Origin of the asymmetric light emission from molecular exciton-polaritons: supplementary material

This document provides supplementary information to "Origin of the asymmetric light emission from molecular exciton-polaritons , " https://doi.org/10.1364/OPTICA.5.001247 . It includes more details on (i) Description of the strong coupling of the cavity mode with excitons of multiple molecules and the elimination of the de-phasing reservoir in the collective case, (ii) details about the incoherent decay of the polariton states in strong coupling, (iii) description of the truncated Hilbert space used for the numerical calculations, (iv) technical aspects of the calculation of the emission and absorption spectra, (v) comparison of the emission spectra of polaritons interacting with dephasing bath characterized by different value of the effective reservoir frequency Ω R and (vi) comparison of absorption and emission spectra of polaritons considering a variable value of g .


STRONG COUPLING OF THE CAVITY MODE WITH EXCITONS OF MULTIPLE MOLECULES
In the main text we present the system Hamiltonian, H tot [Eq. (7) of the main text], containing the molecules interacting with their local dephasing reservoirs and the cavity mode, together with the laser pumping H pump . Here we show how the Hamiltonian can be transformed into the polariton picture. We first introduce the picture of collective excitations of the molecules. To that end we introduce a new set of operators S i = ∑ α c iα σ α , where c iα are coefficients that are elements of a unitary matrix such that (c i1 , c i2 , . . . , c iN ) form a set of N orthonormal vectors. It is convenient to make the choice (c 11 , c 12 , . . . , c 1N ) = 1 ∑ α |g α | 2 (g 1 , g 2 , . . . , g N ) (S1) and the remaining vectors orthonormal to the first vector. With this choice S 1 becomes fully coupled to the plasmonic cavity via a new effective coupling constant g eff = ∑ α |g α | 2 . In the following we consider that all coefficients g α = g are equal (c 1α = 1/ √ N) and recover the result g eff = √ Ng. We further consider the low-excitation limit where the new operators S i become approximately bosonic and independent with the transformation rules where in the second line we used the orthogonality of the coefficient vectors that we assumed to be real. The transformation rules allow rewriting the Hamiltonian as: and the Lindblad terms As in the case of the single molecule, we can proceed to diagonalize the Hamiltonian part involving the bright excitonic mode strongly coupled with the cavity [neglecting for now the inter-molecular coupling in the last line of Eq. (S5)]. We thus generate a new set of annihilation operators of the lower, S − , and the upper, S + , polaritons with θ defined in analogy with the single-exciton case presented in the main text. In the polaritonic picture, the system Hamiltonian becomes Here Eq. (S9a) represents the system of the newly arising polariton states. The coupling of the various polaritonic modes with the dephasing reservoir is given by Eq. (S9b) to Eq. (S9e).
The interaction with the reservoir will lead to population transfer among |+ , |− , and the dark polaritons |D i . Throughout this supplementary material we are going to use S i and S † i to denote the operators S i = |0 D i | and S † i = |D i 0| (for i > 1) for brevity. Notice that in the main text S i (S † i ) is denoted as S D i (S † D i ). The coherent laser pumping is included in Eq. (S9f). The term in Eq. (S9g) of the transformed Hamiltonian introduces mixing of the dark states with the bright modes that leads to formation of the dark-polariton peak in the emission spectra. Equation (S9i) represents additional interactions among the dark modes that are weak for the selected parameters.
The incoherent damping of the dephasing reservoir is in- . The transformation into the basis of the polariton states further changes the form of the incoherent damping of the cavity. In the secular approximation, the cavity damping Lindblad term, L a (ρ), transforms as with respective decay rates The contribution of the intrinsic molecular decay γ σ to the decay of |+ and |− can be neglected compared to the large cavity losses γ a .

A. Elimination of the dephasing reservoir in the collective case
Here we derive the effective Lindblad terms that govern the incoherent processes induced by the interaction of the excitoncavity-mode system with the dephasing reservoir, as discussed in the main text. For simplicity, we further assume that the intermolecular coupling is negligible and only weakly perturbs the dynamics given by Eq. (S9a) to Eq. (S9e). We eliminate the reservoir whose dynamics is given by the Hamiltonian term Eq. (S9b) represents the incoherent interaction between the upper, |+ , and the lower, |− , polariton in close analogy with the single-excitonic case presented in the main text and leads to the Lindblad terms L S † ± S ∓ (ρ) (i.e. the terms L S † + S − (ρ) and L S † − S + (ρ) using the compact notation). We further define as the reservoir modes are equivalent) and note that as the respective bath modes are locally interacting with each molecule and are assumed to be uncorrelated. The respective rates then become which can be found from where we have used the definition of the coefficients c α1 = 1/ √ N and to obtain the final result. From Eq. (S9c) we obtain for the upper polariton the terms where we neglected the Lindblad superators containing the cross-terms. In analogy with γ S † ∓ S ± , the respective rates are In close analogy, from Eq. (S9d) we get for the lower polariton where we neglected the Lindblad superators containing the cross-terms. In analogy with γ S † ∓ S ± , the respective rates are Last, we obtain the pure dephasing and energy transfer among the dark polariton states L S (ρ) (S = cos 2 θS † . These terms do not contribute to the population decay and we will not consider them in the following.

B. Decay of the polariton states
The effective polariton dynamics derived above leads to the following rate equations (without the driving terms) of the polariton populations n where we used the equivalence of the dark-polariton states (neglecting the influence of the inter-molecular coupling) and defined γ S † i S ξ ≡ γ S † D S ξ and γ S † ξ S i ≡ γ S † ξ S D for i > 1 and the index ξ ∈ {+, −}.
The rate equations can be solved for a given initial condition. We present the results in the main text and compare them with the results of the full model.
The total state of the system is defined as a Kronecker product of the cavity mode-exiton and reservoir states |ψ tot = |ψ P−E ⊗ |ψ res .
This basis defines the dimension of the Hilbert space. With the number of molecules N the dimension of the Hilbert space H grows as Dim {H} = Dim {|ψ tot } = (N + 1)(N + 2) 2 /2. Moreover, the superoperator space necessary for the solution of the quantum master equation has the dimension Dim{S } = Dim {H} 4 , which makes the numerical treatment of systems containing larger number of molecules difficult. In the main text we thus present results for up to N = 5 molecules.

CALCULATION OF THE INCOHERENT EMISSION SPECTRA AND THE ABSORPTION SPECTRA
The full model containing the incoherent dynamics presented in the main text can be solved numerically for smaller numbers of molecules. Here we discuss the technical details of the practical implementation of the calculation of the emission spectra. The emission spectra of molecules can be calculated from the two-time correlation function as or equivalently where we assume that the system is in the steady state induced by the pumping laser for t = 0 and a † (τ)a(0) ss = a † (τ)a(0) ss − lim τ→∞ a † (τ)a(0) ss . On the other hand, the absorption spectra are obtained from and we calculate the two-time correlation function with respect to the ground state of the system. We can apply the quantum regression theorem to obtain the effective dynamics of the twotime correlator. We assume the equation of motion for the density matrix (the quantum master equation) in the forṁ with ρ represented by a column vector and L a superoperator matrix constructed from the Hamiltonian and Lindblad terms. The quantum regression theorem (QRT) then states In other words, the time evolution of the two-time correlator O 1 (t + τ)O 2 (t) obeys the same dynamics as the mean-value of the Schrödinger-picture operator O 1 , but with initial condition given by the operator P = O 2 (0)ρ(t) that replaces here the density matrix. More explicitly, the steady-state two-time correlation function reads in this notation (setting t = 0) In the practical implementation we first calculate the initial value of P as P(0) = O 2 (0)ρ(0), where ρ(0) is the steady-state density matrix fulfilling together with The time evolution of P is obtained by integration of the equatioṅ using standard numerical methods. The spectrum is finally obtained by explicit calculation of the Fourier transform of the two-time correlation function as defined in Eq. (S27). Last, we remark that the same method is applicable for calculation of the weak-probe absorption spectra given by simply exchanging the operators O 1 and O 2 and replacing ω → −ω.

LARITON SPLITTING
In the main text we describe the effective dephasing reservoir as a broad damped harmonic oscillator of energyhΩ R = 400 meV, widthhγ B = 400 meV and coupling to the molecular electronic levels via d R ≈ 0.173, yielding the reservoir spectral density Here we briefly discuss the influence of the frequency Ω R on the observed emission spectra. To that end we calculate the polariton emission and absorption spectra [ Fig. S1 (a) and (b), respectively] for N = 4 molecules illuminated at the frequency of the upper polariton (hω L = 2.2 eV) for a constant broadeninḡ hγ B = 400 meV and varyinghΩ R = 100 meV, 200 meV, 300 meV and 400 meV. We adjust d R such that J(0) remains unchanged for all cases. For clarity, all of the spectra in Fig. S1 are normalized to the maximal value and vertically displaced. Figure S1 (a) shows that as Ω R is decreasing (from top to bottom), the emission spectra slightly change symmetry, making the emission from the upper polariton slightly more pronounced but preserving the qualitative picture. On the other hand, the absorption spectra remain practically identical for all Ω R , as shown in Fig. S1 (b).
Next we study the dependence of the polariton emission and absorption on the polariton splitting ω + − ω − = 2 √ Ng. The spectra are calculated considering the parameters of the reservoir hΩ R = 400 meV,hγ R = 400 meV and d R ≈ 0.173 and values of hg ranging fromhg = 100 meV tohg = 300 meV. In all cases we consider N = 4 molecules and tune the pumping laser frequency to the frequency of the upper polariton (ω L = ω 0 + √ Ng). The We adjust d R such that J(0) remains unchanged for all cases. The spectra are calculated forhω c = 2 eV,hg i = hg = 100 meV,hE = 0.1 meV,hγ a = 150 meV andhγ σ = 2 × 10 −2 meV. emission and absorption spectra are plotted in Fig. S2 (a) and (b), respectively. We normalize the spectra to the maximum of the lower-polariton peak and apply a constant vertical offset. Increasing g leads to larger separation of the polariton spectral peaks in both the emission and the absorption spectra. In the emission spectra, the lower-polariton peak is more pronounced than the upper-polariton peak due to the asymmetric population transfer. Interestingly, the relative intensity of the upper-polariton peak with respect to the intensity of the lower-polariton peak is decreased when 2 √ Ng ≈ Ω R , i.e. when the incoherent population transfer |+ → |− becomes resonant [J(ω + − ω − ) is maximized]. In the absorption spectra the upperand lower-polariton peaks are of similar intensity. absorption spectra for N = 4 molecules illuminated at the frequency of the upper polariton for reservoir parametershγ B = 400 meV, hΩ R = 400 meV and d R ≈ 0.173. The spectra are calculated for varyinghg i =hg = 100 meV, 140 meV, 180 meV, 220 meV, 260 meV and 300 meV. The other model parameters arehω c = 2 eV,hE = 0.1 meV,hγ a = 150 meV and hγ σ = 2 × 10 −2 meV.