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Excitonic spectral features in strongly-coupled organic polaritons

Justyna A. Cwik,1 Peter Kirton,1 Simone De Liberato,2 and Jonathan Keeling1

1SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, United Kingdom2School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom

(Dated: December 7, 2015)

Starting from a microscopic model, we investigate the optical spectra of molecules in strongly-coupled organic microcavities examining how they might self-consistently adapt their coupling tolight. We consider both rotational and vibrational degrees of freedom, focusing on features whichcan be seen in the peak in the center of the spectrum at the bare excitonic frequency. In bothcases we find that the matter-light coupling can lead to a self-consistent change of the molecularstates, with consequent temperature-dependent signatures in the absorption spectrum. However,for typical parameters, these effects are much too weak to explain recent measurements. We showthat another mechanism which naturally arises from our model of vibrationally dressed polaritonshas the right magnitude and temperature dependence to be at the origin of the observed data.

I. INTRODUCTION

When matter is strongly coupled to light, the interac-tion cannot simply be thought of in terms of absorptionand emission processes. Instead we must consider theeigenstates of the fully coupled matter-light system. Theparadigmatic example of this is the existence of excitonpolaritons, hybrid matter-light particles formed by thestrong interaction between excitons and photons [1, 2].Matter-light coupling can be engineered by confininglight in optical cavities, so as to modify the density ofstates and the coupling to matter. For weak coupling, ora bad cavity, cavity losses are fast so one can eliminatevirtual processes where photons are in the cavity. Thisgives Fermi’s golden rule, but with the cavity density ofstates modifying the emission rate, as first discussed byPurcell [3]. When coupling is strong, first order pertur-bation theory (i.e. Fermi’s golden rule) fails, as there caninstead be coherent emission and reabsorption of photonsbefore light leaks out of the cavity [4, 5].

A natural context in which strong matter-light cou-pling arises is between organic molecules and light insemiconductor microcavities. Because of the existence ofconjugated π bonds in organic molecules, electronic tran-sitions can acquire large dipole moments [6–8], leading tovery strong coupling to light. When such molecules areplaced in optical microcavities this leads to huge polari-ton splittings [9–12]. These scales allow such experimentsto be performed at room temperature, whereas for manyinorganic materials, cryogenic temperatures are required.The polariton splitting is due to a collective phenomenon:the electronic transitions of many molecules couple toradiation, and as such the polariton splitting grows asthe square root of the molecule density. In contrast, inweak coupling, the rate at which one molecule emits isindependent of whether or not any other molecules arepresent [13].

Much of the recent work on organic microcavity po-laritons (see, e.g. Ref. [14] for a recent review) has beenfocused on condensation and lasing [15–18], involving astrongly pumped system, and the appearance of macro-

scopic quantum coherence. There has however also beensignificant recent work on the effects of matter-light cou-pling in the vacuum state, i.e. without strong pump-ing. Such work aims to understand how the physicaland chemical properties of organic molecules are affectedby strong coupling to electromagnetic modes. Examplesof this include modifying the rates of photochemical re-actions [19], or modifying the transport properties of or-ganic semiconductors [20–22]. More recently, there hasalso been experimental [23] and theoretical [24–26] workon coupling the vibrational state of organic moleculesto infra-red radiation, leading to molecular optomechan-ics. Theoretical work [27] has also studied how strongmatter-light coupling to electronic states can suppressthe effects of disorder and vibronic features in the po-lariton spectrum. Of particular interest for the presentpaper is a recent work from the Ebbesen group [28], inwhich the optical spectra of strongly-coupled organic mi-crocavities were studied by varying molecular concentra-tion and temperature, and paying particular attention tothe relative weights of the resonant features in the ab-sorption spectra: the two polariton peaks, and a thirdpeak at the bare excitonic energy [29].Our aim in this manuscript is to examine the behavior

of such strongly-coupled organic microcavities startingfrom various microscopic models, allowing quantitativepredictions of the extent to which a self-consistent adap-tation of the molecular state, driven by coupling withlight, may occur.To understand the variation of the optical spectra with

both concentration and temperature the models whichwe consider all contain a variable degree of coupling tolight. This is because a molecule that has strictly zerocoupling to light is not visible in the absorption spectrum,while molecules with a small but non-zero coupling willlead to absorption at the bare molecular energy. In or-der that the coupling to light can vary self consistently(in response to the Rabi splitting), it must depend onsome adaptable feature of the state or environment ofthe molecule. i.e. there must be some physical propertythat can vary, which determines the strength of matter-light coupling. We refer to this concept hereafter as

2

“self-consistent molecular adaptation”. We refer to thisprocess as “self-consistent” because the effective matter-light coupling depends on (some aspect of) the molecularstate, and the molecular state is modified because of howits energy depends on the matter-light coupling. In thefirst part of this manuscript we investigate in detail twocandidates that could lead to self-consistent adaptation:rotational and vibrational degrees of freedom, we alsoconsider an extension of these models to generic (classi-cal) aspects of the molecules physical or chemical state.While we do find a temperature dependence of the op-tical spectra, the involved energy scales turn out to beincompatible with the observation of Ref. [28]. In the fi-nal part of the present work, we examine how our modelof vibrationally dressed polaritons naturally predicts aneffect whose energy scale is of the right magnitude toexplain the data. This effect does not involve the renor-malization of the coupling strength, but instead involvesthe effect of vibrational replicas and their coupling tothe excitonic transition on the optical spectra. Our re-sults thus show a first example of the rich, and presentlypoorly understood behavior that can stem from the inter-play of strong matter-light coupling with strong couplingto vibrational/conformational modes of the molecules.

We start by noting that the existence of a peak atthe bare energy of the exciton, brought forward as ev-idence of novel physics in Ref. [28], is not unexpected;such a “residual excitonic peak” has been seen in manycases, for example Ref. [30] discussed theoretically, anddemonstrated experimentally, the appearance of such afeature in a GaAs/AlGaAs heterostructure, containingquantum wells inside a DBR microcavity. While such apeak comes from the spectral weight of the exciton line,it is important to note that this peak cannot be viewedsimply as excitons which do not couple to light: if theydid not couple, they would not be visible in the absorp-tion or transmission spectrum. The origin of the peakcan be understood physically as coming from the subra-diant excitonic states due to inhomogeneous broadening.In a disordered system, the coupling to the photon modepicks out a specific superradiant state, which forms thepolaritons, while the other states — orthogonal to thissuperradiant state — remain at the bare exciton energies.However, because of energetic disorder, the superradiantstate is not an energy eigenstate, and a residual couplingbetween the superradiant and subradiant states exists, sothat the spectral weight of the subradiant states is visiblein the optical spectrum [30–32].

A simple analytical treatment shows that the weight ofthis residual excitonic peak does decrease as the matter-light coupling increases. The existence of this peak isthus consistent with the behavior seen in Canaguier-Durand et al. [28]. However, on its own this explana-tion cannot account for the temperature dependence ob-served, as the residual excitonic peak should be unaf-fected by temperature, unless kBT approaches opticalenergies, of the order of 1eV (for comparison 300K≈25meV). One of the main goals of this paper is to ad-

dress how this temperature dependence may occur.As will become clear in the following, to discuss the

microscopic theory of such effects, it will be crucialto consider the physics of ultrastrong matter-light cou-pling [33, 34], and the breakdown of the rotating waveapproximation (RWA). This requires retaining “counter-rotating” terms in the matter-light coupling Hamilto-nian. These terms, which involve simultaneous creationof pairs of excitations, are typically considered to be non-resonant and so are often neglected. However, if thematter-light coupling is a significant fraction of the bareexciton and photon energies, then these terms have anon-negligible impact. Such behavior has been seen inboth inorganic [35] and organic [11, 36, 37] systems, witha current record of a coupling strength 87% of the bare os-cillator frequency [38]. Our focus in this paper is on themore typical regime where such counter-rotating termscannot be neglected, but remain sufficiently small to betreated perturbatively.The rest of the paper is structured in two main sec-

tions. In section II we consider how temperature de-pendence can arise due to self-consistent adaptation ofthe rotational and vibrational degrees of freedom of themolecules, via a mechanism very similar to that pro-posed in Ref. [28]. For the orientational degree of free-dom, we consider both free molecules, and molecules withrandomly pinned orientations as appropriate in a poly-mer matrix. We will see that in such systems we dopredict a temperature dependence of the residual exci-tonic peak. However, while this effect could potentiallybe observed in other experimental realizations, the en-ergy scales (temperatures) required and the scaling withmolecular concentration are not compatible with the ex-perimental observations reported in Ref. [28].In section III we instead consider a different effect,

arising from the interplay of vibrational modes with thematter-light coupling, which is able to reproduce similarbehavior to that observed in experiments. Specifically wefind that vibrational excitations dress the residual exci-tonic peak in a strongly temperature dependent manner.Moreover, the form of the vibrational dressed spectrumshows that the spectral feature at the exciton energy canhave a more complex interpretation than that previouslyconsidered [30].A brief but self-contained account of the main theoret-

ical methods used throughout this paper is given in theappendices.

II. SELF-CONSISTENT MOLECULAR

ADAPTATION DUE TO STRONG COUPLING

In this section we consider whether self-consistentmolecular adaptation can enhance matter-light couplingby renormalizing the bare matter-light coupling strength.We consider models in which the effective matter-lightcoupling strength of a given molecule depends on theconfiguration of that molecule, such as its orientation, or

3

its vibrational state. We then ask how this same matter-light coupling modifies the energy landscape for the aux-iliary parameters describing the configuration. This leadsto the idea of self consistency — if strong coupling leadsto a reduction of the ground state energy, the energylandscape is deformed so as to favor auxiliary param-eters for which the effective matter-light coupling is aslarge as possible. Our aim is to derive this from a micro-scopic model, and so quantify this effect. In the followingwe consider two potential scenarios involving adaptationof either orientational or vibrational degrees of freedom.If such a self-consistent enhancement of matter-light

coupling occurs, then this can lead to a temperaturedependent effective coupling, and thus to a tempera-ture dependence of the residual excitonic peak. Weshow that such an effect exists, but that its strengthis relatively weak, and that the relevant energy scaleshows no collective enhancement, i.e. the presence of Nm

molecules does not lead to a Nm enhancement of this en-ergy scale, because it must compete with the extensiveentropy gain from orientational or vibrational disorder.As such, while increasing the molecular concentrationwill increase the polariton splitting, it has little effect onthe self-consistent orientation. Changing the bare oscil-lator strength of the molecules does however affect boththe polariton splitting and the self-consistent molecularadaptation energy scale.Before introducing any auxiliary variables, the basic

Hamiltonian which we consider is an extended Dickemodel, including diamagnetic terms:

H =∑

k

ωkψ†kψk +

∑

n

[

ǫn2σzn

+∑

k

gk,nϕk(rn)σxn +

(

∑

k

√

Dkϕk(rn)

)2]

, (1)

where the field ϕk(r) = ψ†ke−ik·r + ψ

keik·r is written

in terms of the bosonic creation and annihilation oper-

ators ψk, ψ†

kdescribing photon modes labeled by their

in-plane momentum k, and energy ωk. The Pauli matri-ces σi=x,y,z

n describe the electronic state of the molecules.For completeness, we therefore also included the diamag-netic terms, arising from the A

2 term of the minimalcoupling Hamiltonian [44].Furthermore, in order to consider varying the cavity

mode volume while respecting the Thomas-Reiche-Kuhnsum rule [39] we use gk,n =

√ǫnDkfn where 0 < fn <

1 is the oscillator strength of the given molecule. Thecoefficient Dk depends on the electric field strength of asingle photon, and the properties of the effective chargesthat respond to the field.If we assume an uniform distribution of molecules, mo-

mentum conservation can be used to write the diamag-

netic term in Eq. (1) as Nm

∑

kDk(ψ†k+ψ−k

)(ψ†−k

+ψk).

This term can then be removed by a Bogoliubov transfor-

mation ψk→ cosh θkψk

+sinh θkψ†k, yielding the effective

Hamiltonian:

H =∑

k

ωkψ†kψk+∑

n

hn, (2)

where the on-site Hamiltonian is given by

hn =ǫn2σzn +

∑

k

gk,nϕ(rn)σxn, (3)

and the renormalized parameters ωk, gk are:

ωk =√

ωk(ωk + 4NmDk), gk,n = gk,n

√

ωk

ωk, (4)

with Nm the number of molecules in the mode volume.In the dipole approximation, for a molecule with a sin-

gle mobile electron Dk = ζ/ωkV with V the mode vol-ume and ζ = h2e2/(4mrε0) quantifying the electronicresponse of a single electron in terms of the vacuum per-mittivity ε0 and its reduced mass mr. For moleculesinvolving many conjugated bonds, the coefficient ζ is re-placed by a sum over all mobile charges. It is importantto note that changing the cavity length changes both themode volume V (and hence both Dk and gk,n) and thespectrum of photon modes ωk. In the following numeri-cal results we will fix the values of the polariton splittinggk,n

√Nm, exciton energy ǫn, and coupling strength f ,

and use these to determine DkNm.Before considering the role of vibrational and orien-

tational degrees of freedom, we consider how the exis-tence of a temperature dependent matter-light couplingstrength, geff,k,n would be seen in the absorption spec-trum. For this purpose, it is sufficient to consider theabsorption spectrum of Eq. (2), and its dependence ongk,n. Appendix A summarizes how the absorption spec-trum can be calculated, including the counter-rotatingterms in the matter-light coupling [40]. In performingthese calculations, as noted above, it is necessary to in-clude disorder in order to see the residual excitonic fea-ture. We will consider disorder in the exciton energies,denoted by the energy distribution h(ǫ) For simplicity weconsider only disorder in the energies and we ignore thesubleading effect of the exciton energy distribution onthe coupling strength gk,n, by using the averaged valueg2k

≡ ∑

n g2k,n/Nm. As discussed in the appendix, we

consider the quantity ak(ν) = −2ℑ[GRk,xx(ν)], in terms

of the retarded Green’sGRk,xx(ν), which is proportional to

the absorption spectrum for a good cavity. The Green’sfunction has the form:

GRk,xx(ν) =

2ωk

ν2 − ω2k + 2ωkΣk,xx(ν) + iωkκ(ν)

, (5)

where the excitonic self energy for Eq. (2), can be writtenas Σk,xx = ΣRWA

k,+−(ν) + ΣRWAk,+−(−ν)∗ with:

ΣRWAk,+−(ν) = −g2kNm

∫ ∞

−∞dǫh(ǫ)

tanh(βǫ/2)

ν + iγ+ − ǫ. (6)

4

Here β = (kBT )−1 is the inverse temperature and γ is the

homogeneous linewidth of the excitons. In the followingwe will present results both for γ = 0+ and for small butnon-zero γ as indicated in the figure captions.

The spectrum is shown in Fig. 1, focusing on the resid-ual excitonic peak, to show its dependence on the polari-ton splitting. As noted above, such a feature has been ob-served and commented on several times before, e.g. [30–32]. When the polariton splitting g

√Nm is increased,

the splitting between the lower and upper polaritons in-creases, and the exciton spectral weight of the feature atthe exciton energy decreases. The asymmetry betweenthe shift of the lower and upper polaritons arises due tothe diamagnetic term Dk renormalizing the photon en-ergy. If the coupling to light is weak, so that the RWAis valid, this asymmetry vanishes. In Fig. 1 (b) we showhow the spectrum is modified by including the effectsof cavity losses and non-radiative excitonic decay as de-scribed in Appendix A. The main effect these processeshave is to broaden and thus reduce the height of the po-lariton peaks, there is also additional broadening to thecentral peak.

Because the only temperature dependence of Eq. (5,6)is via the combination tanh(βǫ/2), the spectrum is tem-perature independent while kBT ≪ ǫ (with ǫ of the orderof the exciton bare energy). Therefore, as noted in theintroduction, the experimentally observed temperaturedependence cannot occur from this mechanism alone, un-less the effective value of gk,n is made temperature de-pendent via its dependence on configuration. We now goon to consider if this effect is plausible.

To consider molecular adaptation, the Hamiltonian inEq. (2) will be modified to include either orientational orvibrational degrees of freedom. These are illustrated inFig. 2. We next introduce these modifications and thendiscuss how they may be treated.

The Hamiltonian including an orientational degree offreedom takes the same form as Eq. (2) with the on-sitepart

hn =ǫn2σzn + Λn(θn) +

∑

k

gk,n cos(θn)ϕk(rn)σxn, (7)

where the angle θn parametrizes the orientation of thedipole moment of the nth molecule with respect to thepolarization of the cavity electric field, thus reducingthe oscillator strength. The term Λn(θn) represents thebare dependence of the Hamiltonian on orientation. Thisterm allows one to model pinning of the orientation θn.For simplicity we consider only a classical orientationaldegree of freedom; the corresponding quantum theorywould require us to also include a rotational kinetic en-ergy term, and diagonalize the resulting Hamiltonian.The effective matter-light coupling strength, which de-pends on the distribution of angles θn adopted by themolecule, can be written as g2

k,n,eff = g2k,n〈〈cos2(θn)〉〉,

where the double angle brackets represent both an en-semble and thermal average.

0

0.1

0.2

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2ν [eV]

a) g√N=0.3eV

g√N=0.5eV

g√N=0.7eV

ak=0(ν)

0

0.1

0.2

0.3

0.4

0.5

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

ak=0(ν)

ν [eV]

FIG. 1. (color online) Evolution of k = 0 absorption spec-trum, a(ν) vs polariton splitting gk=0

√Nm for the system

with no auxiliary degrees of freedom. Plotted for kBT =0.025eV (i.e. T = 290K), f = 0.05, ωk=0 = 2.1eV, and a

truncated Gaussian distribution h(ǫ) ∝ Θ(ǫ)e(ǫ−ǫ0)2/2σ2

withǫ0 = 2.0eV, σ = 0.01eV. Panel (a) shows the results with aperfect cavity while (b) includes the effects of cavity losses atrate κ = 0.075eV and excitonic non-radiative decay at rateγ = 10−4eV also the width of the energy distribution is larger,σ = 0.025eV as discussed in Appendix A.

To consider vibrational degrees of freedom, we againstart from the transformed Hamiltonian, Eq. (2), andnow consider the following modification:

hn =ǫn2σzn +

∑

k

gk,nϕk(rn)σxn+

∑

m

Ωm

(

b†n,mbn,m +

√Sm

2(b†n,m + bn,m)σz

n

)

. (8)

Here bn,m, b†n,m describe the mth harmonic vibrational

mode of molecule n, with the mode having frequencyΩm and its coupling to the electronic state beingparametrized by the Huang-Rhys parameter

√Sm. In

this case, defining the effective oscillator strength is moreinvolved: the effective oscillator strength depends on thematrix element describing the overlap between the vibra-tional states in the ground and excited state manifold.We return to this point in later sections.For both the orientational and vibrational degrees of

freedom, our aim is to find how the matter-light cou-pling is self-consistently modified by these auxiliary de-grees of freedom: i.e. how the presence of matter-lightcoupling modifies the distribution of orientational or vi-brational states, and how this in turn affects the effectivematter-light coupling strength. We consider a case with-out any strong pumping, and with a temperature suchthat kBT ≪ ω, ǫ, which is typically satisfied even at roomtemperature for organic polaritons. As such, the originof the “self-consistent” dependence of configuration on

5

Coupled

Pinned

FIG. 2. (color online) Schematic diagram showing orienta-tional and vibrational degrees of freedom. The top right insetshows the potential energy landscape with (dashed red line)and without (solid black line) a cavity. The other insets show(top) the orientational degree of freedom and (bottom-right)the vibrational degree of freedom illustrating how the cou-pling to light decreases the displacement between the groundand excited manifolds. Shifts due to coupling to the cavityare exaggerated for clarity in all three insets.

the matter-light coupling coupling arises due to the exis-tence of the counter-rotating terms in the original Hamil-tonian: If these terms were neglected then the energyin the ground state sector can be trivially found as theground state would correspond to the empty state, andits energy would therefore not involve the matter-lightcoupling strength at all [33]. The presence of counter-rotating terms means that the ground state sector alsoinvolves an admixture of all even parity sectors, and thedegree of admixture depends on the effective matter-lightcoupling.As discussed below, while exact solutions are possi-

ble in some limiting cases of the orientational problem,these are not generally possible at finite temperature,nor for the vibrational problem. This is because ther-mal or quantum fluctuations of the auxiliary degrees offreedom break translational symmetry, preventing sim-ple exact diagonalization. As such, we proceed using theSchrieffer-Wolff formalism [41], which allows us to con-sider perturbatively the effects of these counter-rotatingterms, and how they modify the energy landscape seenby the auxiliary orientational or vibrational degrees offreedom. For completeness, appendix B provides a briefsummary of the Schrieffer-Wolff formalism. The essen-tial point is to separate H = H0 + H1, where H1 are theterms treated perturbatively. At leading order this givesan effective Hamiltonian:

ˆH ≈ H0 +i

2

[

G, H1

]

, G : [G, H0] ≡ iH1. (9)

Taking the counter-rotating terms as H1, the pertur-bation theory is controlled by the small parametergk,n/(ωk + ǫn), which is indeed small for the physical

parameters we consider [42]. We are thus in a regimewhere the counter-rotating terms cannot be ignored, butwhere they can be included perturbatively. In the fol-lowing sections we apply this approach in turn to theorientational and vibrational degrees of freedom, and seehow the effective matter-light coupling can be derivedself-consistently.

A. Orientational degrees of freedom

As discussed above, we consider first the classicalorientational degrees of freedom θn, subject to a pin-ning potential Λn(θn). If all molecules are identical,Λn(θn) = Λ(θn), then one can find the zero tempera-ture ground state by choosing θn = θ. For the groundstate, all that is required is to find the quantum groundstate of Eq. (7) as a function of θ and then minimize overθ. Since Eq. (7) is translationally invariant in the caseθn = θ, it is possible to find the exact ground state inthe bosonic approximation by Fourier transforming. Thebosonic approximation assumes the occupation of eachexcited molecule is small, a result that is valid unlessgk ≫ ωk, ǫ [43, 44]. However, at finite temperature, evenfor identical molecules, it is crucial to allow independentfluctuations of each θn; assuming θn = θ massively un-derestimates the entropy at finite temperature. At zerotemperature, one may compare the exact solution to theSchrieffer-Wolff perturbative expansion used below, andone finds that these indeed match to leading order.For this rotational case, the form of G required

in Eq. (9) can be found trivially, and the resulting

Hamiltonian can most conveniently be written as ˆH =∑

kωkψ

†kψk+∑

nˆhn, with the molecular Hamiltonian

in the form:

ˆhn = h(0,RWA)

n −∑

k

g2k,n cos

2(θn)

ωk + ǫn,

where h(0,RWA)n is the bare molecular Hamiltonian, in-

cluding the RWA coupling to light and including the pin-ning term Λn(θn).In order to consider the thermal distribution of θn,

we must specify the orientational potential Λn(θn). Weconsider a form

Λn(θn) = λ sin2(

θn − θ0n2

)

, (10)

which tries to pin the molecules at angle θ0n, relative tothe cavity electric field, with strength λ. We may thusconsider both the free orientation case, λ = 0, and thepinned case simultaneously. In what follows we will as-sume that the pinning angles have a uniform distributionsuch as would be found in a polymer matrix. This treat-ment, however, ignores effects which would be importantin systems such as organic crystals in which the pinning

6

angle has a fixed direction. The energy landscape foreach angle θn, given the pinning angle θ0n is then:

E(θn|θ0n) = λ sin2(

θn − θ0n2

)

−∑

k

g2kcos2(θn)

ωk + ǫ. (11)

where for simplicity we have assumed that each moleculehas the same values of ǫ, gk and the only disorder is inthe pinning angles. In the following we will define thequantity:

K0 ≡∑

k

g2k

(ωk + ǫ). (12)

The quantity K0 characterizes the self-consistent energyfavoring alignment of molecules. Replacing the summa-tion by an integral, and inserting the explicit forms of gkand Dk written above we have that

K0 =fǫζ

lc

∫ Λ

0

kdk

(2π)

1

ωk(ωk + ǫ), (13)

where lc is again the cavity length, ζ the combinationdefined following Eq. (2), and Λ = 2π/aBohr is a cutoffreflecting the breakdown of the dipole approximation. Toevaluate such integrals it is useful to write the dispersionin the form ω2

k = (ω20 + c2k2) + 4ζNm/V which allows us

to find the exact result

K0 =fǫζ

2πlcc2ln

(

ǫ+ ωΛ

ǫ+ ω0

)

. (14)

A notable feature of Eq. (14), as anticipated above,is that this quantity does not simply increase as one in-creases the polariton splitting by varying the density ofemitters, Nm/V . There is a sub-leading dependence onthe molecule density, via the renormalization ωk → ωk

given in Eq. (4). However this effect is only significantin the deep strong coupling limit [43, 44]. Physicallythis lack of scaling with Nm is because this “molecularadaptation energy” depends on the shift seen for each

molecule. Inserting typical experimental values for anorganic system Nm/V = 4.2 · 1025m−3, ǫ = ω0 = 2.1eV,lc = 145nm, f = 0.5, aBohr = 2nm, ζ = 4.3eV2nm3

which correspond to the values extracted from Ref. [28],one finds that

K0 = 6.3 · 10−4meV,

which is much smaller than kBT at room temperature.As noted earlier, the effective matter-light coupling

strength is given by g2eff,k = g2k〈〈cos2(θn)〉〉. At zero

temperature, this corresponds to minimizing the energy,leading to λn sin(θn − θ0n) = −2K0 sin(2θn). However,since the value ofK0 given above is such thatK0 ≪ kBT ,it is crucial to consider finite temperatures. The small-ness of the ratio K0/kBT will also allow us to make fur-ther perturbative expansions in the following.

Defining g2eff/g2 ≡ 〈〈cos2(θn)〉〉, this ratio can be cal-

culated as

g2effg2

=1

2πβ

d

dK0

∫

dθ0 ln[Z(λ,K0, θ0)], (15)

where the partition function is

Z(λ,K0, θ0) =

∫

dθ exp

[

βK0 cos2 θ − βλ sin2

(

θ − θ0

2

)]

.

As noted above βK0 ≪ 1, and so we may Taylor expandin this small parameter to get a closed form for geff. As-suming that the pinning angle distribution χ(θ0) = 1/2πis uniform we obtain the simple result for the coupling

g2effg2

=1

2

[

1 +βK0

4

(

1− I2(βλ/2)2

I0(βλ/2)2

)]

(16)

with In(z) the imaginary Bessel function

In(z) =

∫

dθ cos(nθ)ez cos θ. (17)

At this point we have made no assumption about thepinning strength λ, and the value of the effective couplingis controlled by the combination βλ. If the pinning isstrong βλ≫ 1 then we find the asymptotic form

g2effg2

=1

2

(

1 +2K0

λ

)

. (18)

This no longer depends on temperature as the strongpinning limit means entropy become unimportant. Thus,in the λ → ∞ limit the coupling takes on the isotropicvalue of 1/2, corresponding to the uniform distributionof angles θn = θ0n. The effective coupling increases as thepinning λ decreases. In the limit of vanishing pinningλ→ 0, the imaginary Bessel function I2(0) vanishes andso we have:

g2effg2

=1

2

(

1 +βK0

4

)

. (19)

Both Eq. (18) and Eq. (19) indicate that as long asβK0 ≪ 1, the modification and temperature dependenceof g2eff is very small. We note that while in the organicsystems which we focus on here K0 is relatively small, insystems which have very small mode volumes this param-eter could be engineered to be much larger and hence thecoupling strength renormalization would be much morepronounced. i.e., since there is no scaling with Nm, thecrucial feature to see a strong renormalization is to min-imize the mode volume in absolute terms, and not themode volume per molecule. This suggests evanescentlyconfined radiation modes in plasmonic [45] or phonon po-lariton [46] systems may be a promising venue to explorethis physics.

7

B. Vibrational degrees of freedom

In the vibrational case, even without disorder, an exactsolution of Eq. (8) via Fourier transformation is no longerpossible, because the vibrational degrees of freedom aremodeled as quantum degrees of freedom with their ownquantum dynamics. As such, they break translational in-variance — i.e. localized vibrational excitations can scat-ter between different polariton momentum states. Thus,once again we must use the Schrieffer-Wolff formalism. Ifwe start from Eq. (8), solving the equation [G, H0] = iH1

is now more challenging than for the rotational case, asH0 involves terms that couple the electronic state to thevibrational quantum state. The equation can however besolved in the form of a power series,

G =∑

k,n,j

−igk,n(ωk + ǫn)j+1

(

Ojψ†kσ+n e

−ik·rn −H.c.)

, (20)

where the operators Oj are defined by the recursion re-lation

Oj =∑

m

Ωm

[

Oj−1, b†mbm +

√Sm

2(b†m + bm)

]

− Oj−1

√

Sm(bm + b†m)

,

with the base case O0 = 1. This expansion then allowsone to write out the effective Hamiltonian in the sameform as above, but with the molecular Hamiltonian,

ˆhn = h(0,RWA)

n −[

1− σz4

]

∑

k,j

g2k

(ωk + ǫn)j+1(Oj + O†

j),

(21)

with h(0,RWA)n the bare molecular Hamiltonian, includ-

ing the RWA coupling to light, and vibrational terms ofEq. (8).This expression can be considered as a multinomial

power series in the quantities Ωm/(ωk + ǫn) ≪ 1 foreach vibrational mode m. For typical parameters, suchquantities are small, and so we may truncate at first or-der, i.e. keep terms up to j = 1. Beyond j = 1, theexpression becomes considerably more complicated, ascross terms between different vibrational modes appear.Up to j = 1, we find an effective molecular Hamiltonianwhich we write out in full (neglecting constant terms):

ˆhn =

ǫn +K0

2σz +

∑

k

gk,n

(

ψ†ke−ik·rσ−

n +H.c.)

+∑

m

Ωm

(

b†mbm +

√Sm

2(b†m + bm)σz

+K1

[

1− σz2

]

√

Sm(bm + b†m)

)

. (22)

The j = 0 term gave an energy shift of two-level systemswith the same formK0 found previously. The j = 1 termis on the last line, and describes a shift to the vibrationalmodes. The coefficients for these terms are defined by ageneralization of that used in Eq. (12) for the rotationalcase above,

Kj(ǫ) =∑

k

g2k

(ωk + ǫ)j+1,

where we may find an analytic form for the resultingintegral

Kj =fǫζ

2πlcc2j

(

1

(ǫ + ω0)j− 1

(ǫ + ωΛ)j

)

.

We may once again note that none of the terms Kj(ǫ) areproportional to the number of molecules Nm: vacuum-state molecular adaptation is not collectively enhanced.From the form of Eq. (22) it is clear that the term inK1

describes a reduction in the offset between the vibrationalground state in the two electronic states, as the virtualexcitations admix the excited electronic state configura-tion into the ground state. This can be viewed as a reduc-tion of the Huang-Rhys parameter, Sm,eff = Sm[1−2K1].This point is illustrated in Fig. 2. As noted earlier, thedependence of geff on the vibrational degrees of freedomis more complicated: one must calculate the overlap be-tween the vibrational states of the ground and excitedelectronic state manifolds. The reason that the vibra-tional states differ in these manifolds is the existence ofthe terms (b†m+ bm)σz , which correspond to displacementof the vibrational coordinate dependent on the electronicstate of the molecule. As such, a reduction of Sm,eff wouldlead to an enhanced matter-light coupling. However, us-ing the same typical experimental values as before we findthat this dimensionless shift has a value of approximatelyK1 = 2.9 · 10−8. Thus, once again one may concludethat the characteristic scale of any vibrational molecularadaptation (determined by K1) is negligibly small.

C. Other aspects of molecular state

The discussion so far has focused on two specificmicroscopic mechanisms which might have led to self-consistent adaptation of the molecules so as to enhancetheir coupling to light. Here we note those aspects of theabove results which can be easily generalized to other mi-croscopic mechanisms. Examples of such other potentialmechanisms include solvation of the molecule, molecu-lar configuration, and charge transfer state. In somecases, these will require detailed modeling of the spe-cific process, particularly for degrees of freedom with en-ergy scales larger than temperature, where quantum ef-fects become important, as in the example of vibrationalmodes above. However, the basic idea can be illustratedin the simplest case, where some aspect of configurationcan be parametrized by a classical variable.

8

In the case where classical parametrization applies, thedescription is very similar to the discussion of orienta-tional degrees of freedom: One considers a classical vari-able xn which parametrizes some aspect the state formolecule n. Associated with this will be an energy func-tion Λn(xn), and, for self-consistent adaptation to oc-cur, the matter-light coupling must depend on this vari-able as g 7→ gφn(xn). The same perturbative analysisas discussed above will then lead to the effective energyfunction: En(xn) = Λn(xn) − K0φn(xn)

2. This in turnmeans that the self-consistent effective matter-light cou-pling will take the form:

g2n,effg2n

=1

β

d

dK0ln

[∫

dxne−βEn(xn)

]

.

Since this involves the same sum K0 as defined inEq. (12), the same absence of scaling with number ofmolecules occurs i.e. it is the single-molecular, ratherthan collective, coupling which is important. Moreover,since βK0 ≪ 1, then:

g2n,eff ≃ g2n〈〈φn(xn)2〉〉0 +O(βK0)

where 〈〈. . .〉〉0 indicates thermal averaging with the bareenergy function Λn(xn), i.e. any such vacuum-state self-consistent adaptation of molecules is suppressed by thesmall parameter βK0.

III. VIBRATIONAL DRESSING OF THE

EXCITON-POLARITON SPECTRUM

In the previous section we have seen that while self-consistent molecular adaptation due to matter-light cou-pling is possible, neither the strength nor the dependenceupon molecular concentration are consistent with the re-sults reported in Ref. [28]. In this section we show howdirectly calculating the absorption spectrum from themodel of vibrationally dressed polaritons introduced inthe previous section can lead to a rather different mecha-nism which could however be responsible for the featuresobserved. This mechanism arises from the combinationof vibrational dressing of the spectrum and the effectsof disorder. We focus in particular on the peak in thespectrum near the bare excitonic resonance. We will seethat the shape of this feature depends strongly on thevibrational state of the molecules.

This section is divided into two subsections, in sec-tion III A we first discuss how vibrational excitationsshould be included in the calculation of the disorderedpolariton spectrum. This follows the method outlined inappendix B, and so all that is required is to calculate theexcitonic self energy. We then discuss the resulting formof the spectrum in section III B.

A. Self energy of vibrationally dressed excitons

As discussed in appendix A, the absorption, emissionand transmission spectra can all be found in terms ofthe photon retarded Green’s function. In this approach,all the properties of the molecules (i.e. inhomogeneousbroadening, vibrational dressing etc.) are incorporated

via the excitonic self energy Σk,xx(ν). We must there-fore calculate this quantity, defined in Eq. (A1,A2), forthe vibrationally dressed Hamiltonian, Eq. (8). To dothis, it is useful to label the eigenstates as |p ↑〉, |q ↓〉,corresponding to the vibrational eigenstates in the ex-cited electronic state manifold and the ground electronicstate manifold. We denote the energies of these statesas E↑

p , and E↓q and we introduce the overlap matrix el-

ement: αpq = |〈p ↑|σ+|↓ q〉|2. This matrix element de-scribes the extent to which the state with q vibrationalexcitations in the electronic ground manifold overlapswith the state in the excited electronic manifold with pexcitations. In order to find these eigenvalues and eigen-functions we must diagonalize the vibrational problem,

Hvib = Ωb†b + Ω√S2 σz

(

b† + b)

in the electronic ground

and excited states. As discussed in Sec. II B, the renor-malization of these parameters due to virtual pair cre-ation is very small, and so we neglect it in the following.In terms of these quantities we may write the self en-

ergy, including inhomogeneous broadening of excitonicenergies in the form:

ΣRWAk,+−(ν) = −g

2kNm

Z

∫ ∞

−∞dǫh(ǫ)

∑

p,q

αpq

[

e−βE↓q − e−βE↑

p

]

ν + iγ + (E↓q − E↑

p),

(23)

where h(ǫ) is again the distribution of exciton energies,as in Sec. II, and we have again ignored the subleadingeffects of the exciton energy distribution on the couplingstrength, by using g2

kNm ≡∑n g

2k,n. It should be noted

that the energies E↑p , E

↓q depend (linearly) on the energy

ǫ, but that the matrix elements αpq are not dependenton this energy scale.In the following, we model the inhomogeneous broad-

ening of excitons by using the distribution h(ǫ) ∝Θ(ǫ)e(ǫ−ǫ0)

2/2σ2

, i.e. a truncated Gaussian distribution ofexcitonic energies. The truncation has little effect, but isformally required, as negative energy states cannot phys-ically exist, and cause problems in the ultrastrong cou-pling limit [40, 47].

B. Dependence of absorption spectrum on

vibrational state

The exciton spectrum in the absence of vibrationaldressing was shown previously in Fig. 1. Before exploringthe effect of vibrational modes, it is helpful to summarizehow the residual exciton peak arises. Mathematically,

9

the excitonic feature in the polaritonic spectrum can beunderstood directly from the form of Eq. (5) and thedefinition of the absorption spectrum. The peak at theexciton frequency occurs because the imaginary part ofthe retarded Green’s function has a numerator involvingthe imaginary part of the self energy, ℑ[Σk,xx(ν)], andthis self energy has a peak at the excitonic energy. How-ever, the weight of the residual excitonic peak reduces asthe matter-light coupling increases, because the excitonicself energy also appears, squared, in the denominator.It is also important to note that this residual excitonicpeak does not appear in the transmission spectrum: sinceTk(ν) ∝ |GR

k,xx(ν)|2, there is no term in the numerator

of Tk(ν) arising from the exciton self energy. Thus, theresidual exictonic peak is a feature of the absorption (andreflection), but not the transmission spectrum.

0

0.1

0.2

0.3

0.4

0.5

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

ak=0(ν)

ν [eV]

0

20

40

60

1.95 2 2.05

0

0.1

0.2

0.3

0.4

0.5

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

ak=0(ν)

ν [eV]

0

20

1.95 2 2.05

FIG. 3. (color online) The evolution of absorption spectra

with the strength of g√N for a system with a single vibra-

tional mode with S = 0.06,Ω = 0.05eV. The insets show theequivalent absorption spectra for bare molecules. The otherparameters are as in Fig. 1. Panel (a) shows the spectrumfor a perfect cavity while panel (b) includes the effects ofcavity losses and non-radiative excitonic decay exactly as inFig. 1(b).

Figure 3 shows an equivalent set of spectra to Fig. 1,but with vibrational dressing. For comparison in the in-sets we show the absorption spectra of a bare moleculeℑ[Σk,xx(ν)], i.e. the spectra without a cavity. Panel (a)shows the simple case where cavity losses are ignored,while panel (b) shows the more experimentally relevantcase where these effects are included (discussed in ap-pendix A). In the following we use the notation p − qwhich denotes transitions in the molecule from the statewith p vibrational excitations in the electronic groundstate to the state with q vibrational excitations in theelectronic excited state. The presence of the vibrationalmodes causes dramatic changes to the residual excitonicpeak in the polariton spectrum not seen in the bare ex-citonic absorption. For the bare molecule the spectral

weight associated with transition with the “zero phononline”, i.e. the transition denoted 0 − 0 in the notationintroduced above is completely dominant, only a smallamount of weight visible in the sideband which corre-sponds to 0− 1 transition. When the molecule is placedinside a cavity, the vibrational sidebands correspondingto the 1 − 0 and 0 − 1 transitions become much moreprominent. Physically this is because the spectral weightthat had been associated with the 0− 0 transition in thebare molecular spectrum has been moved into the polari-ton peaks of the spectrum. The mathematical form of theGreen’s function makes clear that formation of the polari-ton spectral feature is predominantly at the expense ofwhatever feature dominates the excitonic emission spec-trum; here this is the 0−0 feature. The excitonic featurecorresponds to the “left-over” spectral weight associatedwith the subradiant states. Thus, the vibrational side-bands have been “excavated” by removing the dominantfeature from the zero-phonon line. Including the the ef-fects of cavity losses and non-radiative excitonic decays,as can be seen in Fig. 3 (b), washes out this complexsideband structure of the central peaks and the spectrumas a function of coupling strength looks very similar tothat obtained without coupling to vibrational modes, asin Fig. 1. However as we discuss below, there are stilleffects which can be observed which are a direct conse-quence of the vibrational structure.Fig. 4 illustrates the evolution of spectra with tem-

perature. Panel (a) shows that without coupling to vi-brational modes there is no notable temperature depen-dence. In the presence of the vibrational dressing, astrong temperature dependence appears. At higher tem-peratures, there is a greater thermal occupation of thevibrational modes hence the spectral weight under thevibrational peaks rises. While this has a small effect inthe bare molecular spectrum (where the 0− 0 transitiondwarfs all other features), see inset in (b), it is very pro-nounced in the polariton spectrum and is even visible inthe presence of large cavity losses as in Fig. 4(c).The figures so far have shown results where disorder is

relatively small, and so vibronic replicas can be clearlyobserved for the good cavity, but merge for the bad cav-ity limit. To show that small disorder is not requiredfor the strong temperature dependence to occur Fig. 5shows the effects of large disorder (i.e. inhomogeneousbroadening). Just as seen for the homogeneous broaden-ing in Fig. 4(c), a temperature dependence of the resid-ual excitonic peak is still visible. In this figure, we havealso included a more complicated vibrational spectrum,involving two vibrational modes. Such a system will ex-hibit behavior similar to that of a single mode but witha large vibrational coupling [48].

IV. DISCUSSION

In this paper we have presented two microscopic mod-els which could in principle describe self-consistent molec-

10

a)a)a)a)a)a)a)a)a)a)a)a)

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

ν [eV]

0

0.04

0.08

0.12

0.16ak=0(ν)

0.025

0.05

kBT[eV]

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

ν [eV]

0

0.1

0.2

0.3

0.4

ak

ν

1.95 2 2.050

5

10

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

ν [eV]

0

0.1

0.2

0.3

0.4

ak

ν

FIG. 4. (color online) Evolution of absorption spectra withthe temperature (values shown on the colorscale) for: (a) Asystem with no coupling to vibrational modes, S = 0, (b) Cou-pling to a single vibrational mode with S = 0.02,Ω = 0.05eV.The inset shows the bare molecular spectrum. (c) Includingthe effects of cavity losses and non-radiative excitonic decay.All panels are for g

√N = 0.3eV, and other parameters as in

previous figures.

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

ν [eV]

0

0.4

0.8

1.2

ak=0(ν)

1.8 2 2.20

1

2

3

FIG. 5. (color online) Temperature dependence of absorptionspectrum for a system with strong disorder σ = 0.025eV . Themain panel shows the optical spectrum of the strongly coupledsystem, the inset shows the spectrum of the bare excitons. Inthis case, we include two vibrational modes: S1 = 0.2,Ω1 =0.05eV, and S2 = 0.3,Ω2 = 0.04eV. Other parameters as inFig. 4.

ular adaptation so as to maximize the vacuum-state cou-pling to light. In both cases, the crucial feature of themodel is the counter-rotating terms in the matter-lightcoupling. These allow virtual fluctuations in the groundstate, that lower the ground state energy depending onthe configuration of the molecules. This energy gain isthe only energy gain that can be relevant in the linearresponse regime — i.e. the question of whether the ex-cited states would have lower energy is not of relevancewhile the system is only weakly pumped. We found thatwhile such a mechanism for molecular adaptation doesexist, it does not show any collective enhancement, incontrast to the polariton splitting, and does not there-fore lead to significant molecular adaptation, even whenthe polariton splitting gk

√Nm ∼ ω, ǫ.

The appearance of a residual exciton peak in the po-lariton spectrum would be affected by any such self-consistent molecular adaptation if its scale were suffi-cient. However, for relevant parameters, such effects aredwarfed by a far more dramatic effect, of vibrationaldressing of the residual exciton peak. This leads to apronounced temperature dependence of the feature nearthe exciton energy in the absorption spectrum.While the molecular adaptation energy scale is not col-

lectively enhanced, the basic underlying physics could po-tentially be relevant in single molecule strong coupling,e.g. with plasmonic resonances [49]. In such cases, ratherthan having many molecules in a large mode volume, themode volume is reduced so that strong, or even ultra-strong coupling occurs at the single molecule level, sothat both the polariton energy and the molecular adap-tation energy become large. In such a case, one mayhope to see either reorientation, or renormalization ofthe Huang-Rhys parameter due to strong coupling.Another intriguing direction for future research is to

consider how the physics discussed in this manuscriptinteracts with the physics of polariton condensation andlasing [50]. Polariton condensation has been seen in bothinorganic [51, 52] and organic [15–18] systems. In addi-tion, condensation of photons has been seen for weaklycoupled systems of organic molecules [53]. Theoreticalwork [14, 54–61] has begun to address some of the pecu-liarities of the organic polariton system, including effectsof disorder and of vibrational modes. However, featuresas seen in this paper, resulting from the interplay of thesemay lead to further exotic behavior in the high densitycondensed phase.

ACKNOWLEDGMENTS

We are grateful for comments from T. Ebbesen on anearlier version of this paper. JK acknowledges helpful dis-cussions with David Lidzey and Brendon Lovett. JAC ac-knowledges support from EPSRC. JK and PGK acknowl-edge financial support from EPSRC program “TOPNES”(EP/I031014/1). JK acknowledges support from theLeverhulme Trust (IAF-2014-025). SDL is Royal Society

11

Research Fellow. SDL acknowledges financial supportfrom EPSRC grant EP/M003183/1. PGK acknowledgessupport from EPSRC grant EP/M010910/1. JK, PGKand SDL acknowledge support from the British Councilfor the meeting which initiated this work.Note added: During the final preparation of this

manuscript, another paper [62] appeared, also report-ing the fact that ground-state bond length depends onthe single-molecule coupling g, not the collective couplingg√Nm.

Appendix A: Absorption, Transmission and

Reflection spectrum of exciton-polariton system

In this appendix we summarize the calculation of theabsorption, transmission and reflection spectra. Some

subtleties arise because we wish to calculate the spec-trum of a model with ultrastrong coupling, i.e. withoutmaking the rotating wave approximation. Such resultswere first calculated by Ciuti and Carusotto [40], herewe present a synopsis of these results, as well as a “dic-tionary” to translate the results of that paper into thelanguage of Green’s functions. We begin by definingthe retarded Green’s function [63]. Because we considerboth co- and counter-rotating terms, we must considerboth normal and anomalous Green’s functions. i.e. wemust include number non-conserving terms which appearfor ultrastrong coupling, and thus we consider a matrixGreen’s function:

GRk(t, t′) = −i

(

〈[ψk(t), ψ†

k(t′)]〉 〈[ψ†

−k(t), ψ†

k(t′)]〉

〈[ψk(t), ψ−k

(t′)]〉 〈[ψ†−k

(t), ψ−k(t′)]〉

)

In terms of the Bogoliubov transformed operators, i.e. the operators appearing in Eq. (2) the inverse Green’sfunction takes the form

[GRk(ν)]−1 =

(

ν + iκ(ν)− ωk + Σk,xx(ν) +iκ(ν) + Σk,xx(ν)

−iκ∗(−ν) + Σ∗k,xx(−ν) −ν − iκ∗(−ν)− ωk + Σ∗

k,xx(−ν)

)

,

where Σk,xx(ν) is the self energy for a photon of in-plane momentum k, arising from the excitonic response (discussed

further below), and κ(ν) is the loss rate.

Here, following [40] we have used a frequency depen-dent complex loss rate κ(ν). Frequency dependence isrequired for physical consistency in the case of ultra-strong coupling — Markovian loss and ultrastrong cou-pling would predict a perpetual light source. Frequencydependent loss requires, via the Kramers-Kronig relation,a corresponding Lamb shift, which is incorporated intothe imaginary part of κ(ν). Both Σ and κ are writtenfor the Bogoliubov transformed operators, and so boththese terms incorporate a pre-factor ωk/ωk to accountfor the Bogoliubov transformation of the combination

(ψk+ ψ†

−k).

The specific self energy required, Σk,xx(ν), corresponds

to correlation functions of the σx excitonic operators. Inthe absence of strong (i.e. beyond RWA) excitonic damp-ing (see [40] for the more general case), this self energycan however be related to results in the rotating waveapproximation by:

Σk,xx(ν) =ωk

ωk

(

ΣRWAk,+−(ν) +

[

ΣRWAk,+−(−ν)

]⋆)

, (A1)

where the expression ΣRWAk,+− is the “standard” self energy

that would appear in the rotating wave approximation,depending on the correlation of σ+, σ− operators. Thesecan most easily be found by analytic continuation fromimaginary time to real time, starting from the Matsubara

self energy,

ΣRWAk,+−(iωm) =

1

Z∑

n

g2k,n

∫ β

0

dτe−iωmτ

×∑

p

〈p|σ+n (τ)σ

−n (0)|p〉e−βEp (A2)

and replacing the Matsubara frequency by iωm → ν +i0+. The sum over p appearing here is over all statesof the excitonic system, and Z is the partition function.For the “vacuum” state we consider — i.e. in the absenceof strong pumping — these are the bare exciton states,including the quantum states of any auxiliary degrees offreedom.Using the input-output formalism [64] adapted to the

ultrastrong coupling regime [40], one can write a fre-quency dependent scattering matrix relating input andoutput fields at the left and right sides of the cavity,

Sk(ν) =

(

1− iφk,L(ν) −i√

φk,L(ν)φk,R(ν)

−i√

φk,L(ν)φk,R(ν) 1− iφk,R(ν)

)

where φk,σ(ν) = κ′σ(ν)GRk,xx(ν) with κ

′L,R(ν) are the real

parts of the loss rates arising from the left and right mir-rors and the quantity GR

k,xx(ν) relates to the matrix re-

tarded Green’s function as GRk,xx(ν) = ITGR

k(ν)I where

IT =(

1 1)

. This structure means the Bogoliubov trans-

formation corresponds to GRk,xx(ν) = (ωk/ωk)G

Rk,xx(ν)

12

and the Bogoliubov transformed Green’s function takesthe form:

GRk,xx(ν) =

2ωk

ν2 − ω2k+ 2ωkΣk,xx(ν) + iωkκ(ν)

. (A3)

One can then find the transmission Tk(ν), reflectionRk(ν) and absorption Ak(ν) coefficients by consider-ing the modulus square of various coefficients. ClearlyTk(ν) = κ′L(ν)κ

′R(ν)|GR

k,xx(ν)|2 is independent of whichdirection light is incident from, while the absorption co-efficient Ak,σ=L,R(ν) takes the form

Ak,σ(ν) = −κ′σ(ν)[

2ℑ[

GRk,xx(ν)

]

+ κ(ν)∣

∣GRk,xx(ν)

∣

∣

2]

,

(A4)with κ(ν) = (κL(ν) + κR(ν))/2. The prefactor in thisexpression shows the obvious dependence on the trans-missivity of the input mirrors.

In order to separate mirror transmissivity dependentfeatures from the “intrinsic” properties of the ultra-strong coupling we will consider below the two quantitiestk(ν) = |GR

k,xx(ν)|2 as being proportional to the trans-

mission, and ak(ν) = −2ℑ[GRk,xx(ν)] as controlling the

absorption in the limit of a good cavity, i.e. κ(ν) → 0.The quantity ak(ν) differs from the full absorption as itneglects interference effects at the input mirror.

0

0.2

0.4

0.6

0.8

1

1.2

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Ak=

0(ν)

/κ

ν [eV]

κ=0.050eVκ=0.075eVκ=0.100eV

FIG. 6. (color online) Absorption spectrum, as defined by thefull expression, Eq. (A4), plotted for S1 = 0.25, S2 = 0.35,Ω1 = 0.06eV, Ω2 = 0.05eV, kBT = 0.035eV and various cav-ity loss rates, κ. Other parameters are as in Fig. 5. As dis-cussed in the text, in order that the polaritonic peaks have afinite width, one must include the effects of excitonic absorp-tion or non-mirror cavity losses. For this figure, we includean excitonic linewidth γ = 10−4eV.

For comparison to the results in Ref. [28] where silvermirrors were used, we present in Figures 1, 3, and 4 theeffects of large cavity linewidth. Figure 6 also shows howthe full absorption spectrum given by Eq. (A4) evolveswith varying linewidth. In calculating these spectra forlarge linewidth, an issue arises regarding the form of theabsorption spectrum: The equations written above as-sume that the only photon loss is due to escape throughthe mirrors. This means that the κ(ν) appearing explic-itly in Eq. (A4) (describing interference effects from the

mirror) is the same as the κ(ν) appearing in the denomi-nator of the photon Green’s function, Eq. (A3), and onemay check that for any frequency where ℑ[Σk,xx(ν)] issmall, this causes a near cancellation between the twocontributions. For the Gaussian exciton density of statesh(ǫ) used in this paper, this cancellation almost com-pletely suppresses the polariton peaks. Such a cancel-lation in the absorption spectrum can be expected onphysical grounds: if the only loss channel for photonsis the mirrors, then there is no absorption. All photonsthat enter eventually leave. In a real device there areother photon loss sources (absorbers, scattering by sur-face roughness). Similarly, in a real device, the excitonshave a non-zero rate of non-radiative decay. The effectof this is included by retaining a non-zero value of γ inthe denominator of the self-energy, Eq. (23). This leadsto Lorentzian tails of ℑ[Σk,xx(ν)], giving a finite weightto the polariton peak in the absorption spectrum; suchan effect was included by Houdre et al. [30]. We followthis approach in plotting Fig. 6.

Appendix B: Schrieffer-Wolff transformation

In section II we make use of the Schrieffer-Wolff ap-proximation; for completeness we provide here a briefexplanation of this formalism. The approach is based ondividing the Hamiltonian into two parts, H = H0 + H1,where the term H1 takes one between different “sectors”.In our case, these sectors correspond to different numbersof polaritons — i.e. H1 is the “counter-rotating” partof the Hamiltonian which simultaneous creates a photonand excites a molecule.The aim of the Schrieffer-Wolff formulation is to make

a unitary transformation ˆH = eiGHe−iG such that thetransformed Hamiltonian no longer has any coupling be-tween sectors. Physically, this corresponds to eliminat-ing the effect of virtual pair creation and destruction,and deriving how such virtual processes renormalize theHamiltonian within a given sector.If the Hamiltonian H1 can be treated perturbatively

by replacing H1 → ηH1 with η a small parameter, thenone can consider a series solution G =

∑∞n=1 η

nG(n). Inorder to make the first-order terms in η vanish, one mustchoose [G(1), H0] = iH1. This then leads (setting η = 1)to the expression:

˜H = H0 +

i

2

[

G(1), H1

]

+H.o.t (B1)

where the higher order terms involve all G(n>1). Stop-ping at leading order gives the expression in Eq. (9), cor-responding to the leading order effects of virtual pair cre-ation and annihilation. To solve [G,H0] = iH1 in prac-tice is straightforward if one knows the eigenspectrum of

H0 =∑

nE(0)n |n〉〈n|, which then allows one to write

G = i∑

n,m

|n〉〈n|H1|m〉〈m|E

(0)m − E

(0)n

.

13

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