THE JOURNAL OF CHEMICAL PHYSICS 142, 184707 (2015)

Optical microcavities enhance the exciton coherence length and eliminate vibronic coupling in J-aggregates F. C. Spano Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, USA

(Received 2 March 2015; accepted 17 April 2015; published online 12 May 2015) The properties of polaritons in J-aggregate microcavities are explored using a Hamiltonian which treats exciton-vibrational coupling and exciton-photon coupling on equal footing. When the cavity mode is resonant with the lowest-energy (0-0) transition in the J-aggregate, two polaritons are formed, the lowest-energy polariton (LP) and its higher-energy partner (P1), separated by the Rabi splitting. Strong coupling between the material and cavity modes leads to a decoupling of the exciton and vibrational degrees of freedom and an overall reduction of disorder within the LP. Such effects lead to an expanded material coherence length in the LP which leads to enhanced radiative decay rates. Additional spectral signatures include an amplification of the 0-0 peak coincident with a reduction in the 0-1 peak in the photoluminescence spectrum. It is also shown that the same cavity photon responsible for the LP/P1 splitting causes comparable splittings in the higher vibronic bands due to additional resonances between vibrationally excited states in the electronic ground state manifold and higher energy vibronic excitons. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4919348] I. INTRODUCTION

For some time now, microcavities have been used to mix the material eigenstates (or excitons) with optical cavity field modes in inorganic1–4 and organic materials5–21 resulting in the creation of hybrid matter-radiation states known as excitonpolaritons (or polaritons for short). Recent interest in polaritons has largely centered on Bose-Einstein (BE) condensation, which has primarily been observed in inorganic semiconductor microcavities.2,3 Typically, organic molecular crystals and films are more disordered; however, polariton lasing with a reduced threshold compared to ordinary photon lasing has been demonstrated in anthracene microcavities by KenaCohen and Forrest.12 Very recently, BE condensation was reported in ladder-type conjugated polymers with narrow, vibronically resolved transitions.20,21 In the simplest case, a material two-level system coupled to a resonant cavity mode results in upper and lower polariton branches separated by the so-called Rabi frequency, which is proportional to the amplitude of the cavity mode electric field. The cavity mode frequency and amplitude can therefore be viewed as control variables with which to create material-radiation states with targeted properties. A particularly relevant example to the current work is known as exciton hybridization, where the strong cavity mode mixes two or more exciton states together. This has been demonstrated for the vibronic excitons responsible for 0-0 and 0-1 absorption in thin films of 3,4,7,8 napthalenetetracarboxylic dianhydride.10 Cavity-induced mixing of excitons from two different J-aggregating dyes has also been observed.7,9 One of the most important properties of excitons is the range over which they can maintain wave-like properties—the so-called coherence length, L coh.22–31 The latter is a complicated function of the exciton bandwidth, vibronic coupling, disorder, and temperature. Enhanced coherence lengths lead 0021-9606/2015/142(18)/184707/12/$30.00

to ballistic transport which may enhance the efficiency of organic-based electronic devices such as solar cells.32–34 However, due to the inherent disorder in conjugated polymer films, coherence lengths are limited to only 1-2 nm35,36 as deduced from steady-state photoluminescence (PL) measurements, although larger coherence lengths at early times may be responsible for the sub-picosecond charge generation observed in conjugated polymer photovoltaic devices.32–34 Dubin et al. have provided striking evidence of micrometers-long coherence lengths in polydiacetylene wires grown in situ within the monomer crystal.27,37 Due to steric hindrance, such wires can be considered as perfectly straight and therefore devoid of the torsional disorder that plagues conventional spin-cast polymer films. The attainment of such long coherence lengths in polymer wires may involve the creation of polaritons,38 as cavity modes confined by the regular crystal lattice surrounding a given wire may mix with the excitons. Macroscopic spatial coherence is also the hallmark of the more exotic BE condensate20,39,40 where exciton-photon coupling exceeding the material and photon dissipation rates results in a macroscopic wave function below a critical temperature. The BE condensate is related to the so-called Dicke superradiant phase,41–43 characterized by similarly large coherence lengths. In the present work, we show how the exciton coherence length L coh in J-aggregates can be enhanced by strong coupling to a microcavity electromagnetic field. Such aggregates have been the subject of numerous experiments probing the optical properties of molecular aggregates inside microcavities.6–9,14–18 The exciton coupling in J-aggregating dyes, such as pseudoisocyanine chloride,44 causes the absorption spectrum to narrow and red-shift relative to the monomer spectrum.45 Such aggregates are disordered due to environmental fluctuations which are manifest as an inhomogeneous distribution of molecular transition frequencies.46,47 Disorder serves to localize excitons; hence, coherence lengths in J-aggregates

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are typically much smaller than the physical length of the aggregate. In a J-aggregate microcavity, the formation of polaritons, which, by definition, have well-defined wave vectors, is complicated by the loss of translational symmetry. Several works have been devoted to determining the nature of the microcavity excitations in a disordered J-aggregate, and in particular, the level of macroscopic coherence as a function of the in-plane wave vector.48–50 Here, we take the aggregate dimensions to be much smaller than the wavelength of the cavity field and confine our attention to the microscopic intraaggregate coherence. The main optical transition in J-aggregating dyes as well as many conjugated molecules is strongly coupled to one or more vibrational modes with energies in the vicinity of 0.17-0.2 eV, giving rise to pronounced vibronic progressions in absorption and emission. Vibronic coupling increases the effective mass of exciton, thereby increasing the ease at which the vibronic excitons can be localized by disorder. Recently, it was shown that L coh for the band bottom exciton, from which emission originates in J-aggregates, scales as the ratio of the 0-0 to 0-1 vibronic line strengths in the PL spectrum,29,51 thereby allowing a straightforward method for determining L coh directly from the PL line shape. However, the situation is more complex inside a microcavity where exciton-photon coupling induces mixing between the vibronic excitons. Such mixing leads to additional polariton branches; for example, in Ref. 10, three polariton branches were observed resulting from the mixing of the cavity field and the first two vibronic excitons. Vibronic exciton mixing should also have a profound effect on the PL line shape and, in particular, the relative vibronic band intensities. We begin our analysis with an investigation of the impact of exciton-photon coupling on the polariton energies of disorder-free J-aggregates. To account for vibronic coupling, we employ a Holstein-style material Hamiltonian expressed in a one- and two-particle basis set which has been very successful in capturing the photophysics of molecular aggregates.52 We next consider the effect of exciton-photon coupling on the PL spectral line shape, with particular emphasis on how intensity is redistributed within the vibronic progression. The effect of site-energy disorder is also included. Essentially, we find that sufficiently strong cavity-exciton coupling can effectively eliminate the diagonal disorder as well as vibronic coupling within the material part of the lower-branch polariton. Overall, we demonstrate that for experimentally attainable Rabi splittings of ≈0.1 eV and for typical disorder widths of ≈0.1 eV, the value of L coh is enhanced several-fold inside a resonant microcavity.

J. Chem. Phys. 142, 184707 (2015)

FIG. 1. Microcavity containing J-aggregates. The aggregate length is assumed to be small compared to the wavelength of the cavity mode.

A J-aggregate is typically realized as a head-to-tail arrangement of rod-shaped chromophores, with each chromophore hosting an electronic (S0 → S1) transition responsible for UVVis absorption. In this arrangement, the resonant Coulombic coupling between chromophores causes the k = 0 exciton, which possesses most of the oscillator strength, to shift to lower energies, resulting in a red-shift of the main peak in the absorption spectrum54 and superradiance at sufficiently low temperatures.46,55–57 The S0 → S1 transition in many molecules that form J-aggregates is often strongly coupled to a symmetric intramolecular vibrational mode as depicted in Fig. 2. Vibronic coupling leads to a pronounced vibronic progression in the isolated-molecule absorption and PL spectra. The vibrational frequency, ωvib, is approximately 0.15-0.18 eV/~ for a vast number π-conjugated molecules. Upon optical excitation, a given molecule relaxes along the associated nuclear coordinate to a new equilibrium position in the S1 potential. The relaxation energy is equal to the Huang-Rhys (HR) factor, λ2, multiplied by a vibrational quantum. The HR factor λ2 governs the relative shift of the two nuclear wells (S0 and S1), both assumed to be harmonic and of identical curvature in our analysis. We first consider aggregates in the absence of disorder, focusing on the interplay of vibronic coupling and cavitymaterial coupling. The Hamiltonian for an ensemble of N A disorder-free J-aggregates, each containing N chromophores,

II. MODEL

We consider an ensemble of linear J-aggregates positioned at or near the anti-node of the electromagnetic field inside an optical microcavity as depicted in Fig. 1. As in Ref. 53, we assume the aggregates to be identical and aligned, with the aggregate length taken to be much smaller than the wavelength of the cavity mode so that phase variations of the optical field over a given aggregate can be ignored. We further neglect interactions between aggregates.

FIG. 2. Ground state (S0) and excited state (S1) nuclear potentials for a single molecule. S1 vibrational quanta are indicated with a tilde overstrike. The 0-v emission transitions are indicated by the black dashed arrows.

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reads as H0 = H M + Hrad + H M −rad,

(1)

where H M , Hrad, and H M-rad, represent the material part, the radiation part, and the coupling between them, respectively. Since all aggregates are identical, noninteracting, and confined to dimensions smaller than a cavity wavelength, it is convenient to write the Hamiltonian in a subspace spanned by the collective states,53 which are in-phase linear combinations of individual aggregate states summed over all N A aggregates. For example, the collective electronic state |n⟩ N is defined as |n⟩ ≡ N A−1/2 i=1A |n⟩i , where |n⟩i represents the state in which the nth chromophore of the ith aggregate is electronically excited (S1) while all of the other chromophores in the ensemble remain in their ground (S0) states. Only the inphase collective states so defined can couple to the cavity mode when the ensemble has dimensions smaller than an optical wavelength (as we take to be the case here). In the collective subspace, the material part is the represented by Frenkel-Holstein Hamiltonian, H M = ~ω0−0

N 

|n⟩ ⟨n| +

n=1

+ ~ωvib

 n



J0{|n + 1⟩ ⟨n| + |n⟩ ⟨n + 1|}

n

b†n bn +~ωvib



{λ(b†n + bn )+λ 2} |n⟩ ⟨n| .

n

(2) The first term accounts for the electronic excitation energy, ~ω0−0, corresponding to the adiabatic (“0-0”) S0 → S1 transition within each molecule. The second term accounts for the electronic coupling between molecules within a given aggregate, which, for simplicity, is truncated to nearest-neighbors. For a J-aggregate, the nearest neighbor (nn) coupling, J0, is negative. The remaining terms in H M account for the vibronic coupling involving the aforementioned symmetric vibrational mode with energy, ~ωvib. The collective operators b†n and bn create and annihilate vibrational quanta, respectively, with N b†n ≡ N A−1/2 i=1A b†n,i . Here, b†n,i creates a vibrational quantum in the (S0) ground nuclear potential well of the nth chromophore on aggregate i. bn is the Hermitian conjugate of b†n . The energy of the microcavity field is accounted for in the second term in Eq. (1), † Hrad = ~ωcavacav acav.

(3)

† Here, acav (acav) creates (annihilates) a cavity photon of energy,

~ωcav. The final term in Eq. (1) represents the interaction between the cavity mode and the material states,53 ~Ω RM   † |g⟩ ⟨n| acav + |n⟩ ⟨g| acav , H M −rad = 2 n

(4)

where |g⟩ is the pure electronic ground state in which all N molecules on all N A aggregates are electronically unexcited and Ω RM is the molecular vacuum Rabi frequency given by  2µ N A~ωcav M ΩR ≡ . (5) ~ 2V ε 0 Here, µ is the transition dipole moment for an individual molecule within the aggregate, and V is the volume of the

contained radiation within the cavity. We refer to Ω RM as the molecular Rabi frequency, since √ it does not contain the aggregate enhancement factor of N, as shown in Sec. III. In the ground state of the material Hamiltonian, H M , all molecules are electronically unexcited with no vibrational quanta in the S0 potential. The ground state is denoted as |g; 0⟩ ≡ |g⟩ ⊗ |0, . . . , 0⟩ ,

(6)

where the list of zeros indicates the vibrational vacuum state for all N × N A molecules in the ensemble. Excited states are classified as either pure vibrational excitations or electronic/vibrational excitations (excitons). Vibrational excitations confined to the ith aggregate are denoted by |g; vr , vs , . . .⟩i = |g⟩ ⊗ |0, 0, vr , . . . , vs . . . , 0⟩i vr ≥ 1, vs ≥ 1, . . . ,

(7)

where vr is the number of quanta in the ground state (S0) potential of the rth chromophore (r = 1, 2, ..., N). It is also implicit in |g; vr , vs , . . .⟩i that all molecules in aggregates other than the ith aggregate are in their vibrational ground states. The states  in Eq. (7) have energies given by ~ωvib l vl . The collective electronic excitations of H M are denoted as |k, α⟩, |k, α⟩ ≡ N A−1/2

NA 

|k, α⟩i ,

(8)

i=1

with energy ~ωk,α . Here, |k, α⟩i represents the state in which the ith aggregate hosts a single electronic excitation (exciton) with wave vector k and index α,52 while all other aggregates remain in their electronic and vibrational ground states. The wave vector k is a good quantum number since periodic boundary conditions within a given aggregate are assumed (N + 1 = 1). The dimensionless wave vector takes the values k = 0, ±2π/N, . . . , π, and the index α (=1, 2, . . .) increases in order of increasing state energy for a given k. Hence, the lowest-energy excitons form a band of N states, denoted by |k, α = 1⟩i . Generally, the aggregate states |k, α⟩i can be expanded in oneparticle and two-particle states,52,58–61 1   ik n k,α |k, α⟩i = √ e cv˜ |n, v˜ ⟩i N n v˜ =0,1, ... 1    +√ N n v˜ =0,1, ... l  |n, v˜ ; n + l, v ′⟩i . × eik n cvk,α (9) ˜ ,l, v ′ v ′=1,2, ...

Here, |n, v˜ ⟩i represents an electronic excitation on the nth chromophore of aggregate i, with v˜ (=0, 1, 2, . . .) vibrational quanta in the shifted S1 nuclear well (see Fig. 2). The twoparticle states, |n, v˜ ; n + l, v ′⟩i , consist of a vibronic excitation on chromophore n (of aggregate i) and a purely vibrational excitation with v ′ (=1, 2, . . .) vibrational quanta residing in the S0 well of the chromophore at n + l. Our two-particle basis set is restricted to states in which both the vibronic and vibrational excitations are confined to the same aggregate. Fig. 3 shows the energy bands corresponding to J-aggregates with N = 30, obtained by a numeral analysis of the Hamiltonian H M , parameterized using typical values for the

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III. RABI SPLITTING IN THE LOWEST POLARITON BRANCH

As we are concerned with vacuum Rabi splitting, only dressed states with at most one cavity photon need to be considered in the analysis of the Hamiltonian in Eq. (1). Moreover, since the aggregate dimensions are much smaller than the cavity length, L cav, only aggregate excitons with k = 0 will couple to the cavity mode to form polaritons. In this section, we consider the case when the cavity mode frequency, ωcav, is resonant with the 0-0 electronic transition of the J-aggregate. The 0-0 transition, with frequency, ωk=0,α=1, corresponds to an excitation from the vibrationless ground state, |g, 0⟩, to the lowest energy exciton, |k = 0, α = 1⟩. In a resonantly tuned cavity, ωcav ≈ ωk=0,α=1, the dressed states, |g; 0; 1cav⟩ and |k = 0, a = 1; 0cav⟩, have the same energy and are coupled by H M -rad through the matrix element √ N F~Ω RM . (11) | ⟨g; 0; 1cav |H M−rad | k = 0, α = 1; 0cav⟩ | = 2 FIG. 3. Exciton bands for a linear J-aggregate with vibronic coupling. The electron-vibrational coupling involves a symmetric intramolecular mode with ~ω vib = 0.17 eV and HR factor, λ2 = 1. The aggregate contains N = 30 molecules with nearest-neighbor coupling, J0 = −~ω vib/4. Energies (reported relative to ~ω 0−0) are obtained from a numerical analysis of the H M in Eq. (2) using periodic boundary conditions. Only the first three vibronic bands are shown. Red arrows indicate emission pathways from the k = 0 exciton to the vibrationally excited states in the ground electronic manifold.

vibrational frequency (ωvib = 0.17 eV/~), the HR factor (λ2 = 1), and the nn coupling (J0 = −~ωvib/4).61 The energy levels are roughly divided into vibronic bands defined by the number of quanta v˜ (=0, 1, 2, . . .) characterizing the dominant single-particle component in the included band states. The lowest energy band includes the states labeled |k, α = 1⟩, dominated by single particle states with v˜ = 0. The positive band curvature, consistent with the k = 0 exciton lying at the band minimum, is the defining feature of J-aggregates. The 2 width of the v˜ th band is approximately given by (λ 2 v˜ e−λ /˜v !) W , where W is the free-exciton bandwidth, W = 4| J0|. We emphasize that the band structure of Fig. 3 is only present in the weak to intermediate exciton coupling limit where W . λ 2ωvib.52 In order to analyze the complete Hamiltonian H0 in Eq. (1), we utilize the “dressed state” basis set, consisting of products of the material eigenstates of H M and the radiation eigenstates of Hrad. The latter are simply the photon number states, |ncav⟩, where ncav is the number of cavity photons. Dressed states are therefore eigenstates of H M + Hrad. States in which there are no electronic or vibrational excitations are represented as |g; 0; ncav⟩ ≡ |g; 0⟩ ⊗ |ncav⟩. States that remain electronically unexcited but with additional vibrational quanta include |g; vr , vs , . . . ; ncav⟩i ≡ |g; vr , vs , . . .⟩i ⊗ |ncav⟩ , r, s = 1, 2, . . . N,

(10a)

while states hosting a single electronic excitation include |k, α; ncav⟩ ≡ |k, α⟩ ⊗ |ncav⟩ .

(10b)

Dressed states with the same total number of photons plus excitons are coupled to each other via H M−rad.

Here, F is the generalized Franck-Condon (FC) factor51  ⟨0| v˜ ⟩ |2, F≡| cvk=0;α=1 (12) ˜ v˜

where ⟨0| v˜ ⟩ is a vibrational overlap integral between the ground vibrational state in the S0 potential and the state with v˜ quanta in the shifted excited state potential (S1). F ranges from exp(−λ2) in the weak exciton coupling limit (i.e., N uncoupled chromophores) to exp(−λ2/N) in the strong exciton coupling limit.52 For weak exciton-photon coupling, the dressed states, |g; 0; 1cav⟩ and |k = 0, a = 1; 0cav⟩, are roughly equally admixed in the lower and upper polaritons with the associated Rabi splitting given by √ L P/P L P/P Ω R 1 ≈ N F Ω RM Ω R 1 σ, or when the total number of chromophores satisfies  2 σ N × NA ≫ . Ω RM (N A = 1)

FIG. 7. The effective HR factor (from Eq. (28)) and the free-exciton admixture to the LP as a function of the molecular Rabi frequency. Open and closed circles correspond to the disorder-free J-aggregate from Fig. 4. Open triangles represent the mean free-exciton admixture determined by averaging over an ensemble of 2000 site disordered J-aggregates with standard deviation of σ = 0.5ω vib.

just the material part of the LP wave function, is given by AFE ≡

⟨k = 0; 0; 0cav |ΨL P ⟩|2 ,  1 − |⟨g; 0; 1cav |ΨL P ⟩|2 − |⟨g; 1r ; 1cav |ΨL P ⟩|2 − . . . r

(30) where the sum in the denominator is over all radiation states having a single cavity photon but with any number of vibrational quanta within the S0 ground state potential. Fig. 7 shows how AFE increases towards unity with increasing Rabi frequency, consistent with the cavity-induced elimination of vibronic coupling within the lower-branch polariton. VI. THE EFFECTS OF DISORDER

The results of Sec. V show that strong coupling between the exciton and the cavity mode results in a polariton in which the material part is practically a free exciton, uncoupled to vibrations. As the Rabi frequency increases, the cavity field increasingly admixes the k = 0 free-exciton in forming a lower-branch polariton. The selective mechanism should also operate when disorder is present, as we verify in what follows. To accommodate site disorder, we consider the Hamiltonian  Hdis = H0 + ~ ∆n |n⟩ ⟨n| , (31)

In the opposite limit of very strong disorder, each aggregate is independent of all others and we recover the free-space limit. We can treat the general case by assuming the Rabi frequency is a function of disorder, Ω RM (σ) = Ω RM ( N˜ A(σ)), where N˜ A(σ) replaces N A in Eq. (5). N˜ A(σ) is the approximate number of resonantly coupled J-aggregates in the microcavity and ranges from unity in the limit of strong disorder to N A in the limit of strong coupling. In a more sophisticated model, Ω RM (σ) would be solved for self-consistently, but for the purposes of the present work, we treat it as an external parameter. Fig. 8 shows the reduced PL spectra for disordered J-aggregates within a microcavity obtained by averaging over 2000 configurations of disorder. The k ∥ = 0 cavity mode is tuned to the 0-0 transition evaluated in the absence of disorder, i.e., ωcav = ωk=0,α=1. H0 in Eq. (31) is parameterized exactly as was done in Fig. 3 and the standard deviation characterizing the diagonal disorder is σ = 0.5ωvib = 0.085 eV, a typical value for molecular aggregates as well as conjugated polymer films. Spectra are shown for the same range of Rabi frequencies as reported in Fig. 5(b). In all spectra, the 0-0 peak has been scaled by the inverse of N˜ A(σ). In general, the 0-0/0-1 ratio increases while the spectra significantly red-shift and narrow with increasing excitonphoton coupling. In the weak coupling limit (black dashed curve), the spectrum is essentially the free-space spectrum. The FWHM is approximately 0.60ωvib which√ is about one-half of the FWHM of the disorder distribution, 2 2 ln 2 σ(=1.17ωvib). The discrepancy is mainly due to the narrowing experienced by the lowest-energy site in a Gaussian distribution of N sites,29 although exciton motional narrowing63 also contributes to a much smaller extent. The 0-0/0-1 ratio in the black-dashed spectrum is about two; the sharp reduction from the value of 10

n

with H0 defined in Eq. (1). In Eq. (31), ∆n represents the detuning in the molecular transition frequency for the nth molecule. In what follows, each ∆n is chosen randomly from a Gaussian distribution of standard deviation σ. The ∆n within a particular aggregate are chosen independently from each other. However, because we have assumed identical J-aggregates, the configuration of disorder remains the same for each aggregate. This means that the N A dependence of the molecular Rabi frequency, Ω RM , in Eq. (5) and the 0-0 intensity in Eq. (22a) remains intact. If, however, we assume each aggregate is independently disordered, then Eqs. (5) and (22) still remain valid

FIG. 8. Calculated reduced PL spectra for disordered J-aggregates (J0 = −~ω vib/4, ω vib = 0.17 eV, λ2 = 1) containing N = 10 chromophores for various Rabi frequencies. The 0-0 peak is reduced by a factor of N A (σ ). The standard deviation of the disorder distribution is σ = 0.5ω vib and each spectrum is an average over 2000 configurations of disorder. The cavity mode frequency is ω cav = ω cav,0 with ω cav,0 identical to that used in Figs. 3 and 4(b). Inset shows the associated coherence function (see text) with the coherence numbers Ncoh indicated.

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when disorder is absent (see Fig. 5(b)) is caused by disorderinduced localization. This can be approximately understood by replacing N in Eq. (27) by the (smaller) coherence number Ncoh (see below),51 IP0−0 L IP0−1 L

=

Ncoh × N˜ A(σ) . λ2

(32)

When the coupling to the cavity mode is much stronger (Ω RM = ωvib, red curve in Fig. 8), the line width narrows by more than a factor of 2; the FWHM is now only 0.26ωvib. The spectrum strongly resembles its disorder-free counterpart in Fig. 5(b) including the spectral red-shift due to Rabi splitting as well as the 0-0/0-1 ratio which rises to approximately 20 N˜ A(σ), driven mainly by the substantial increase in the 0-0 emission peak. Such behavior is due to an increasing admixture of the free exciton in Eq. (29) in the lowest energy polariton. Fig. 7 shows how the mean of the free-exciton admixture, ⟨AFE⟩C , evaluated by averaging over 2000 configurations of disorder, increases with Rabi frequency. ⟨AFE⟩C rises from the value of 0.1 when cavity coupling is absent to about 0.8 when Ω RM = ωvib, only slightly smaller than the value of 0.9 when disorder is entirely absent. The effect is more dramatic when Ω RM = 1.5ωvib; here, the admixture of the k = 0 free exciton for the disordered aggregate is only 4% less than that of the disorder-free aggregate. We conclude that strong-coupling effectively eliminates disorder and vibronic coupling within the lower-branch polariton.

The corresponding exciton coherence length is then L coh = Ncoh − 1. As an important limiting case, inserting the free exciton in Eq. (29) into Eq. (33) yields C(r) = 1, independent of r, indicating full delocalization (Ncoh = N) with no vibronic relaxation whatsoever. Within the LP, the coherence drops to C(r) = 1/2, since it contains 50% admixture of the free exciton. Note that Eq. (34) still yields Ncoh = N. The inset of Fig. 8 shows that the coherence functions for the disordered J-aggregates for various Rabi frequencies. The width of the coherence function increases dramatically with increasing coupling to the cavity mode, consistent with the increased admixture of the vibrationally uncoupled (“free”) k = 0 exciton (see Fig. 7). The coherence numbers evaluated from Eq. (34) increase from 2.4 in the limit of very weak coupling to 9.9 for strong coupling. Since N = 10, the excitonic part of the lower-branch polariton is almost completely delocalized on each aggregate in the strong coupling limit. The inset of Figure 8 also shows that C(0) increases with cavity mode coupling, reflecting the anticipated decoupling of the exciton to vibrations. For the highest coupling shown, C(0) is approximately 0.45, just shy of the theoretical maximum of 0.5 for the polariton in Eq. (15). Based on our analysis, the material part of the lower-branch polariton is about 90% dominated by the k = 0 free-exciton for the strongest Rabi frequency. The ability of the cavity mode to “select” the coherent exciton in the presence of vibronic coupling and disorder is rather remarkable and suggests the possibility of eliminating or at least greatly reducing the effects of disorder and vibronic coupling in molecular aggregates.

VII. INTRA-AGGREGATE EXCITON COHERENCE

If the narrowing of the PL spectrum with increasing Rabi frequency observed in Sec. VI is due to an increasing contribution of the k = 0 free-exciton, then this should be reflected in an increasing intra-aggregate excitonic coherence length. In this section, we investigate the spatial coherence of the material part of the (lower-branch) polariton in disordered J-aggregates. For a thermal distribution of lower-branch polaritons responsible for emission, the intra-aggregate coherence function pertaining to the excitonic component is given by29,51,64    † , (33) C(r) ≡ ⟨ΨL P | Bn Bn+r |ΨL P ⟩ n

C,T

Bn†

≡ |n; vac⟩ ⟨g; vac| creates an excitation on the nth where site with no quanta in the unshifted (S0) nuclear potential, and ⟨· · · ⟩C,T represents a dual configurational and Boltzmann average. Under periodic boundary conditions, the index r takes on the values, r = 0, ±1, ±2, . . . , N/2 assuming N is even. In addition, the value of n + r must lie between 1 and N; hence, if n + r > N, n + r is replaced by n + r − N. Conversely, if n + r < 1, n + r is replaced by n + r + N. The coherence function contains two primary sources of information: its width determines the intra-aggregate coherence length, and its “height,” C(0), indicates the level of nuclear relaxation on the electronically excited chromophore. The total number of coherently connected chromophores, Ncoh, is obtained from the coherence function, via51  Ncoh = C(0)−1 |C(r)|. (34) r

VIII. DISCUSSION/CONCLUSION

We have shown that strong coupling between a microcavity mode and a molecular aggregate can decouple electronic and nuclear degrees of freedom and dramatically reduce disorder in the lower-branch polariton. In free-space J-aggregates with typical disorder widths (σ ≈ 0.1 eV) and exciton bandwidths (4J0 ≈ 0.2 eV), low-energy excitons are coherent over 2-3 chromophores. As shown in Fig. 8, the coherence number more than doubles (black curve to blue curve)√ when the vacuum Rabi splitting rivals the disorder width, NΩ RM ≈ 2σ. These estimates were obtained within the backdrop of vibronic coupling to the ubiquitous symmetric stretching mode with a HR factor of √ unity. In the limit of very strong cavity-exciton coupling ( NΩ RM >> σ, λ 2ωvib), the excitonic part of the LP approaches a (k = 0) free exciton with a (intraaggregate) coherence length which spans the entire aggregate. The pronounced increase in the free-exciton admixture to the LP with increasing Rabi frequency is readily appreciated from Fig. 7. The simultaneous incorporation of exciton-phonon and exciton-photon coupling leads to interesting effects. For example, when the cavity photon is near resonance with the lowest-energy transition in the aggregate, it causes the usual Rabi splitting and the creation of the LP and P1 polaritons. However, the same photon is also approximately resonant with a transition between an excited vibrational state (with v > 0 vibrations) in the electronic ground state manifold and a k = 0 vibronic exciton in the v˜ = v band. This allows higher

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polaritons, P2, P3, etc., to form with splittings that are comparable to the LP/P1 splitting, as demonstrated in Fig. 4(b). It is also quite interesting that the higher-energy polaritons have a strong two-particle material component. Two-particle states define the polaron radius62,64 and are essential for a quantitative description of the photophysical response of molecular aggregates, especially the photoluminescence.52,58,59 We will investigate the two-particle polaritons in greater detail in a future work. J-aggregates in free-space are superradiant at sufficiently low temperatures due to the enhancement of the 0-0 transition strength by the number of coherently coupled molecules.23,25,45,46,55–57 When placed inside a microcavity the radiative decay rate scales as Ncoh N˜ A(σ). The intra-aggregate coherence number, Ncoh, is enhanced over its free-space value as shown in Fig. 8. The additional enhancement by N˜ A(σ), is due to the cavity-induced inter-aggregate coherence. (There is also an additional factor arising from the cavity-modulated density of photon states.13) For strong cavity-exciton coupling, the radiative rate enhancement increases to N × N A despite the presence of disorder and vibronic coupling. The creation of the LP in this case is related to the so-called superradiant phase transition.41,42 The cavity-induced changes in the vibronic coupling are reflected in changes to the PL line shape. For an ensemble of disordered J-aggregates, the ratio of 0-0 and 0-1 line strengths is approximately given by 0−1 IP0−0 L /I P L ≈

N˜ A(σ)Ncoh , λ 2eff

(35)

which reduces to ≈Ncoh/λ 2 in the free-space limit in agreement with Ref. 51. The PL ratio in Eq. (35) derives from coherently enhanced 0-0 emission and incoherent 0-1 emission. The ratio can also be expressed in terms of the Rabi splitting, with the right hand side of Eq. (35) equal to Ncoh[Ω RM /Ω RM (N A = 1)]2/λ 2eff . Taking a measured Rabi splitting of 0.1 eV6–9,14–18 0−1 and ~Ω RM (N A = 1) ≈ 0.4 meV from Eq. (14) gives IP0−0 L /I P L ≈ 105/λ 2eff which can easily exceed 106 for a substantially reduced HR factor. Hence, only the 0-0 peak should be observable in the PL spectrum, which appears to be consistent with most studies,6–9,14–18 although in order to avoid cavity enhancement13 of the 0-0 and/or 0-1 emission, one should employ an orthogonal detection scheme. Finally, we point out that the PL spectrum discussed so far assumes that emission originates from the LP. At higher temperatures, excited states may also contribute as long as kT is comparable to the Rabi splitting. Interesting effects should also occur in H-aggregates, where, unlike in J-aggregates, the band-bottom exciton has k = π and is therefore only weakly fluorescent. By tuning the cavity mode into resonance with the higher-energy k = 0 exciton, it should be possible to Rabi-split the lower-branch polariton to energies below the uncoupled k = π exciton, thereby converting a weakly fluorescent H-aggregate into a superradiant “J-like” aggregate. We are unaware of any discussion of this effect in the literature, but it may be useful in optimizing the performance of light-emitting diodes, as one practical application.

It would be interesting to apply the current approach, which treats exciton-photon and exciton-phonon coupling on equal footing, to study the effect of vibronic coupling on BE condensation and lasing in organic systems. Important steps in this direction have already been taken by Bittner et al.40,65 and Cwik et al.66 It is also of interest to examine nonlinear optical properties inside microcavities. Herrera et al.53 have recently shown theoretically that a single pump photon inside a J-aggregate microcavity is sufficient to optically switch a weak probe field. The current techniques can be used to study optical switching in the so-called “lambda” configuration where the base levels involve the electronic ground state |g; 0, . . . 0⟩ and its first vibrational excited state. Such studies will form the basis of future work. ACKNOWLEDGMENTS

F.C.S. is supported by the National Science Foundation, Grant No. DMR-1203811. F.C.S. would like to acknowledge fruitful discussions with Felipe Herrera. 1C.

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