ν is a state-dependent

Rabi frequency. The zero of energy is deﬁned by the

energy of the vibrational ground state (ν= 0) in the

cavity vacuum. Equation (11) corresponds to a multi-

level quantum Rabi (MLQR) model, which reduces to

the quantum Rabi model for a two-level system [50–52],

when the vibrational space is truncated to ν= 0,1, and

the energy reference rescaled.

The vacuum ﬁeld amplitude E0is considered here as a

tunable parameter that determines the light-matter cou-

pling strength. In a cavity with small mode volume, the

mode amplitude E0can be large and tunable by fabrica-

tion [12]. Moreover, the cavity detuning ∆ ≡ωc−ω10 is

another energy scale that can be tuned by fabrication.

For convenience, we deﬁne the state-independent Rabi

frequency

g≡g10 =E0h1|d(q)|0i.(12)

Although we use the single parameter gto quantify light-

matter coupling strength throughout, we emphasize that

dipole transitions ν↔ν0in Eq. (11) have in general

diﬀerent coupling strengths.

III. SPECTRUM OF VIBRATIONAL

POLARITONS

In order to gain some physical intuition about the

structure of vibrational polaritons, in Fig. 3we illustrate

the light-matter coupling scheme implied by the uncou-

pled basis |νi|ni, where |niis a cavity Fock state. We

can associate a complete vibrational manifold {|νi;ν=

0,1,2, . . .}to every Fock state of the cavity |ni. The

ground level in each vibrational manifold (ν= 0) has

energy nωcin the Fock state |ni, and the dissociation

energy E∞becomes

E∞=De+nωc,(13)

Only in the cavity vacuum (n= 0), the bond dissociation

energy coincides with the value expected for a Morse os-

cillator in free space. In general, the energy required to

break a chemical bond depends on quantum state of the

cavity ﬁeld.

Vibrational manifolds with diﬀerent Fock states can

couple each other via the light-matter term in Eq. (9).

Since parity is broken for vibrational states due to an-

harmonicity, the only quasi selection rule that holds is

∆n=±1, because the free cavity Hamiltonian ˆ

HCcom-

mutes with parity. Therefore, vibrational states |νiand

|ν0ithat diﬀer by one photon number can admix due

to light-matter coupling. Because of anharmonicity, ad-

mixing of vibrational states with |ν−ν0| ≥ 1 is allowed.

The amount of admixing that can occur between vibra-

tional eigenstates in diﬀerent manifolds is ultimately de-

termined by the electric dipole function d(q).

The number bare states |νi|nithat can potentially

admix to form vibrational polariton eigenstates grows

as the total energy increases. Figure 3b shows that

5

FIG. 3: (a) Illustration of resonant light-matter coupling be-

tween a Morse oscillator with dissociation energy Deand

a quantized cavity ﬁeld with photon number n(unbound).

Each Morse potential corresponds to the uncoupled subspace

|νi |ni. (b) Low energy couplings involving the subspace S1=

{|1i |0i,|0i |1i} at E≈ω10, and S2={|2i |0i,|1i |1i,|0i |2i}

at E≈2ω10. Dipole coupling within S1leads to the forma-

tion of the lower and upper polaritons, and coupling within S2

to the formation of a polariton triplet. (c) High energy cou-

plings involving S6at E≈6ω10. State |6i |0iis red shifted

with respect to |0i |6iby δ6, for ω10 =ωc. The highlighted

levels can strongly admix.

for the lowest Fock states, resonant coupling at energy

E≈ω10 only involves the subspace S1={|1i|0i,|0i|1i}

for g/ω10 1. This coupling results in the formation of

the so-called lower polariton (LP) and upper polariton

(UP), which are observable in linear spectroscopy [6,9].

They can be written as

|Ψ1i=α|0i|1i − β|1i|0i(14a)

|Ψ2i=β|0i|1i+α|1i|0i(14b)

where |Ψ1iand |Ψ2icorrespond to LP and UP, respec-

tively. The orthonormal coeﬃcients αand βdepend on

gand ∆. |Ψ1iand |Ψ2iin Eq. (14) coincide with the

ﬁrst excitation manifold of the Jaynes-Cummings model

[53]. Figure 3b also shows that for g/ω10 1, reso-

nant coupling at energy E≈2ω10 only involves the sub-

space S2={|2i|0i,|1i|1i,|0i|2i}, leading to the forma-

tion of three polariton branches, as discussed below. For

g/ω10 ∼0.1 coupling of bare states |νi|nibeyond S1and

S2is allowed by counter-rotating terms in Eq. (9).

In Fig. 3c, we consider the coupling between vibra-

tional manifolds around energy E≈6ω10. If the molecu-

lar vibrations were harmonic, vibrational states |νiwould

have energy νω10. Due to anharmonicity, vibrational lev-

els in free space have energy

ων=ν ω10 −δν,

where δν>0 is the shift from a harmonic oscillator level,

shown in Fig. 1for a Morse oscillator. For ν= 6, the an-

harmonic shift δ6is not negligible in comparison with ω10,

which means that for the smaller couplings g/ω10 1,

the number of bare states |νi|nithat can resonantly ad-

mix is relatively limited. This resembles the role of an-

harmonicity in limiting the eﬃciency of vibrational lad-

der climbing using laser pulses [47,54].

On the other hand, Fig. 3c suggests that for larger

coupling ratios g/ω10 there is a greater number of quasi-

degenerate bare states |νi|nithat are energetically avail-

able to admix within an energy range 2δ. As the to-

tal energy increases, the density of quasi-degenerate bare

states that can strongly admix within a bandwidth δE

grows. We show below that this complex coupling struc-

ture leads to a large density of true and avoided crossings

in the excited polariton manifold, even for relatively low

values of the coupling ratio g/ω10.

In Fig. 4, we show the spectrum of anharmonic vi-

brational polaritons as a function of gand ∆. Figure

4a shows that the system has a unique non-degenerate

ground state |Ψ0i(GS). The ﬁrst excited manifold fea-

tures a LP-UP doublet that scales linearly with gover

the range of couplings considered (R2= 1.000 for log-

log ﬁt). However, the LP-UP splitting is not symmetric

around E=ω10, which is the energy of the degenerate

bare states |1i|0iand |0i|1ifor ωc=ω10. Fig. 4a shows

the polariton triplet around E= 2ω10, associated with

light-matter coupling within the subspace S2discussed

above (see Fig. 3b). Multiple true and avoided crossings

occur at energies E≥2ω10 over the entire range of cou-

plings considered. The density of energy crossings grows

with increasing energy.

In Fig. 4b we compare the energies of the lowest ﬁve

excited states obtained by three levels of theory: (i) the

MLQR model in Eq. (11); (ii) the quantum Rabi model

for a two-level vibration involving states ν={0,1}; (iii) a

three-level quantum Rabi model with ν={0,1,2}which

takes into account the anharmonicy shift of the transition

ν= 1 →ν= 2, i.e., ω21 =ω10 −δ2. The latter was used

in Ref. [38] to interpret the intracavity diﬀerential ab-

sorption spectrum of W(CO2)6[55]. By construction, the

qubit model can qualitatively match the asymmetric LP-

UP splitting around E=ω10 over the range g/ω10 ≤0.1,

but deviations occur for larger coupling strengths. Since

g/ω10 = 0.1 is conventionally regarded as the onset of

the ultrastrong coupling regime [31], the deviations of the

two-level model from MLQR for g/ω10 >0.1, can be at-

tributed to the inability of the truncated two-level model

to capture counter-rotating overtone couplings properly.

By increasing the dimensionality of the vibrational basis

6

FIG. 4: (a) Spectrum of anharmonic vibrational polaritons as a function of the coupling strength g/ω10, for resonant coupling

ωc=ω10. The ground state (GS), lower (LP) and upper (UP) polaritons are highlighted. (b) Spectrum of the lowest ﬁve

excited polaritons obtained by three levels of theory: the multi-level quantum Rabi model of Eq. (11) (solid line), the quantum

Rabi model for a two-level vibration ν={0,1}(open circles) and an anharmonic three-level quantum Rabi model with

ν={0,1,2}(dashed line). (c) Vibrational polaritons spectrum as a function of the cavity detuning from the fundamental

frequency ∆ = ωc−ω10 . The GS and LP are highlighted. We set g= 0.2ω0. Energy is in units of ω10.

by one additional state (ν= 2), the three-level quantum

Rabi model matches better the LP-UP spectrum pre-

dicted by the MLQR model, but is unable to correctly

capture the splitting of the polariton triplet around en-

ergy E= 2 ω10, except for the smallest coupling ratios

(g/ω10 1).

The comparison between models in Fig. 4b suggests

that for the excited vibrational polaritons considered,

the onset of ultrastrong coupling–where counter-rotating

terms in the light-matter interaction becomes important–

occurs at much smaller values of gthan those expected for

a qubit, and can involve oﬀ-resonant coupling to higher

vibrational levels with ν≥3. For excited polaritons with

energies E≥3ω10, few-level truncations of the material

Hamiltonian (e.g. Ref. [38]) fail to capture the multiple

true and avoided crossings that the Hamiltonian allows.

We further discuss these excited state crossings in Section

V.

In Fig. 4c, we show the polariton spectrum as a func-

tion of detuning ∆ ≡ωc−ω10, for g/ω10 = 0.2. Several

true and avoided crossings develop in the excited mani-

fold. When ∆ ∼g, the energetic ordering of the excited

polaritons can change in comparison with the resonant

regime (∆/g 1). For example, there is an avoided

crossing at E≈2.1ω10 near ∆ ≈ −0.1ω10. The upper

polariton (UP) also crosses with the next excited polari-

ton level at ∆ ≈ −0.28 ω01. This raises concerns regard-

ing the assignment of spectral lines in linear and non-

linear cavity transmission spectroscopy for light-matter

coupling in the dispersive regime |∆|/g &1.

IV. VIBRATIONAL POLARITONS IN

NUCLEAR COORDINATE SPACE

In molecules and materials, the strength of a chemical

bond is commonly associated with its vibration frequency

ω0via the relation

ω0=pk/µ, (15)

where kis the bond spring constant and µis the reduced

mass of the vibrating nuclei. Stronger bonds (higher k)

thus lead to higher vibrational frequencies. This simple

argument has also been used to discuss the bonding char-

acter of vibrational polaritons under strong coupling [6].

In this Section, we show that the description of the bond-

ing strength of vibrational polaritons is far more complex

than the commonly used spring model suggests.

In order to analyze vibrational polaritons in nuclear

coordinate space, keeping photons in Hilbert space, let

us expand the eigenstates of Eq. (11) in the uncoupled

basis {|νi|ni} as

|Ψji=X

ν,n

cj

νn |νi |ni,(16)

where cj

νn are orthonormal coeﬃcients associated with

the j-th eigenstate. We can rewrite Eq. (16) by com-

bining vibrational components associated with a given

photon number nas

|Ψji=X

n|Φj

ni|ni,(17)

where |Φj

ni=Pνcj

νn |νi. The state |Φj

nican be inter-

preted as a vibrational wavepacket conditional on the

cavity photon number. Its nuclear coordinate represen-

tation is simply given by the projection

Φjn (q) = hq|Φj

ni.(18)

7

FIG. 5: Conditional probability densities |Φ6n(q)|2for the ex-

cited polariton eigenstate |Ψ6i, for coupling strengths g/ω10 =

0.002 (a) and g/ω10 = 0.2 (b). Coordinates are in units of the

bare equilibrium bond length qe. All densities are normalized.

For concreteness, we show in Fig. 5a set of normal-

ized conditional probability distributions |Φjn(q)|2with

n≤4, for the excited polariton eigenstate |Ψ6iunder

resonant light-matter coupling. Since the energy of ex-

cited polariton |Ψ6itends asymptotically to E6≈3ω10

as g/ω10 →0, one could expect the normalized proba-

bility distribution |Φ6n(q)|2to resemble the behaviour

of the Morse oscillator eigenfunction with ν= 3 for

g/ω10 1. Figure 5(lower panel) shows that indeed

the vacuum component (n= 0) of |Ψ6iqualitatively

matches the node structure of the bare Morse oscillator

state |ν= 3ifor g/ω10 = 0.002. However, for the cou-

pling ratio g/ω10 = 0.2, the nuclear density of the cavity

vacuum |Φ60(q)|2behaves qualitatively diﬀerent from a

Morse eigenfunction. Similar deviations from the bare

Morse behavior occurs also for nuclear components with

higher photon numbers (n≥1).

Figure 5also shows that the nuclear densities |Φ6n(q)|2

associated with n≥1 can also approximately resemble

the node pattern of a bare Morse oscillator with the ap-

FIG. 6: Probability amplitudes |cνn|2in the uncoupled basis

|νi |nifor excited polariton eigenstates |Ψ6i(a) and |Ψ8i(b),

as a function the coupling strength g/ω10 . Curves are labelled

by the quantum numbers (ν, n). We set ωc=ω10.

propriate number of excitations, for small values of g/ω10.

For example, since the energy of |Ψ6itends to E≈3ω10

as g→0, its wave function should have components in

the uncoupled basis |νi|nisuch that ν+n= 3 at zero

detuning (ωc=ω10). For g/ω10 = 0.002, Fig. 5shows

that indeed for n= 1 the nuclear density |Φ61(q)|2of

state |Ψ6ihas a node structure similar to the bare Morse

eigenstate |ν= 2i, i.e., it has two nodes. The nuclear

densities associated with n= 2 and n= 3 also seem to

satisfy a conservation rule for the total number of excita-

tions (ν+n). This rule however is broken for the n= 4

nuclear wave packet Φ64(q) (Fig. 5, upper panel), which

has a node structure similar to the bare Morse eigenstate

|ν= 1i, corresponding to a total number of excitations

ν+n= 5 for all values of g/ω10 considered.

In order to assess the contribution of each photon-

number-dependent nuclear wave packet Φjn(q) on the

j-th polariton eigenstate |Ψji, we show in Fig. 6the

probability amplitudes |cνn|2[see Eq. (16)] as a func-

tion of the coupling ratio g/ω10 , for the excited polari-

ton eigenstates |Ψ6iand |Ψ8i. These two excited states

tend asymptotically to the energy E≈3ω10 as g→0,

and therefore can be expected to be mainly composed

of uncoupled states |νi|nisuch that ν+n= 3, for res-

onant coupling. Figure 6shows that indeed occurs for

g/ω10 1. In this small coupling regime, the selected

8

FIG. 7: Mean bond length hˆqiand mean cavity photon num-

ber hˆa†ˆaias a function of coupling strength g/ω10 (a,c) and

cavity detuning ∆ (b,d), for the system ground state (dashed

line), lower polariton (solid line) and upper polariton (dot-

dashed line). We set ∆ = 0 in panels a,c and g/ω10 = 0.1 in

panels b,d. Energy is in units of ω10.

polariton eigenstates can be approximately written in the

basis |νi|nias

|Ψ6i≈|3i|0i(19)

and

|Ψ8i ≈ a|0i|3i+b|1i|2i,(20)

where |a|2≈ |b|2= 0.5. As the coupling strength reaches

the regime g/ω10 ∼0.1, the near resonant coupling be-

tween vibrational manifolds with higher photon numbers

in Fig. 3leads to the emergence of wave function com-

ponents with lower vibrational quantum numbers. For

instance, for g/ω10 = 0.2 the excited state |Ψ6iis ap-

proximately given by

|Ψ6i ≈ a|1i|2i+b|0i|3i+c|2i|0i+d|1i|4i(21)

where |a|2∼ |b|2>|c|2 |d|2. In other words, the

state evolves from a bare Morse oscillator |ν= 3iin vac-

uum [see Eq. (19)], into a state with a lower mean

vibrational excitation and higher mean photon number

as g/ω10 grows. On the other hand, the state |Ψ8iat

g/ω10 = 0.2 can be written as

|Ψ8i ≈ a|3i|0i+b|2i|1i+c|0i|3i+d|1i|4i,(22)

where |a|2≈1/2>|b|2>|c|2 |d|2, which also devel-

ops components with lower vibrational quanta and higher

FIG. 8: Mean bond length hˆqiand mean cavity photon num-

ber hˆa†ˆaias a function of coupling strength g/ω10 (a,c) and

cavity detuning ∆ (b,d), for excited polaritons |Ψ6i(solid

line) and |Ψ8i(dashed line). We set ∆ = 0 in panels a,c and

g/ω10 = 0.1 in panels b,d. Energy is in units of ω10 .

photon numbers in comparison with Eq. 20. The emer-

gence of uncoupled components with ν+n6= 3 in Eqs.

(21) and (22) is a consequence of the counter-rotating

terms in Eq. (11). Although the results in Figs. 5and 6

were obtained for speciﬁc polariton eigenstates, we ﬁnd

that they qualitatively describe the behavior of most ex-

cited polaritons |Ψjiwith energies Ejω10, i.e., above

the LP and UP frequency region.

We can also understand the structure of vibrational

polaritons in coordinate space and Fock space by analyz-

ing the dependence of the mean bond distance hˆqiand

the mean photon number hˆa†ˆaiwith the coupling pa-

rameter gand cavity detuning ∆, for selected polariton

eigenstates. In Fig. 7, we compare the evolution of these

observables with gand ∆ for the system ground state

|Ψ0i(GS), the lower polariton state |Ψ1iand the upper

polariton state |Ψ2i. In the regime g/ω10 1, both LP

and UP have the approximately the same bond length,

given by

hˆqi ≈ 1

2(h0|ˆq|0i+h1|ˆq|1i),(23)

with expectation value taken with respect to Morse eigen-

states |νi.

Figure 7a shows that as the coupling strength in-

creases, the bond length of the LP decreases, reaching

values even lower than the bond length of the Morse

ground state |ν= 0i. On the other hand, the bond

length of the UP grows with increasing coupling strength.

9

FIG. 9: (a) Spectral region with an avoided crossing (circled in grey) involving the excited polaritons |Ψ9i(blue) and |Ψ10i

(red). (b) and (c) Main components of |Ψ9iand |Ψ10i, respectively, in the uncoupled basis |νi |ni. Curves are labelled by the

quantum numbers (ν, n). We set ωc=ω10.

The value of hˆqifor the UP is upper bounded by the

bond length of the ﬁrst excited Morse state |ν= 1i. In

other words, for resonant coupling the molecular bond

in the LP state becomes stronger relative to the UP

with increasing coupling strength, although both polari-

ton states experience bond hardening in comparison with

the Morse eigenstate |ν= 1i, which is in the same energy

region as LP and UP (E/ω10 ≈1).

Bond hardening should be accompanied by the cre-

ation of virtual cavity photons, and bond softening by the

decrease in the mean photon number. Figure 7b shows

that the GS, LP and UP states follow this behaviour as

a function of g/ω10 , for resonant coupling. We show in

panels 7b,d that for detuned cavities, the compromise be-

tween bond strength and cavity photon occupation also

holds. Within the range of system parameters consid-

ered, we ﬁnd that this compromise also holds for higher

excited vibrational polaritons, as Fig. 8shows for states

|Ψ6iand |Ψ8i.

Bond hardening of vibrational polaritons can be under-

stood by recalling that an eigenstate |Ψjiin the vicin-

ity of a bare Morse energy level Eν0in general has non-

vanishing components in the uncoupled basis |νi|niwith

ν < ν 0[see Eq. (16)]. These low-νcomponents con-

tribute to the stabilization of the molecular bond even at

high excitation energies.

V. ENERGY CROSSINGS IN THE EXCITED

POLARITON MANIFOLD

We discussed in Section III how the density of polariton

levels increases with energy, ultimately due to the large

number of near-degenerate uncoupled subspaces |νi|ni

(see Fig. 3). Light-matter coupling leads to the forma-

tion of closely-spaced polariton levels that can become

quasi-degenerate at speciﬁc values of gand ∆. As the

Hamiltonian parameters (g, ∆) are tuned across the de-

generacy point, the polariton levels may undergo true

or avoided crossing. For a Hamiltonian like the quan-

tum Rabi model for the qubit [51,52,56] and its multi-

level generalizations [57], parity is a conserved quantity.

Therefore polaritons in the quantum Rabi model have

well-deﬁned parity and level crossings are analyzed in

the usual way: states with opposite parity undergo true

crossing under variation of a Hamiltonian parameter. In

particular, the crossing of the ground state with the lower

polariton state at g/ωc= 1 marks the onset of the deep

strong coupling regime [31,58,59].

Parity conservation in the quantum Rabi model ulti-

mately emerges from the even character of the under-

lying microscopic Hamiltonians that describe the mate-

rial system and the cavity ﬁeld. The harmonic oscillator

Hamiltonian that describes the cavity ﬁeld is invariant

under the transformation ˆa→ −ˆa, and therefore com-

mutes with parity (as any harmonic oscillator Hamilto-

nian). For the material system, let ˆqand ˆprepresent po-

sition and momentum operators in the material Hamil-

tonian ˆ

HM. Then polariton eigenstates of the coupled

light-matter system would only have well-deﬁned par-

ity if ˆ

HMis invariant under the parity transformation

ˆq→ −ˆqand ˆp→ −ˆp. The Morse potential in Eq. (1)

is not invariant under the transformation q→ −qand

qe→ −qe, and therefore breaks parity, which is the ori-

gin of vibrational overtones. Polariton eigenstates of the

MLQR model therefore do not have well-deﬁned parity.

Even parity is not a good quantum number, the vibra-

tional polariton spectrum still exhibits true and avoided

level crossings as the Hamiltonian parameters gand ∆

vary. We can track the origin of these crossings into

an eﬀective photonic parity selection rule imposed by

the light-matter interaction term in the total Hamilto-

nian [Eq. (11)], which reads ∆n=±1. For two near-

degenerate polariton levels Ejand Ek(k6=j), there will

be a strong avoided crossing between them only if the

largest probability amplitudes cνn of their wavefunctions

10

FIG. 10: (a) Spectral region with true crossings involving the excited polaritons |Ψ22i(blue) |Ψ23 i(red), and |Ψ21iand |Ψ23 i

(both crossing regions circled in grey). (b) and (c) Main components of |Ψ22iand |Ψ23i, respectively, in the uncoupled basis

|νi |ni. Curves are labelled by the quantum numbers (ν, n). We set ωc=ω10.

|Ψjiand |Ψkiin the uncoupled basis |νi|ni, diﬀer by

one photon number [see Eq. 16]. Otherwise, the levels

will cross as the Hamiltonian is varied through the de-

generacy. We show this explicitly in Figs. 9and 10, with

examples of avoided and true crossings, respectively, in

the excited polariton manifold.

In Figure 9a we highlight an avoided crossing be-

tween excited polaritons |Ψ9iand |Ψ10i, as the coupling

strength is g/ω10 ≈0.19. Panels 9b and c show that the

largest uncoupled components to the left of the avoided

crossing are {|1i|2i,|0i|3i} for |Ψ9iand {|1i|3i,|0i|4i}

for |Ψ10i, which indeed diﬀer by one photon number.

Past the avoided crossing, the state |Ψ10 idominantly

acquires |0i|3icharacter.

In Fig. 10a we highlight a pair of level crossings as

g/ω10 increases. For g/ω10 '0.1, the excited polariton

states |Ψ22iand |Ψ23 iundergo the ﬁrst crossing. Fig.

10b shows that to the left of the crossing point, the largest

uncoupled components of |Ψ22iis |5i |1i, while for |Ψ23 i

the largest components are {|1i|5i,|2i|4i}. Since |Ψ22i

and |Ψ23ithus predominantly satisfy ∆n > 1, they do

not interact via by the light-matter term as gis varied

across the degeneracy. Note that Fig. 10b,c the state

does undergo a small change of character at the two cross-

ing points in Fig. 9a. This occurs because states |Ψ22 i

and |Ψ23ido have uncoupled components that interact

via the photonic selection rule ∆n±1, but their weight

in the eigenfunction is comparatively small.

VI. DIPOLE SELF-ENERGY

For simplicity, we have neglected the dipole self-energy

term throughout. This term which arises via the transfor-

mation from minimal-coupling light-matter interaction

to the multipolar interaction through the Power-Zineau-

Wolley (PZW) transformation UPZW = exp[iRdxP(x)·

A(x)] which eliminates the vector potential A(x) from

the theory. Even though the multipolar formalism is

non-covariant, it has been widely used to describe light-

molecule interaction in the non-relativistic regime [48].

For light-matter coupling in which counter-rotating

terms become important, it has been argued that the

dipole self-energy contribution should be taken into ac-

count in order to describe polaritons correctly [31]. In

molecular polariton problems, self-energy terms have

been given ad-hoc model treatments in previous work

[14,39]. There, the dipole self-energy is considered to be

proportional to the Rabi frequency g, which is in remark-

able contrast with the PZW frame, in which a material

Hamiltonian [see Eq. (6)] contains a dipole self-energy

contribution even when light-matter coupling is pertur-

bative.

In principle, relating the macroscopic polarization den-

sity P(x) with the molecular electric dipole operator

d(x0) at position x0in the medium would require an

ab-initio quantum electrodynamics formulation of ﬁeld

quantization in dispersive and absorptive media [60],

which is beyond the scope of our work. We ignore the

contribution of the dielectric background and assume

that for a single polar vibration the following ansatz holds

ˆ

Hself =Z|P(x)|2dx ≡γX

νhν|ˆ

d(q)|νi2|νihν|,(24)

where |νiare the anharmonic vibrational eigenstates. In

other words, we assume that self-energy leads to a state-

dependent blue shift of every vibrational level. We can

thus build a new total Hamiltonian ˆ

H0=ˆ

H+ˆ

Hself , where

His the MLQR model from Eq. (11). The parameter γ

is introduced to control the numerical magnitude of ˆ

Hself

relative to ˆ

H.

In Fig. 11a, we show the polariton spectrum with in-

creasing γ, for gand ∆ ﬁxed. As expected [14,39], ˆ

Hself

only results in a state-dependent positive energy shift in

the polariton levels. Once the dipole-shifted ground state

11

FIG. 11: (a) Vibrational polariton spectrum as a function

of the dipole self-energy parameter γ, for g= 0.2ω10 and

ωc=ω10. (b) Comparison of the polariton spectrum as

a function of the coupling strength g/ω10 using the MLQR

model without (solid line) and with (dashed line) the dipole

self-energy term with γ= 0.15. Energy is given relative to

the ground state (GS), in units of ω10.

energy EGS is subtracted from the energies of H0, Fig.

11b shows that the energy spectrum of H0has the same

qualitative behavior with increasing gas the spectrum

of ˆ

H. Quantitative deviations from the MLQR spectrum

due to self-energy become important for relative energies

(E−EGS)&2ω10 , when the magnitude of the parameter

γis comparable with the coupling ratio g/ω10 .

VII. CONCLUSION AND OUTLOOK

In order to understand the microscopic behaviour of an

individual anharmonic molecular vibration coupled to a

single infrared cavity mode, we introduce and analyze the

multi-level quantum Rabi (MLQR) model of vibrational

polaritons [Eq. (11)]. We derive the model Hamiltonian

starting from the exact anharmonic solutions of a free-

space Morse oscillator, and treat light-matter interaction

within the Power-Zineau-Wolley multipolar framework

[48], which includes the dipole self-energy. The model

takes into account counter-rotating terms in the light-

matter coupling and allows the analysis of vibrational

polaritons both in Hilbert space and nuclear coordinate

space. Phase-space representations of the photon state

follow directly from the QED formulation of the model

[61]. Such phase-space analysis would be closely related

to previous coordinate-only treatments of photon-nuclei

coupling [28,39,62], although a systematic comparison

has yet to be done.

The model is consistent with previous work based on

few-level vibrational systems [38], and therefore is also

able to describe the spectral features observed in lin-

ear and nonlinear transmission spectroscopy [26,55,63],

which due to the relatively weak intensities involved, can

only probe up to the second excited polariton triplet

around E≈2ω10, where ω10 is the fundamental vi-

bration frequency. Few-level vibrational truncations are

however unable to capture the dense and complex po-

lariton level structure predicted by the MLQR model at

energies E&3ω10. The system Hamiltonian allows the

emergence of an ensemble of avoided and true crossings

as the Rabi frequency gand cavity detuning ∆ are tuned.

The density of these level crossings increases with energy.

These crossings are governed by a pseudo-parity selection

rule in the photonic degree of freedom (details in Sec. V).

The nuclear coordinate analysis of vibrational polari-

tons within the MLQR model unveils a few general trends

accross the entire energy spectrum. First, it is no longer

possible to deﬁne a unique bond dissociation energy in

an infrared cavity as is commonly done in free space.

The dissociation energy depends on the quantum state of

the cavity ﬁeld. Second, within any given energy range

Ej+ ∆E, it is always possible to ﬁnd a vibrational po-

lariton eigenstate with small mean photon number hˆa†ˆai

and large mean bond distance hˆqi, and vice-versa. Third,

the bond distance hˆqiof an arbitrary vibrational po-

lariton state with energy Ej, never exceeds the bond

length of a free-space Morse eigenstate |νiwith similar

energy (Eν≈Ej). In other words, the formation of vi-

brational polaritons inside the cavity leads to a type of

bond-hardening eﬀect that may have consequences in the

reactivity of chemical bonds.

The generalization of the multi-level quantum Rabi

model developed here to the many-molecule and multi-

mode scenarios is straightforward. Since it is formulated

in the energy eigenbasis, treating the dissipative dynam-

ics of vibrational polaritons due to cavity photon decay

and vibrational relaxation is also straightforward to for-

mulate within a Markovian approach [64]. The dynam-

ics of vibrational polaritons in the many-body regime

has been previously discussed in Refs. [33,65], using

truncated vibrational subspaces. The main qualitatively

new eﬀect that the many-body system introduces to the

problem, is the formation of collective molecular states

that are not symmetric with respect to particle permu-

tations. These so-called “dark exciton states” [66] arise

naturally from state classiﬁcation by permutation sym-

metry in the Hilbert space of the Dicke model [67,68].

It has been shown originally within a quasi-particle ap-

proach for systems with macroscopic translational invari-

ance [69], and later using a cavity QED approach [32,70–

72], that totally-symmetric and non-symmetric collective

molecular states can strongly admix due to ever-present

inhomogeneous broadening of molecular energy levels,

inhomogeneities in the light-matter interaction energy

across the medium, or any local coherent term such as

intramolecular electron-vibration coupling (in the case

of electronic strong coupling [72]). In general, the role

of quasi-dark collective states in determining the rate of

chemical reactions and also spectroscopic signals of vi-

brational polaritons is yet to be fully understood.

12

Acknowledgments

We thank Guillermo Romero, Blake Simpkins and Jef-

frey Owrutsky for discussions. This work is supported by

CONICYT through the Proyecto REDES ETAPA INI-

CIAL, Convocatoria 2017 no. REDI 170423, FONDE-

CYT Regular No. 1181743, and also thank support by

Iniciativa Cient´ıﬁca Milenio (ICM) through the Millen-

nium Institute for Research in Optics (MIRO).

[1] H Jeﬀ Kimble. The quantum internet. Nature,

453(7198):1023, 2008.

[2] Jeremy L. O’Brien, Akira Furusawa, and Jelena

Vuˇckovi´c. Photonic quantum technologies. Nature Pho-

tonics, 3(12):687–695, December 2009.

[3] H. Mabuchi and A. C. Doherty. Cavity quantum

electrodynamics: Coherence in context. Science,

298(5597):1372–1377, 2002.

[4] Alexandre Blais, Ren-Shou Huang, Andreas Wallraﬀ,

S. M. Girvin, and R. J. Schoelkopf. Cavity quantum

electrodynamics for superconducting electrical circuits:

An architecture for quantum computation. Phys. Rev.

A, 69:062320, Jun 2004.

[5] R. Miller et al. Trapped atoms in cavity qed: coupling

quantized light and matter. J. Phys. B: At., Mol. Opt.

Phys., 38(9):S551, 2005.

[6] Thomas W Ebbesen. Hybrid Light-Matter States in a

Molecular and Material Science Perspective. Accounts of

Chemical Research, 49:2403–2412, 2016.

[7] Antoine Canaguier-Durand, Elo¨ıse Devaux, Jino George,

Yantao Pang, James A. Hutchison, Tal Schwartz, Cyr-

iaque Genet, Nadine Wilhelms, Jean Marie Lehn, and

Thomas W. Ebbesen. Thermodynamics of molecules

strongly coupled to the vacuum ﬁeld. Angewandte

Chemie - International Edition, 52(40):10533–10536,

2013.

[8] J. P. Long and B. S. Simpkins. Coherent coupling be-

tween a molecular vibration and fabry–perot optical cav-

ity to give hybridized states in the strong coupling limit.

ACS Photonics, 2(1):130–136, 2015.

[9] B. S. Simpkins, Kenan P. Fears, Walter J. Dressick,

Bryan T. Spann, Adam D. Dunkelberger, and Jeﬀrey C.

Owrutsky. Spanning Strong to Weak Normal Mode Cou-

pling between Vibrational and FabryP´erot Cavity Modes

through Tuning of Vibrational Absorption Strength. ACS

Photonics, 2(10):1460–1467, 2015.

[10] Atef Shalabney, Jino George, Hidefumi Hiura, James A.

Hutchison, Cyriaque Genet, Petra Hellwig, and

Thomas W. Ebbesen. Enhanced Raman Scattering from

Vibro-Polariton Hybrid States. Angewandte Chemie In-

ternational Edition, 54(27):7971–7975, 2015.

[11] A. Shalabney, J. George, J. Hutchison, G. Pupillo,

C. Genet, and T. W. Ebbesen. Coherent coupling of

molecular resonators with a microcavity mode. Nature

Communications, 6(Umr 7006):1–6, 2015.

[12] Rohit Chikkaraddy, Bart de Nijs, Felix Benz, Steven J.

Barrow, Oren A. Scherman, Edina Rosta, Angela Deme-

triadou, Peter Fox, Ortwin Hess, and Jeremy J. Baum-

berg. Single-molecule strong coupling at room tempera-

ture in plasmonic nanocavities. Nature, 535(7610):127–

130, 07 2016.

[13] Felix Benz, Mikolaj K. Schmidt, Alexander Dreismann,

Rohit Chikkaraddy, Yao Zhang, Angela Demetriadou,

Cloudy Carnegie, Hamid Ohadi, Bart de Nijs, Ruben

Esteban, Javier Aizpurua, and Jeremy J. Baumberg.

Single-molecule optomechanics in picocavities. Science,

354(6313):726–729, 2016.

[14] Jino George, Thibault Chervy, Atef Shalabney, Elo¨ıse

Devaux, Hidefumi Hiura, Cyriaque Genet, and

Thomas W. Ebbesen. Multiple Rabi Splittings un-

der Ultrastrong Vibrational Coupling. Physical Review

Letters, 117(15):153601, 2016.

[15] Robrecht M.A. Vergauwe, Jino George, Thibault Chervy,

James A. Hutchison, Atef Shalabney, Vladimir Y. Tor-

beev, and Thomas W. Ebbesen. Quantum Strong Cou-

pling with Protein Vibrational Modes. Journal of Phys-

ical Chemistry Letters, 7(20):4159–4164, 2016.

[16] Anoop Thomas, Jino George, Atef Shalabney, Marian

Dryzhakov, Sreejith J. Varma, Joseph Moran, Thibault

Chervy, Xiaolan Zhong, Elo¨ıse Devaux, Cyriaque Genet,

James A. Hutchison, and Thomas W. Ebbesen. Ground-

State Chemical Reactivity under Vibrational Coupling to

the Vacuum Electromagnetic Field. Angewandte Chemie

- International Edition, 55(38):11462–11466, 2016.

[17] Manuel Hertzog, Per Rudquist, James A. Hutchison,

Jino George, Thomas W. Ebbesen, and Karl B¨orjesson.

Voltage-Controlled Switching of Strong Light-Matter In-

teractions using Liquid Crystals. Chemistry - A European

Journal, 23(72):18166–18170, 2017.

[18] Daqing Wang, Hrishikesh Kelkar, Diego Martin-Cano,

Tobias Utikal, Stephan G¨otzinger, and Vahid Sandogh-

dar. Coherent Coupling of a Single Molecule to a

Scanning Fabry-Perot Microcavity. Physical Review X,

7(2):021014, apr 2017.

[19] Vivian F. Crum, Shaelyn R. Casey, and Justin R. Sparks.

Photon-mediated hybridization of molecular vibrational

states. Physical Chemistry Chemical Physics, 20(2):850–

857, 2018.

[20] Thibault Chervy, Anoop Thomas, Elias Akiki, Robrecht

M. A. Vergauwe, Atef Shalabney, Jino George, Elo¨ıse

Devaux, James A. Hutchison, Cyriaque Genet, and

Thomas W. Ebbesen. Vibro-Polaritonic IR Emission in

the Strong Coupling Regime. ACS Photonics, 5(1):217–

224, 2018.

[21] Matthew Du, Raphael F. Ribeiro, and Joel Yuen-Zhou.

Remote control of chemistry in optical cavities. Chem,

5(5):1167 – 1181, 2019.

[22] Hidefumi Hiura, Atef Shalabney, and Jino George. Cav-

ity Catalysis Accelerating Reactions under Vibrational

Strong Coupling . Chemrxiv.Org, 2018.

[23] A. Thomas, L. Lethuillier-Karl, K. Nagarajan, R. M. A.

Vergauwe, J. George, T. Chervy, A. Shalabney, E. De-

vaux, C. Genet, J. Moran, and T. W. Ebbesen. Tilting

a ground-state reactivity landscape by vibrational strong

coupling. Science, 363(6427):615–619, 2019.

[24] Robert J. Gordon, Langchi Zhu, and Tamar Seideman.

Coherent control of chemical reactions. Accounts of

Chemical Research, 32(12):1007–1016, 1999.

13

[25] Javier del Pino, Johannes Feist, and F. J. Garcia-

Vidal. Signatures of vibrational strong coupling in ra-

man scattering. The Journal of Physical Chemistry C,

119(52):29132–29137, 12 2015.

[26] A. D. Dunkelberger, B. T. Spann, K. P. Fears, B. S. Simp-

kins, and J. C. Owrutsky. Modiﬁed relaxation dynamics

and coherent energy exchange in coupled vibration-cavity

polaritons. Nature Communications, 7:1–10, 2016.

[27] Adam Dunkelberger, Roderick Davidson, Wonmi Ahn,

Blake Simpkins, and Jeﬀrey Owrutsky. Ultrafast Trans-

mission Modulation and Recovery via Vibrational Strong

Coupling. The Journal of Physical Chemistry A,

122(4):965–971, feb 2018.

[28] Johannes Feist, Javier Galego, and Francisco J. Garcia-

Vidal. Polaritonic Chemistry with Organic Molecules.

ACS Photonics, 5(1):205–216, 2018.

[29] Raphael F. Ribeiro, Luis A. Mart´ınez-Mart´ınez,

Matthew Du, Jorge Campos-Gonzalez-Angulo, and Joel

Yuen-Zhou. Polariton chemistry: controlling molecu-

lar dynamics with optical cavities. Chemical Science,

9(30):6325–6339, 2018.

[30] Manuel Hertzog, Mao Wang, J¨urgen Mony, and Karl

B¨orjesson. Strong lightmatter interactions: a new di-

rection within chemistry. Chemical Society Reviews,

48(3):937–961, 2019.

[31] Anton Frisk Kockum, Adam Miranowicz, Simone De

Liberato, Salvatore Savasta, and Franco Nori. Ultra-

strong coupling between light and matter. Nature Re-

views Physics, 1:19–40, 2019.

[32] Felipe Herrera and Frank C. Spano. Cavity-controlled

chemistry in molecular ensembles. Phys. Rev. Lett.,

116:238301, Jun 2016.

[33] Javier del Pino, Johannes Feist, and Francisco J. Garcia-

Vidal. Quantum theory of collective strong coupling of

molecular vibrations with a microcavity mode. 2015.

[34] Merav Muallem, Alexander Palatnik, Gilbert D. Nessim,

and Yaakov R. Tischler. Strong Light-Matter Coupling

and Hybridization of Molecular Vibrations in a Low-Loss

Infrared Microcavity. Journal of Physical Chemistry Let-

ters, 7(11):2002–2008, 2016.

[35] Prasoon Saurabh and Shaul Mukamel. Two-dimensional

infrared spectroscopy of vibrational polaritons of

molecules in an optical cavity. J. Chem. Phys., 144(12),

2016.

[36] Hoi Ling Luk, Johannes Feist, J. Jussi Toppari, and Ger-

rit Groenhof. Multiscale Molecular Dynamics Simula-

tions of Polaritonic Chemistry. Journal of Chemical The-

ory and Computation, 13(9):4324–4335, 2017.

[37] Luis A. Mart´ınez-Mart´ınez, Raphael F. Ribeiro, Jorge

Campos-Gonz´alez-Angulo, and Joel Yuen-Zhou. Can Ul-

trastrong Coupling Change Ground-State Chemical Re-

actions? ACS Photonics, 5(1):167–176, 2018.

[38] Bo Xiang Wei Xiong Blake S. Simpkins Jeﬀrey C. Owrut-

sky Raphael Florentino Ribeiro, Adam D. Dunkelberger

and Joel Yuen-Zhou. Theory for Nonlinear Spectroscopy

of Vibrational Polaritons. Journal of Physical Chemistry

Letters, 9(13):3766–3771, 2018.

[39] Johannes Flick, Michael Ruggenthaler, Heiko Appel, and

Angel Rubio. Atoms and molecules in cavities, from weak

to strong coupling in quantum-electrodynamics (qed)

chemistry. Proceedings of the National Academy of Sci-

ences, 114(12):3026–3034, 2017.

[40] Johannes Flick, Heiko Appel, Michael Ruggenthaler, and

Angel Rubio. Cavity Born-Oppenheimer Approximation

for Correlated Electron-Nuclear-Photon Systems. Jour-

nal of Chemical Theory and Computation, 13(4):1616–

1625, 2017.

[41] Philip M. Morse. Diatomic Molecules According to the

Wave Mechanics. II. Vibrational Levels. Physical Review,

34(1):57–64, jul 1929.

[42] V. S. Vasan and R. J. Cross. Matrix elements for morse

oscillators. The Journal of Chemical Physics, 78(6):3869–

3871, 1983.

[43] John C. Light and Tucker Carrington Jr. Discrete-

Variable Representations and their Utilization, pages

263–310. John Wiley & Sons, Inc., 2007.

[44] Daniel T. Colbert and William H. Miller. A novel discrete

variable representation for quantum mechanical reactive

scattering via the smatrix kohn method. The Journal of

Chemical Physics, 96(3):1982–1991, 1992.

[45] Wolfgang Demtr¨oder. Molecular physics: theoretical

principles and experimental methods. John Wiley & Sons,

2008.

[46] P Jakob and BNJ Persson. Infrared spectroscopy of over-

tones and combination bands. The Journal of chemical

physics, 109(19):8641–8651, 1998.

[47] Marcus, G., Zigler, A., and Friedland, L. Molecular vi-

brational ladder climbing using a sub-nanosecond chirped

laser pulse. Europhys. Lett., 74(1):43–48, 2006.

[48] David Parker Craig and Thiru Thirunamachandran.

Molecular quantum electrodynamics: an introduction to

radiation-molecule interactions. Courier Corporation,

1998.

[49] Lukas Novotny and Bert Hecht. Principles of nano-

optics. Cambridge university press, 2012.

[50] T. Werlang, A. V. Dodonov, E. I. Duzzioni, and C. J.

Villas-Bˆoas. Rabi model beyond the rotating-wave

approximation: Generation of photons from vacuum

through decoherence. Phys. Rev. A, 78:053805, Nov 2008.

[51] D. Braak. Integrability of the rabi model. Phys. Rev.

Lett., 107:100401, Aug 2011.

[52] F. A. Wolf, F. Vallone, G. Romero, M. Kollar, E. Solano,

and D. Braak. Dynamical correlation functions and the

quantum rabi model. Phys. Rev. A, 87:023835, Feb 2013.

[53] Edwin T Jaynes and Frederick W Cummings. Compari-

son of quantum and semiclassical radiation theories with

application to the beam maser. Proceedings of the IEEE,

51(1):89–109, 1963.

[54] Oliver K¨uhn, J¨orn Manz, and Yi Zhao. Ultrafast ir

laser control of photodissociation: single- vs. multi-pulse

schemes. Phys. Chem. Chem. Phys., 1:3103–3110, 1999.

[55] Bo Xiang, Raphael F. Ribeiro, Adam D. Dunkelberger,

Jiaxi Wang, Yingmin Li, Blake S. Simpkins, Jeﬀrey C.

Owrutsky, Joel Yuen-Zhou, and Wei Xiong. Two-

dimensional infrared spectroscopy of vibrational polari-

tons. Proceedings of the National Academy of Sciences,

115(19):4845–4850, 2018.

[56] Qing-Hu Chen, Chen Wang, Shu He, Tao Liu, and Ke-

Lin Wang. Exact solvability of the quantum rabi model

using bogoliubov operators. Phys. Rev. A, 86:023822,

Aug 2012.

[57] Victor V. Albert. Quantum rabi model for n-state atoms.

Phys. Rev. Lett., 108:180401, May 2012.

[58] J. Casanova, G. Romero, I. Lizuain, J. J. Garc´ıa-Ripoll,

and E. Solano. Deep strong coupling regime of the jaynes-

cummings model. Phys. Rev. Lett., 105:263603, Dec 2010.

[59] P. Forn-D´ıaz, L. Lamata, E. Rico, J. Kono, and

E. Solano. Ultrastrong coupling regimes of light-matter

14

interaction. arXiv:1804.09275, 2018.

[60] Ludwig Kn¨oll, Stefan Scheel, and Dirk-Gunnar Welsch.

Qed in dispersing and absorbing media. In Jan Perina,

editor, Coherence and Statistics of Photons and Atoms,

Wiley Series in Lasers and Applications. Wiley-VCH,

2001.

[61] H Carmichael. Statistical Mehods in Quantum Op-

tics 1: Master Equations and Fokker-Planck Equations.

Springer Berlin / Heidelberg, 1999.

[62] Johan F. Triana, Daniel Pel´aez, and Jos´e Luis Sanz-

Vicario. Entangled Photonic-Nuclear Molecular Dynam-

ics of LiF in Quantum Optical Cavities. Journal of Phys-

ical Chemistry A, 122(8):2266–2278, 2018.

[63] Merav Muallem, Alexander Palatnik, Gilbert Daniel Nes-

sim, and Yaakov R Tischler. Strong light-matter coupling

and hybridization of molecular vibrations in a low-loss

infrared microcavity. The journal of physical chemistry

letters, 2016.

[64] H P. Breuer and F Petruccione. The theory of open quan-

tum systems. Oxford University Press, 2002.

[65] Artem Strashko and Jonathan Keeling. Raman

scattering with strongly coupled vibron-polaritons.

arXiv:1606.08343, 2016.

[66] M. Litinskaya, P. Reineker, and V. M. Agranovich.

Fast polariton relaxation in strongly coupled organic

microcavities. Journal of Luminescence, 110(4 SPEC.

ISS.):364–372, 2004.

[67] Barry M. Garraway. The dicke model in quantum optics:

Dicke model revisited. Philosophical Transactions of the

Royal Society of London A: Mathematical, Physical and

Engineering Sciences, 369(1939):1137–1155, 2011.

[68] Peter Kirton, Mor M. Roses, Jonathan Keeling, and

Emanuele G. Dalla Torre. Introduction to the dicke

model: From equilibrium to nonequilibrium, and vice

versa. Advanced Quantum Technologies, 2(1-2):1800043,

2019.

[69] Marina Litinskaya and Peter Reineker. Loss of coherence

of exciton polaritons in inhomogeneous organic microcav-

ities. Phys. Rev. B, 74:165320, Oct 2006.

[70] Felipe Herrera and Frank C. Spano. Absorption and

photoluminescence in organic cavity qed. Phys. Rev. A,

95:053867, May 2017.

[71] Felipe Herrera and Frank C. Spano. Dark vibronic po-

laritons and the spectroscopy of organic microcavities.

Phys. Rev. Lett., 118:223601, May 2017.

[72] Felipe Herrera and Frank C. Spano. Theory of nanoscale

organic cavities: The essential role of vibration-photon

dressed states. ACS Photonics, 5:65–79, 2018.