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SCIENCE ADVANCES | RESEARCH ARTICLE mode and resonantly coupled to the cavity with strength g. The molecules are driven by a laser described by a Gaussian pulse enve- regimes, respectively. Experiments B1 and B2 operate in this regime (Fig. 3B). Decay and coupling rates The parameters needed in the theory calculations are the cavity leakage rate , the dephasing rate z, the nonradiative decay rate −, the interaction strength g, and the temporal width of the instrument response function R. Note that the temporal width of the pump pulse is fixed at = 20 fs. For the dephasing rate, we note that as one enters the strong coupling regime, exciton delocalization suppresses the effect of dephasing (47). To approximately capture this effect, we assume that the dephasing rate scales with the number of mole- cules as z = z0(N5%⁄N), where z0 is taken to be constant, and N5% is the number of molecules in the 5% cavity. The experimental uncer- tainty in N is estimated to be 10%. The cavity lifetime T comes into the model in both R = T and the cavity leakage rate = 1/T. From transfer matrix modeling on the 1 and 0.5% cavities (where polariton effects are small), we estimate that T ≈ 306 fs. However, on the basis of the measured finesse of the cavities, we estimate that T = 120 fs. Transfer matrix modeling assumes perfectly smooth mirrors, while measured finesse includes inhomogeneous broadening effects, neither of which we want to include in and R. In the following optimization, we therefore assume that T ∈ [120,306] fs, with lower values more likely due to transfer matrix calculations being prone to error. For T values within this range, the remaining three parameters in the model (z0, −, and g) were found through a global chi-squared optimization, simultaneously optimizing over all experiments. Un- _ 1_ _t−t0 2 0 − () lope (t ) = √_2 e 2 and a carrier frequency L. We work in the frame of the laser carrier frequency, and so write † N[ħz †− +] † H(t) = ħ a a + ∑j=1 ─2 j + g(a j + a j ) + iħ(t) (a − a) (3) where = − L is the detuning of the cavity frequency from the laser driving frequency. The LFO molecules are initially in the ground state, and the laser is on resonance ( = 0). Cumulant expansion The energy density of the cavity containing identical molecules with transition energy is E(t)= ħ_ [⟨z(t)⟩+1]. In general, the equation zz2 of motion (∂/∂t) ⟨ ⟩= Tr [ ̇ ] depends on both the first-order mo- ments ⟨x, y, z⟩ and ⟨a⟩ and higher-order moments, leading to a hierarchy of coupled equations. Within mean field theory, the second-order moments are factorized as ⟨AB⟩ = ⟨A⟩⟨B⟩, which closes the set of equations at first order. This approximation is valid at large N, as corrections scale as 1/N. To capture the leading order effects of finite sizes, we make a second-order cumulant expansion (34–36), i.e., we keep second-order cumulants ⟨⟨AB⟩⟩ = ⟨AB⟩ − ⟨A⟩ ⟨B⟩ and assume that the third-order cumulants vanish, which allows us to rewrite third-order moments into products of first- and second-order moments (46). In our experiments, the number of molecules in the cavity is large (>1010), and we find that higher-order correlations are negligible. We give the equations of motion up to second order in the Supplementary Materials. Operating regimes certainties in these fitting parameters were then estimated by using The decay-dominated (purple region in Fig. 3, A and B) regime oc- where for a three-parameter optimization and k total data points, curs when the collective light-matter coupling is weaker than the _ ∆ ≈ 3.51/(k − 3) (48). In the Supplementary Materials, we present a figure showing the minimum reduced chi-squared value as a func- tion of T, and for each point, we show the optimal set of parameters (z0, −, and g) along with the 68% confidence intervals. From this, and by comparison of the experimentally measured and theoretically calculated reflectivity for each parameter set, we concluded that the lifetime most representative of the data was T = 120 fs, with decay channels, g √Nr′ < {, z, −}, where r ′ = max (1, r). In this regime, the time scale of cavity dynamics is slow relative to the decay rate. Figure 3C shows a typical time dependence in this regime, indicating how the model parameters affect the dynamics. In this regime, the increase in the effective coupling relative to the decay strength sees an N2 superextensive scaling of the energy and power density, while rise time remains constant. Experiment A3 operates near the boundary of this regime (Fig. 3A). _ In the coupling-dominated (green region in Fig. 3, A and B) re- gime, the effective collective light-matter coupling g √Nr′ > {z, −, } dominates over the decay channels. In this regime, the time scale of cavity dynamics is fast relative to the decay rate, and we observe √_N-superextensive power scaling and 1/√_N dependence of rise time, while the maximum energy density remains constant. While power scaling is superextensive in both regimes, the origin of this differs: For the decay-dominated regime, this is the result of the superextensive energy scaling, while for the coupling-dominant regime, it is the result of a superextensive decrease in the rise time. Experiments A1 and A2 operate in this regime (Fig. 3A). − = (0.0141+0.0031) meV, g = (10.6+2.2) neV, and z = (1.68+0.25) meV. −0.0024 −1.3 0 −0.18 In the crossover between the regimes (purple-green), the col- SUPPLEMENTARY MATERIALS Supplementary material for this article is available at https://science.org/doi/10.1126/ sciadv.abk3160 REFERENCES AND NOTES 1. M. Gross, S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission. Phys. Rep. 93, 301–396 (1982). 2. N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, Observation of Dicke superradiance in optically pumped HF Gas. Phys. Rev. Lett. 30, 309–312 (1973). 3. H. M. Gibbs, Q. H. F. Vrehen, H. M. J. Hikspoors, Single-pulse superfluorescence in cesium. Phys. Rev. Lett. 39, 547–550 (1977). 4. J. Feldmann, G. Peter, E. O. Gobel, P. Dawson, K. Moore, C. Foxon, R. J. Elliott, Linewidth dependence of radiative exciton lifetimes in quantum-wells. Phys. Rev. Lett. 59, 2337–2340 (1987). 5. B. Deveaud, F. Clerot, N. Roy, K. Satzke, B. Sermage, D. S. Katzer, Enhanced radiative recombination of free excitons in GaAs quantum wells. Phys. Rev. Lett. 67, 2355–2358 (1991). 6. T. Itoh, M. Furumiya, Size-dependent homogeneous broadening of confined excitons in cucl microcrystals. JOL 48-49, 704–708 (1991). 7. S. Deboer, D. A. Wiersma, Dephasing-induced damping of superradiant emission in J-aggregates. Chem. Phys. Lett. 165, 45–53 (1990). lective coupling falls between the cavity decay rate and the TLS _ _−z− dephasing rate, {, }< g√Nr′ < . In Fig. 3 (A and B), is small such that g√Nr′ ≫ − for all values of N, and so, there is no boundary labeled for this decay rate. In this case, capacity and rise time can simultaneously scale super- and subextensively, but at a rate slower than in the decay and coupling-dominated Quach et al., Sci. Adv. 8, eabk3160 (2022) 14 January 2022 6 of 7 the reduced ~2 distribution to find the 68% confidence interval of the model parameters. This corresponds to the range ~2 ≤ ~2 + ∆, min See the Supplementary Materials for more details. Downloaded from https://www.science.org on June 26, 2022PDF Image | Superabsorption organic microcavity Toward a quantum battery
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