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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).
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