Precise determination of excitation energies in condensed-phase molecular systems based on exciton-polariton measurements
PPrecise determination of excitation energies in condensed-phase molecular systemsbased on exciton-polariton measurements
Nguyen Thanh Phuc and Akihito Ishizaki
Department of Theoretical and Computational Molecular Science,Institute for Molecular Science, Okazaki 444-8585, Japan andDepartment of Structural Molecular Science, The Graduate University for Advanced Studies, Okazaki 444-8585, Japan
The precise determination of the excitation energies in condensed-phase molecular systems isimportant for understanding system-environment interactions as well as for the prerequisite inputdata of theoretical models used to study the dynamics of the system. The excitation energies areusually determined by fitting of the measured optical spectra that contain broad and unresolvedpeaks as a result of the thermally random dynamics of the environment. Herein, we propose amethod for precise energy determination by strongly coupling the molecular system to an opticalcavity and measuring the energy of the resulting polariton. The effect of thermal fluctuationsinduced by the environment on the polariton is also investigated, from which a power scaling lawrelating the polariton’s linewidth to the number of molecules is obtained. The power exponent givesimportant information about the environmental dynamics.
Embedded in a high density of environmental particles,the excitation energies of molecules in condensed phasecan be modified from their values in gas phase by thestatic influence of various kinds of system-environmentinteractions including electrostatic interaction and hy-drogen bonding [1–6], as well as the effects of molecularconformation [7, 8]. Therefore, a precise determinationof the excitation energies of condensed-phase molecularsystems is significant for the understanding of system-environment interactions. Moreover, the energy valuesare prerequisite as input data for almost all theoreticalmodels used to study the dynamics of molecular sys-tems [9, 10]. The excitation energies of condensed-phasemolecular systems are usually determined by fitting ofthe measured optical spectra. The optical spectra, how-ever, often contain broad and unresolved peaks as a re-sult of the thermally random dynamics of the environ-ment interacting with the molecular system. Moreover,to extract excitation energy information from the opti-cal spectra, it is necessary to develop a theory of opticalspectra that addresses the often sophisticated spectraldensity of the environment. A variety of approximationsare sometimes used to reduce the complexity of the calcu-lations [9, 11, 12]. Consequently, it is desirable to developan alternative approach that can precisely determine theexcitation energies of condensed-phase molecular systemswithout requiring detailed information about the envi-ronmental random dynamics.In this Letter, we propose a method for the precise de-termination of the excitation energies of condensed-phasemolecular systems by strongly coupling the molecules toan optical cavity and measuring the energy of the po-lariton that results from the hybridization of the de-grees of freedom of light and matter. Strong coupling ofmolecules to an optical cavity has already been realizedin many experimental platforms [13–33]. It has led toa variety of interesting phenomena and important appli-cations including the control of chemical reactivity [34–41], enhancement of transport [42–47], nonlinear optical properties of organic semiconductors with applications tooptoelectronic devices [48–51], and polariton lasing andcondensate [52–55]. The underlying mechanism that al-lows a precise determination of the excitation energies ofcondensed-phase molecular systems is that the polaritonappears as a sharp peak in optical spectrum under theinflence of strong coupling between the cavity mode andthe electronic excitations of molecules inside the cavity.This is related to the effect of vibronic or polaron de-coupling found in the Holstein-Tavis-Cummings (HTC)model that describes molecules with a single vibrationalmode that are coupled to an optical cavity [36, 56–59] andthe extended model [60]. However, since the polariton isa collective superposition of a large number of electronicexcitations of molecules [see Eq. (4)] and therefore can bevulnerable to decoherence, the effect of thermal fluctua-tions induced by the environment on the polariton stateat finite temperatures, which is not captured in the HTCmodel and its extension, is a nontrivial and importantissue.By investigating the effect of thermal fluctuations in-duced by the environment on the polariton, a power scal-ing law relating the polariton’s linewidth to the numberof molecules coupled to the cavity is determined. Thepower exponent strongly depends on the environment’sdynamic time scale. As such, information on environ-mental dynamics can be extracted from the polaritonspectrum obtained for a variable number of molecules.Since the polariton contains both light and matter de-grees of freedom, its energy can be obtained by ei-ther cavity-transmission or molecular-absorption spec-troscopies. In the latter, the polariton needs to be in abright state with respect to molecular absorption. How-ever, this condition is not satisfied if there are pairsof molecules with the opposite orientations, such as inthe case of random orientations. The effective Rabi fre-quencies for an ensemble of identical molecules or molec-ular complexes with random orientations are derived.The molecular-absorption and cavity-transmission spec- a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t troscopies are calculated for several cases of molecularsystems with different types of orientations, in which thepotential of using polariton for precise determination ofexcitation energies in condensed-phase molecular systemsis demonstrated. Effect of environmental thermal fluctuations on the po-lariton’s linewidth.–
We consider a system of N identicalmolecules whose electronic excitations are coupled to asingle mode of an optical cavity (Fig. 1) via dipole inter-action ˆ H mc = (cid:126) Ω R N (cid:88) m =1 (cid:0) | e m (cid:105)(cid:104) g m | ˆ a + | g m (cid:105)(cid:104) e m | ˆ a † (cid:1) , (1)where Ω R is the so-called single-emitter Rabi frequencythat characterizes the coupling strength between the cav-ity and a molecule, ˆ a denotes the annihilation operator ofthe cavity photon, and | g m (cid:105) and | e m (cid:105) represent the elec-tronic ground and excited states, respectively, of the m thmolecule. In this case, we assume that all molecules in-side the cavity have the same orientation such that theirRabi couplings are equal. Molecules with different orien-tations will be considered later.Each molecule in condensed phase is assumed tobe coupled to an independent environment, which ismodeled by an ensemble of harmonic oscillators ˆ H e = (cid:80) Nm =1 (cid:80) ξ (cid:126) ω m,ξ ˆ b † m,ξ ˆ b m,ξ , where ω m,ξ and ˆ b m,ξ representthe frequency and annihilation operator of the ξ modeof the environment surrounding the m th molecule. Thedynamics of the environment at a finite temperature in-duces energy fluctuations in the electronic excited statesof the molecules as given by the Hamiltonian [61]ˆ H me = N (cid:88) m =1 (cid:126) ω + (cid:88) ξ g m,ξ (cid:16) ˆ b † m,ξ + ˆ b m,ξ (cid:17) | e m (cid:105)(cid:104) e m | . (2)Here, (cid:126) ω is the molecule’s excitation energy and g m,ξ denotes the coupling strength between the m th moleculeand the ξ mode of the environment. The dynamicsof the environment is characterized by the relaxationfunction Ψ m ( t ) = (2 /π ) (cid:82) ∞ d ωJ m ( ω ) cos( ωt ) /ω , where J m ( ω ) = π (cid:80) ξ g m,ξ δ ( ω − ω m,ξ ) is the spectral density.When the spectral density is given by the Drude-Lorentzform, J m ( ω ) = 2 λ m τ m ω/ ( τ m ω + 1), the relaxation func-tion has an exponential form, Ψ m ( t ) = 2 λ m exp( − t/τ m ),where λ m is the environmental reorganization energy,which is usually employed to characterize the system-environment coupling strength, and τ m is the character-istic timescale of the environmental relaxation or reorga-nization process [62]. The time evolution of the system’sreduced density operator can be solved in a numericallyaccurate fashion using the hierarchical equations of mo-tion approach for example [63].The molecular absorption spectrum can be expressedin terms of the system’s dynamical quantities as [64] A ( ω ) = Im (cid:26) i (cid:126) (cid:90) ∞ d te iωt Tr (cid:2) ˆ µ G ( t )ˆ µ × ˆ ρ (cid:3)(cid:27) , (3) FIG. 1: Schematic illustration of a system of condensed-phasemolecules coupled to a single mode of an optical cavity (lightmagenta) with frequency ω c . Each molecule (green sphere)with the electronic excitation energy (cid:126) ω interacts with anindependent surrounding environment represented by blue el-lipsoids. The thermal dynamics of the environments at a finitetemperature T induces energy fluctuations in the moleculesthat are characterized by the reorganization energy λ andthe relaxation time scale τ . The coupling strength betweenthe cavity and a molecule is given by the single-emitter Rabifrequency Ω R . Due to the finite transmitivity of the cavitymirrors (grey plates), the cavity photon has a loss rate of κ . where ˆ µ = (cid:80) Nm =1 ( µ m | e m (cid:105)(cid:104) g m | + µ ∗ m | g m (cid:105)(cid:104) e m | ) is the to-tal transition dipole moment operator with µ m being thematrix element of the transition dipole moment of the m th molecule, and ˆ µ × ˆ ρ ≡ ˆ µ ˆ ρ − ˆ ρ ˆ µ . Here, the densityoperator ˆ ρ = | G (cid:105)(cid:104) G | with | G (cid:105) = (cid:81) Nm =1 | g m (cid:105) ⊗ | (cid:105) c be-ing the ground state of the cavity-molecule system inwhich all the molecules are in their electronic groundstates and the cavity is in the vacuum state. The super-operator G ( t ) describes the time evolution of the systemin Liouville space. In the following numerical demonstra-tion, we set the parameters of the molecular system andthe environment to be ω = 12400 cm − , λ = 50 cm − , τ = 100 fs, and T = 300 K, which are typical valuesin photosynthetic pigment-protein complexes [9, 10]. Forsimplicity, we assume that the parameters of the envi-ronments are equal. The cavity frequency is taken to be ω c = 12450 cm − , i.e., with a detuning of 50 cm − fromthe molecule’s excitation energy. The cavity’s Q -factor( κ = ω c /Q ) is set to be Q = 10 . When many moleculesare coupled to a single mode of the optical cavity, thecollective Rabi frequency √ N Ω R rather than the single-emitter Rabi frequency Ω R determines the polariton en-ergy.To investigate the effect of thermal fluctuation due tothe environment on the polariton’s linewidth, we calcu-late the full width at half maximum of the lower-polaritonpeak, which represents the polariton with an energyless than both the molecule’s excitation energy and thecavity frequency in the molecular-absorption spectrum.It is determined that in the absence of molecule-cavitycoupling, the molecular absorption peak has a broadlinewidth of approximately 291 cm − , which is largerthan the typical separation between absorption peaks ofdifferent molecules [9]. In contrast, when the moleculesare coupled to the cavity mode, the polariton peak inthe absorption spectrum has a much smaller linewidththat decreases with an increase of the number N ofmolecules coupled to the cavity. The N -dependence ofthe polariton’s linewidth is determined to follow a powerscaling law ∆ ν LP = ∆ N − α with ∆ = 138 ± − and α = 0 . ± .
02 (for the collective Rabi frequency √ N Ω R = 0 . | LP (cid:105) = c N (cid:89) m =1 | g m (cid:105) ⊗ | (cid:105) c + c | B (cid:105) ⊗ | (cid:105) c , (4)where c and c are coefficients of the superposition thatsatisfy | c | + | c | = 1. Here, | n (cid:105) c denotes the Fock statewith n cavity photons and | B (cid:105) = (1 / √ N ) (cid:80) Nm =1 | e m (cid:105) isthe so-called bright state, which is a superposition of elec-tronic excitations of all molecules coupled to the cavity.Given that each of the molecular electronic excitations | e m (cid:105) is coupled to the environmental modes via the in-teraction given by Eq. (2), the lower polariton | LP (cid:105) withthe aforementioned structure can be regarded as havingan effective interaction in which the number of modesincreases by a factor of N but the coupling strength toeach mode decreases by a factor of 1 /N . As such, theeffective spectral density J LP ( ω ) and the reorganizationenergy λ LP for the lower polariton are modified by a fac-tor of 1 /N because the spectral density is proportionalto the mode density and the coupling strength squared.The effect of the environment on the linewidth ofthe lower polariton in the absorption spectrum dependson the environmental dynamics. In the inhomoge-neous broadening limit √ k B T λ LP (cid:29) τ − , which corre-sponds to the slow environmental dynamics, the line-shape has a Gaussian form with the linewidth given by √ k B T λ LP [61]. As a result, the polariton’s linewidthshould follow an N − / scaling. In the opposite limit ofhomogeneous broadening √ k B T λ LP (cid:28) τ − , which cor-responds to the fast environmental dynamics, the line-shape has a Lorentzian form with the linewidth given by k B T λ LP τ . As a result, the polariton’s linewidth shouldfollow as N − scaling.To examine the validity of the preceding analysis, inwhich it was assumed that the thermal fluctuations donot affect the structure of the lower polariton, we nu-merically calculate the N -dependence of the polariton’slinewidth in the absorption spectrum for different valuesof the reorganization energy λ . It was determined thatthe power scaling law of ∆ ν LP ∝ N − α is well-satisfiedwith a strong λ -dependence of the power exponent asshown in Fig. 2. Consequently, environmental dynamicsinformation can be extracted from the power exponent α obtained by measuring the lower polariton energy fora variable number of molecules coupled to the cavity. It FIG. 2: Dependence of the power exponent α in the powerscaling ∆ ν LP ∝ N − α of the lower polariton’s linewidth withrespect to the number of molecules coupled to the cavity onthe dimensionless quantity τ √ k B T λ that characterizes theenvironment’s dynamic motion. Here, λ is the reorganizationenergy, τ is the relaxation time scale, and T is the temperatureof the environment ( k B is the Boltzmann constant). The red(green) line indicates the value of α = 1 ( α = 0 . α in the inhomogeneous (homogeneous)broadening limit τ √ k B T λ (cid:28) τ √ k B T λ (cid:29)
1) under theassumption that the environmental thermal fluctuations donot affect the structure of the lower polariton. is evident that α generally decreases with an increase of λ and seems to approach a steady value close to α = 1( α = 0 .
5) in the inhomogeneous (homogeneous) broaden-ing limit. The remaining deviation should, however, beattributed to the effect of environmental thermal fluctu-ations on the structure of the lower polariton.
Polariton energy.–
The energy of the lower polaritoncan be obtained by diagonalizing the Hamiltonian of thecavity-molecule system, yielding ω LP = 12 (cid:20) ω c + ω − (cid:113) ( ω c − ω ) + N Ω (cid:21) . (5)For a sufficiently large Rabi frequency, √ N Ω R (cid:29) | ω c − ω | , it reduces to ω LP (cid:39) ( ω c + ω − √ N Ω R ) /
2. There-fore, by repeating the measurement of the lower polaritonenergy with variable molecular density, in which N is var-ied, or with variable number of photons in the cavity, bywhich Ω R is varied, we can obtain the molecular exci-tation energy (cid:126) ω via a simple linear fitting, given thatthe cavity frequency ω c is known, for example, based onthe transmission spectroscopy measurement of the barecavity.The deviation | ν LP − ω LP | of the position of the lower-polariton peak in the absorption spectrum from its en-ergy as a function of N was investigated. It was deter-mined that the deviation increases with an increase of thenumber of molecules coupled to the cavity prior to satura-tion, following the function | ν LP − ω LP | = A − Be − γN [65].Using the exponential fitting procedure, we determine thesaturation value of | ν LP − ω LP | to be A (cid:39)
27 cm − , whichis smaller than ∆ ν LP (cid:39)
55 cm − for N = 5 (for the samecollective Rabi frequency √ N Ω R = 0 . ν LP and | ν LP − ω LP | on the Rabi frequencyΩ R was also investigated. It was determined that ∆ ν LP decreases with an increase of Ω R before it saturates for asufficiently large collective Rabi frequency [65]. The sat-uration value is determined by the number of moleculescoupled to the cavity. The deviation | ν LP − ω LP | followsa power scaling | ν LP − ω LP | = C Ω − η R [65].Next, we consider a more general system of molecu-lar complex composed of molecules with different excita-tion energies and transition dipole moments. Both themagnitude and the sign of the Rabi coupling can dif-fer from one molecule to another in the system. In thefollowing numerical demonstration, we consider a sys-tem of N = 3 molecules with excitation energies ω =12400 cm − , ω = 12500 cm − , and ω = 12600 cm − .The Rabi frequencies associated with the three moleculesare taken to be √ N Ω (1)R = 0 . (2)R = Ω (1)R √ /
2, andΩ (3)R = − Ω (1)R / √
2. We also consider coupling betweenelectronic excitations of different molecules given by theHamiltonianˆ H mm = (cid:88) m (cid:54) = n (cid:126) V mn | e n (cid:105)(cid:104) g n | ⊗ | g m (cid:105)(cid:104) e m | , (6)where the coupling matrix elements V mn between the m th and the n th molecules satisfy V mn = V ∗ nm . In thiscase we take V mn = 50 cm − .There is a relatively sharp and isolated peak in themolecular absorption spectrum that corresponds to thelower polariton [65]. The linewidth of the peak is deter-mined to be ∆ ν LP (cid:39)
85 cm − , which has a comparablemagnitude to the linewidth of the polariton peak in theinvestigated case of N = 3 identical molecules. Besidesthe lower-polariton peak, there is a broad peak that con-tains the absorption spectra of the upper-polariton aswell as two remaining energy eigenstates. In the case ofidentical molecules, these two energy eigenstates are darkstates with respect to molecular absorption and thus donot appear in the absorption spectrum. Due to the differ-ence in the excitation energy and the Rabi coupling be-tween the molecules, as well as the inter-molecular elec-tronic couplings, they are no longer fully dark states.However, given that these eigenstates consist mainly ofthe degrees of freedom of matter, their linewidths arebroad compared with those of the polaritons.The energy of the lower polariton is obtained by diag-onalizing the Hamiltonian of the molecule-cavity system,which in this case is a 4 × ω c Ω (1)R / (2)R / (3)R / (1)R / ω V V Ω (2)R / V ω V Ω (3)R / V V ω . (7)The deviation of the position of the lower-polariton peakin the absorption spectrum from the lower polariton en- ergy was determined to be | ν LP − ω LP | (cid:39)
19 cm − , whichis smaller than ∆ ν LP . By repeating the measurement ofthe lower-polariton energy with variable molecular den-sity and/or variable cavity frequency, for example, by ad-justing the distance between two mirrors and using thegenetic algorithm for a multivariable fitting [9, 66–68],the excitation energies of the molecules can be deter-mined or at least the accuracy of their values obtainedusing other approaches can be evaluated.In the case of an ensemble of identical molecules withdifferent orientations coupled to a single mode of an opti-cal cavity, the energy of the polariton can be obtained, asin the case of one molecule, by using an effective Rabi fre-quency Ω effR = (cid:114)(cid:80) Nm =1 (cid:12)(cid:12)(cid:12) Ω ( m )R (cid:12)(cid:12)(cid:12) , where Ω ( m )R is the Rabicoupling associated with the m th molecule. If the orien-tation of the molecules are random, using (cid:104) cos θ (cid:105) θ = 1 / effR = Ω R (cid:112) N/ R being the Rabi fre-quency of one molecule.Similarly, if an ensemble of identical molecular com-plexes with different orientations is coupled to a sin-gle mode of an optical cavity, the energy of the po-lariton can be obtained, as in the case of one molec-ular complex, by using effective Rabi frequencies forenergy eigenstates (excitons) of the molecular complexΩ i, effR = (cid:114)(cid:80) Nm =1 (cid:12)(cid:12)(cid:12) Ω i, ( m )R (cid:12)(cid:12)(cid:12) . Here i represents the excitonenergy eigenstates of each molecular complex, and Ω i, ( m )R represents the Rabi coupling associated with the i th ex-citon in the m th molecular complex. If the orientationof the molecular complexes is random, the effective Rabifrequencies reduce to Ω i, effR = Ω i R (cid:112) N/ i R beingthe Rabi frequency of the i the exciton in one molecularcomplex. Cavity transmission spectrum.–
We have demonstratedthat the sharp and isolated peak of the lower polaritonappears in the molecular absorption spectrum, which canbe used for precise determination of the excitation ener-gies of molecules. However, if there are pairs of identi-cal molecules with opposite orientations such that theirRabi couplings have the same magnitude but the oppo-site signs, the polariton state would become a dark statewith respect to molecular absorption [65]. This situa-tion is encountered especially in a system of identicalmolecules or identical molecular complexes with randomorientations. In this case, given that the polariton alwaysinvolves the degrees of freedom of the cavity, its energycan be obtained from cavity transmission spectroscopymeasurements.For a numerical demonstration of the cavity transmis-sion spectrum, we consider a system of N = 4 identicalmolecules that form two pairs of molecules with oppo-site orientation. As a result, the Rabi couplings satisfyΩ (2)R = − Ω (1)R and Ω (4)R = − Ω (3)R . In this case, we take √ N Ω (1)R = 0 . (3)R = Ω (1)R √ /
2. The other pa-rameters of the system and the environment are the sameas those of the aforementioned system that was inves-tigated. There is a relatively sharp peak of the lowerpolariton with a linewidth of ∆ ν LP (cid:39)
67 cm − , whichhas a comparable magnitude to that of the molecular-absorption spectrum of a system of N = 4 moleculeswith the same orientation [65]. There is also a verysmall and flat transmission spectrum at approximately12400 cm − due to the three energy eigenstates of thesystem that consists mainly of the degrees of freedomof matter. In the absence of thermal fluctuation fromthe environment, these energy eigenstates do not appearin the cavity transmission spectrum. Their signals inthe cavity transmission spectrum should therefore be at-tributed to the thermal fluctuation of the environment,which affects the structures of the energy eigenstates byinducing small mixing of the degrees of freedom of lightand matter. Conclusions.–
We have demonstrated that a precisedetermination of the excitation energies of condensed-phase molecular systems is possible by strongly couplingthe molecules to an optical cavity and measuring the en-ergy of the polariton, which is a mixture of light andmatter degrees of freedom. The polariton’s linewidth is determined to exhibit a power scaling with respect to thenumber of molecules coupled to the cavity mode. Thepower exponent strongly depends on the environment’sdynamic time. Therefore, the environmental dynamicsinformation can be extracted from the polariton spec-trum measured for a variable number of molecules. Theexciton-polariton-based approach proposed here is thefirst step in the development of new methods for precisemeasurement and/or control of various physical proper-ties of condensed-phase molecular systems [69, 70], whichis significant from the perspective of both fundamentalscience and technological application.
Acknowledgments
This work was supported by JSPS KAKENHIGrant Numbers 19K14638 (N. T. Phuc), 17H02946and 18H01937, and JSPS KAKENHI Grant Number17H06437 in Innovative Areas “Innovations for Light-Energy Conversion (I LEC)” (A. Ishizaki). [1] J. Eccles and B. Honig, Charged amino acids as spectro-scopic determinants for chlorophyll in vivo. Proc. Natl.Acad. Sci. USA. , 4959–4962 (1983).[2] A. Damjanovic, H. M. Vaswani, P. Fromme, and G. R.Fleming, Chlorophyll excitations in Photosystem I ofSynechococcus elongatus. J. Phys. Chem. B , 10251–10262 (2002).[3] D. Spangler, G. M. Maggiora, L. L. Shipman, and R.E. Christoffersen, Stereoelectronic properties of photo-synthetic and related systems. 2. Ab initio quantum me-chanical ground state characterization of magnesium por-phine, magnesium chlorin, and ethyl chlorophyllide-a. J.Am. Chem. Soc. , 7478–7489 (1977).[4] J. N. Sturgis and B. 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