LETTER pubs.acs.org/JPCL
Modeling of Resonant Hole-Burning Spectra in Excitonically Coupled Systems: The Effects of Energy-Transfer Broadening Mike Reppert* Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
bS Supporting Information ABSTRACT: Resonant hole-burned (HB) spectra provide detailed insight into optical line shapes and single-site absorption spectra. In this work, a new method is presented for modeling low-fluence resonant HB spectra in excitonically coupled systems. Our calculations make use of the line shape theory of Renger and Marcus [J. Chem. Phys. 2002, 116, 9997 10019] to include explicitly the effects of excitonic interactions and energy transfer. We find that an accurate treatment of lifetime broadening is crucial for reproducing resonant hole shapes and zerophonon action spectra. Representative spectra are calculated for a series of simple dimers, with a description and physical explanation of the main features of the calculated spectra. Qualitatively, the results are in excellent agreement with experimental literature data. It is anticipated that the new method will provide both more detailed insight into the excitonic information encoded in resonant HB spectra and a means of testing theoretical treatments of excitonic optical line shapes. SECTION: Kinetics, Spectroscopy
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ow-temperature, frequency domain, laser-based spectroscopic methods such as hole-burning1,2 (HB), fluorescence line narrowing3 5 (FLN), and difference fluorescence line narrowing6 10 (ΔFLN) have over the years provided considerable insight into the electronic structure and energy-transfer dynamics of excitonically coupled systems such as photosynthetic complexes.11,12 Due to fast excitation energy transfer (EET) within these systems, these techniques provide selective probes for low-energy states (which are of most interest from an energy-transfer perspective) in addition to insight into the excitonic interactions between high- and low-energy pigments and can provide detailed information on basic physical parameters such as vibrational spectral densities and Huang Rhys factors (S). Nonetheless, modeling studies of HB and FLN spectra in photosynthetic complexes have traditionally followed the approaches developed for dilute glassy systems,1,12 15 that is, ignoring the explicit effects of energy transfer and excitonic delocalization. Only recently16 19 have computational studies based on the experimental nonresonant HB spectra of the CP43 and CP47 proximal antenna complexes of Photosystem II16,17 as well as on model systems18,19 revealed that explicit inclusion of these effects can have unique, and sometimes surprising, influences on the appearance of HB spectra. Therefore, a clear understanding of these effects is crucial for an accurate interpretation of experimental HB data generated for photosynthetic antenna pigment complexes and reaction centers (RCs). Previous modeling studies16 19 have been strictly limited to nonresonant (non-line-narrowed) HB spectra. Although useful for characterizing basic excitonic interactions between highand low-energy pigments, the low-resolution (bulk) nature of r 2011 American Chemical Society
nonresonant measurement significantly limits the information content of the spectra compared to high-resolution resonant HB data because detailed features of the absorption line shape are, to a large extent, averaged out in the bulk HB spectrum. As a result, previous studies have made use of a quite simplistic description of transition line shapes in which a calculated pure electronic absorption spectrum is simply convolved with an assumed (or experimentally determined)17 single-site absorption spectrum. In the present work, we extend these modeling studies in two important ways. First, we replace the simple “convolution” treatement of transition line shapes with the density matrix description of Renger and Marcus20 in order to more accurately reflect the influences of electron phonon coupling and energytransfer lifetime broadening on the calculated spectra. Second, using this improved line shape description, we extend our analysis to resonant (line-narrowed) and zero-phonon action (ZPA) HB spectra. The new method provides not only a means for more precise analysis of HB data but also, thanks to the siteselective nature of resonant HB spectra, a precise means of testing the reliability of various excitonic theories in reproducing optical line shapes. Figure 1 shows a schematic representation of the method used in this work to calculate resonant and nonresonant HB spectra. Inhomogeneous disorder is taken into account by ensemble averaging over pigment site energies chosen randomly from a Gaussian distribution characteristic of each pigment (the simulations Received: September 10, 2011 Accepted: October 10, 2011 Published: October 10, 2011 2716
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Figure 1. Schematic illustration of the simulation method used in this work for resonant HB spectra. See text for details.
presented in this work are averaged over 107 realizations of disorder for each dimer, that is, 2 107 generated site energies). Each randomly generated set of (“pre-burn”) site energies can be considered as a realization of disorder for one particular dimer of pigments in an ensemble. For each dimer, the Hamiltonian generated by the pre-burn site energies and coupling constant (V) is diagonalized, and an excitonic line shape is calculated using either the density matrix approach of Renger and Marcus20 or an assumed (static) single-site spectrum (convolution method). The excitonic lineshapes from all states are then summed to produce the pre-burn absorption spectrum. The calculated line shapes depend in both cases on the temperature (T, set to 5 K in these calculations), electron phonon coupling strength S (which, although not considered here, has for some systems21 been suggested to itself vary with site energy), and the phonon spectral density (or equivalently, the one-phonon profile, in HB terminology11). In the more flexible density matrix theory, the line shape also depends on the pigment site energies and an assumed correlation radius for protein phonon modes (see Supporting Information for further comparison of the present method with the formulas used in previous work). One key advantage to this theory is that, while the energy-transfer rate is obtained in the standard Redfield approach (and for convenience, we refer to it in the text as a Redfield theory), the absorption line shape expression includes in addition to energy-transfer broadening a well-resolved phonon sideband (PSB), as observed in experiment.20 A detailed description of this theory has already been presented in ref 20, and some practical comments on its numerical application are provided in the Supporting Information for this document. Importantly, this theory implicitly assumes weak electron phonon coupling (S < 1, as is frequently observed in photosynthetic systems), and for stronger coupling, different line shape theories must be employed (see refs 22 24 for recent discussions of the applicability and spectral features of various approaches). Fortunately, because the method presented here for calculating HB spectra does not depend specially on the line shape functions employed, it can easily be adapted for other theories and conditions by substituting the appropriate expression for the excitonic absorption line shape, leaving the treatment of the HB process unchanged. Qualitatively, such changes are not expected to produce new effects in the HB spectra; only quantitative features (line widths, PSB intensity, etc.) should be altered. Indeed, given the site-selective nature of the HB measurement, the
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comparison of simulated and experimental resonant HB spectra could provide a unique check of the reliability of various theories in reproducing optical line shapes in excitonic systems. As depicted in Figure 1, after the pre-burn spectrum has been generated, a decision is made whether to “burn” the selected complex; in the experiment, this corresponds to a random event depending on both whether or not the given complex is excited by the laser and whether a particular excitation event leads to HB. For our purposes (because we are interested only in the lowfluence limit and not absolute kinetics), only the first dependency is relevant; the second leads to a slow-down in the overall HB rate but does not affect the calculated line shapes. In addition, we must now distinguish between the resonant and nonresonant HB processes. For nonresonant HB calculations, we can assume that HB occurs in every realization of disorder because any realization for which HB does not occur will not contribute to the HB line shape. For resonant HB, however, we are not allowed this luxury; because the probability of excitation is now a function of the absorption line shape, we must explicitly consider the probability that a given complex undergoes an absorption event. For example, a complex that has a zero-phonon line (ZPL) absorption peak resonant with the laser excitation frequency will be more likely to be excited, and hence undergo HB, than a complex that is nonresonant or resonant only through the PSB. More precisely, the probability of a HB event is proportional to the value of the (properly normalized) excitonic line shape function evaluated at the excitation frequency. For finite excitation line widths, the excitonic line shape is replaced by the convolution of the excitonic line shape with the laser line shape function; in our simulations, the calculation resolution of 1 cm 1 is larger than typical excitation line widths in HB experiments; therefore, the laser line width can be safely neglected. Once a burn probability is obtained, the decision whether to burn the complex is made by generating a random number r1 between 0 and 1; if r1 is lower than the burn probability, the complex is selected for burning. Otherwise, the absorption line shape is added unmodified to the bulk post-burn absorption spectrum. If the complex is selected for burning, the next step is to determine which pigment should be burned. This is accomplished by assuming that the excitation relaxes quickly (before burning) to the lowest excitonic state; the squared eigenvector coefficients c21,i for each pigment to the first (lowest) excitonic state then represent the probability that the excitation is localized on the corresponding pigment and hence that it will be burned. Again, the actual pigment selection is accomplished by generating a random number r2 between 0 and 1. For n pigments, a sequence is generated of n monotonically increasing numbers 0 < P1, P2, ..., Pn = 1, in which Pi = c21,1 + c21,2 + ... + c21,i represents the probability that the excitation is localized on one of the first i pigments. The pigment to be burned is the unique pigment m for which Pm 1 < r2 e Pm. Because the probability that r2 is in the ith interval is equal to the squared eigenvector coefficient c21,i, this provides an efficient method for randomly choosing a burn pigment with probabilities determined by the contribution of each pigment to the lowest exciton state. Finally, the site energy of the selected burn pigment is replaced with a new site energy generated randomly from the pigment’s site energy distribution function (SDF). In this way, the burn process “tweaks” the site energy of a single pigment to give it a new post-burn site energy that is uncorrelated with its pre-burn site energy. Again, the excitonic line shape for the complex is calculated and added to the bulk post-burn absorption spectrum. One could also consider 2717
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Figure 2. Simulated nonresonant (nonline-narrowed) HB spectra using convolution and Redfield methods. In all frames, the strictly positive spectra are absorption and the derivative-shaped curves HB; black curves represent overall absorption or HB, while red and blue represent the lower- and higher-energy excitonic components, respectively. (A) Uncoupled dimer (V = 0 cm 1) calculated in the convolution method; the site energies (14778 and 14922 cm 1) and SDF widths (both 162 cm 1 full width) were chosen to reproduce the absorption spectrum of the coupled dimer in the next frame. (B) Coupled dimer (V = 50 cm 1) calculated in the convolution method; the two pigments have degenerate site energy distributions with fwhm = 212 cm 1 and center = 14850 cm 1. (C) Calculated spectra with the same parameters as those in (B) but using Redfield line shapes.
altered distributions of photoproduct (shifted distributions were already employed for nonresonant spectra of the CP43 complex16), potentially including correlation with the pre-burn site energy. The only change to the method in this case is to employ a different (possibly correlated) distribution function when selecting post-burn site energies. Although such effects could be very important in determining detailed line shape features in resonant spectra, our calculations so far suggest that qualitative spectral features will be similar to those presented here. As a first comparison between the various theories employed here, Figure 2 shows three sets of calculated absorption and nonresonant HB spectra. Frames B and C are bulk absorption/ HB spectra for a degenerate coupled dimer using the convolution (B) and Redfield (C) approaches; in both cases, the (Gaussian) pigment SDF has a fwhm of 212 cm 1 and is centered at 14850 cm 1. Frame A is calculated for a nondegenerate uncoupled dimer (or rather, a dimer coupled by F€orster EET but free of excitonic character as considered in ref 18) with site energies (14778 and 14922 cm 1) and SDF widths (both 162 cm 1 full width) chosen to mimic the bulk absorption spectrum of (B). The three frames are chosen to illustrate the contributions of various effects to the HB spectrum. Comparison of (A) and (B) (which have identical single-complex line shapes) reveals the effect of excitonic coupling on the HB spectrum; note the clear excitonic band appearing between the upper and lower states in (B) (more commentary on these features can be found in refs 11 and 19). On the other hand, (B) and (C) (identical electronic coupling but different line shapes) reveal the effects of lifetime broadening on the spectra; while the basic excitonic features are essentially similar between the two curves, the inclusion of lifetime broadening in the Redfield simulation alters the absorption and HB spectra in similar ways, smearing out the high-energy features. Note the shift between the spectra in
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Figure 3. Simulated ZPA spectra for the same parameter sets as those in Figure 2. The absorption curves from that figure are shown for comparison (black = total absorption; red = lower excitonic state; blue = upper excitonic state). The ZPA spectrum consists of a series of resonant holes (inverted in the figure) at burn frequencies spaced by 50 cm 1 across the absorption band. (A) Uncoupled dimer (V = 0 cm 1) calculated in the convolution method. (B) Coupled dimer (V = 50 cm 1) calculated in the convolution method. (C) Calculated spectra with the same parameters as those in (B) but using Redfield line shapes.
(A) and (B) compared with (C); this is because for identical site energy inputs (vertical transition energies in the Redfield simulation and ZPL energies in the convolution method), the Redfield simulation curves are shifted by approximately (though not exactly due to delocalization) the pigment reorganization energy (∼63 cm 1 in the present case) relative to the convolution method simulations. For comparison, Figure 3 shows a series of calculated resonant HB spectra for excitation wavelengths between 14500 and 15200 cm 1 (the HB spectra are inverted as in a ZPA spectrum for comparison with absorption). The most striking feature here is that for the convolution method, sharp resonant holes are produced for all wavelengths across the inhomogeneous absorption band, in sharp contrast to the behavior observed in both the Redfield and uncoupled convolution calculation and to the behavior seen in experiment, where the ZPA spectrum is often used as a measure of the frequency distribution function of the lowest-energy state (see refs 25 29 for just a few examples from photosynthesis). This brings us to an important conclusion; while the “coupled” convolution calculation correctly reproduces experimental features in nonresonant HB spectra16,17,19 and the “uncoupled” convolution calculation accurately reproduces experimental ZPA features,27 only the Redfield approach is able to reproduces both types of spectra simultaneously. At first glance, the anomalous ZPA spectrum obtained from the coupled convolution calculation might seem surprising. In our calculations, HB is always assumed to occur in the lowest excitonic state; it would seem then that excitation at high frequencies should produce little resonant response because the excitation is transferred to a lower state before burning occurs. Nonetheless, the distinctive behavior under the coupled convolution method is not difficult to understand; because in a coupled complex a burned pigment may contribute to multiple excitonic states, the transition frequencies of all excitonic states are altered in a post-burn absorption spectrum (depending on the contribution of the burned pigment to the excitonic state). 2718
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Figure 4. Calculated resonant HB spectra for the parameters of Figures 1 and 2 but different burn frequencies (λb). In all frames, the net HB spectrum is shown in black, with HB in the lower (upper) excitonic state in red (blue). The total absorption spectrum is shown in gray for reference. (A) and (C) λb = 14700 cm 1; spectra calculated using the convolution method and V = 0 (A) and 50 cm 1 (C). (B) and (D) Convolution method with λb = 14900 cm 1 and V = 0 (B) and 50 cm 1 (D). (E) Redfield method with λb = 14650 cm 1 and V = 50 cm 1. (F) Redfield method with λb = 14850 cm 1 and V = 50 cm 1.
As a result, in the case of a coupled dimer, even high-frequency excitation results in a significant resonant response because the transition frequencies of both excitonic states are altered in the post-burn spectrum. In the Redfield simulations, however, this resonant response is largely masked by lifetime broadening in the upper excitonic state, rendering it too broad to be distinguished in the ZPA spectrum. Conceptually, this is an important distinction; the lack of sharp resonant response at high energies in the coupled case is not because the higher excitonic states are unperturbed by the burn event; rather, it is because the resonant response is sufficiently broadened by fast energytransfer rates so as to spread the resulting hole across a large absorption range. This point is clarified somewhat in Figure 4, which shows detailed views of the HB spectra from Figure 3 for two sets of excitation frequencies (the resonant zero-phonon holes are cut off to reveal the PSB hole structure). The left (right) panel shows holes burned at excitation frequencies of 14700 cm 1 (14900 cm 1) for (A) (D) (convolution method) and 14650 cm 1 (14850 cm 1) for frames (E) and (F). The shift of 50 cm 1 is in order to match (approximately) the relative position of the resonant burn on the bulk absorption spectrum due to the shift between the convolution method and Redfield simulations. Note that in the middle frames (coupled dimer, convolution method), sharp holes are burned not only at the excitation wavelength but at frequencies (100 cm 1 (twice the coupling strength) from the excitation frequency. Recall that the transition frequencies for a simple coupled dimer are given by λL/U = εo - (1/2)(δ2 + 4V2)1/2, with V as the coupling strength, εo the average of the monomer site energies, and δ their difference. The minimum separation between the transition energies for the two excitonic states (regardless of δ) is then 2V. It is this cutoff in the minimum transition-energy difference (leading to an absence of nonresonant HB response within (100 cm 1 of the burn frequency) that gives rise to sharp (nonresonant) features in the HB spectrum. As above, in the Redfield calculation, these sharp features are removed by EET broadening of the transitions. It is significant
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Figure 5. Simulated HB spectra in the Redfield approach for excitonically coupled dimers with coupling constants of 50 (left panel) and 100 cm 1 (right panel) and various angles between their transition dipole moments (see labeling in figure). As in Figure 4, in all frames, the net HB spectrum is plotted in black, with component lower and upper excitonic states plotted in red and blue, respectively. The corresponding absorption spectra are shown as dashed gray lines for reference.
as well that the Redfield calculation also produces a more realistic resonant hole shape than either of the convolution calculations; note in (A) and (C) the strong real-PSB hole (to the blue of the zero-phonon hole), which is present in the convolution simulations but largely masked by photoproduct in the Redfield simulation (in better agreement with typical experimental results for photosynthetic systems1,29 31). Likewise, in (B), the uncoupled convolution method gives rise to a strongly red shifted photoproduct peak, while the coupled calculations (both D and F) seem to give more realistic photoproduct distributions. In this regard, note that although the pre- and post-burn site energies of the burned pigment are uncorrelated in all cases, the pre- and postburn transition energies are correlated in the excitonic calculations because the fixed energy of the unburned pigment (influencing εo) “tethers” the pre- and post-burn values of λL/U to one another, helping to prevent strong shifts in the photoproduct absorption. The calculations shown thus far have been for perpendicular dimers, in which the oscillator strengths of the upper and lower states are identical, regardless of the site energies. To give an overview of the effect of nonperpendicular transition dipoles, we plot in Figure 5 resonant HB spectra calculated in the Redfield approach for two sets of coupling constants (50 and 100 cm 1) and for various angles between the pigment transition dipole moments (0, 45, 90, 135, and 180°). Perhaps the most important point to notice here is that for parallel transition dipole moments where the oscillator strength of the lower excitonic state is strongly suppressed, the great bulk of the HB response occurs at higher energy, nonresonant from the excitation frequency. Although we do not here present a detailed analysis of any experimental data, one may note that this effect is in excellent agreement with the HB data recently obtained for the watersoluble chlorophyll protein (WSCP) complex.29,32 Similar features (i.e., a sharp resonant hole coupled to a broad nonresonant hole at higher energy) have likewise been observed in more complex photosynthetic systems, for example, the LHCII28 and FMO complexes.33 The model presented here provides, to our knowledge, the first quantitative framework for describing such features in the analysis of experimental data. 2719
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The Journal of Physical Chemistry Letters Two other features in Figure 5 are immediately worthy of comment. First, note the virtual absence of PSB holes in several frames for this excitation frequency (50 cm 1 higher than that in Figure 4); in some cases, the nonresonant response is in fact entirely positive (balanced by the negative resonant hole). This finding highlights the need for careful consideration of photoproduct (and photoproduct distributions) in modeling resonant HB shapes, as has recently been discussed in the context of hole growth kinetics (HGK).31 Second, note the mixing of the sharp resonant hole with response from EET-broadened upper exciton bands in the 50 cm 1 perpendicular dimer case, also in agreement with recent experimental results for the WSCP dimer.29 It should be emphasized that in our simulations, these broad resonant holes are not the result of HB in an upper excitonic state (i.e., in which the HB process competes with energy transfer, as in ref 31); rather, the broad feature results from coupling between pigments contributing to multiple excitonic states with HB occurring always in the lowest excitonic state. Finally, a few words on future extensions and applications of the model are in order. Although in this Letter we focus on lowfluence, nonphotochemical HB spectra, the model can easily be adapted for photochemical HB and (at increased computational expense) higher fluence spectra. Photochemical HB is especially straightforward because the only alteration necessary here is to shift the distribution of the photoproduct (and possibly coupling constants) from the original SDF to the frequency range of the chemically converted species; the treatment of excitonic effects is unaltered. A similar approach applies to triplet bottleneck HB (as already employed for the nonresonant triplet bottleneck HB spectrum of the CP43 complex16) in which the transition frequency of the burned pigment is shifted far from the absorption range of the original system, effectively decoupling it from the other pigments. Indeed, as discussed in the introduction of the method, even for the nonphotochemical HB discussed here, it may be important for some systems to consider photoproduct distributions that are different from the pre-burn SDF; for example, one could consider systematic shifts (as employed in the modeling of nonresonant HB spectra of the CP43 complex16) or narrow distributions around the pre-burn transition frequency as employed recently for HGK calculations for the B800 band of LH2.31 Higher fluence effects (including distributions of HB rates) can be included by allowing multiple excitation events in the simulation (with computation time scaling with the number of burn events considered). In addition, the method can, without modification, be applied to the modeling of resonant FLN and ΔFLN spectra simply by calculating the line-narrowed fluorescence before and after burning. Such a calculation would be of great interest because to this point, the effects of energy transfer and excitonic coupling on ΔFLN spectra have not been explored in detail. On the other hand, such calculations will require higher resolution (better than 1 cm 1) to ensure accurate determination of ZPL and sideband intensities (both PSB and energy-transfer contributions) because, due to the double-excitation effect in the ΔFLN technique, the resolution requirements for these calculations are more stringent than those for resonant HB spectra.34 Of more interest to those outside the HB community, one may easily imagine the application of this method as a precise means of comparing various excitonic line shape theories with experimental data.20,22 24 As noted above, although the theory of Renger and Marcus20 is employed here, incorporation of alternate line shape theories is quite straightforward. While bulk
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measurements (such as absorption, emission, and circular dichroism) already provide a coarse-grained test of the reliability of these theories, high-level modeling of resonant HB spectra would provide a much more detailed test of the ability of various theories to reproduce optical features such as line widths, effective electron phonon coupling strength, and PSB shapes as well as a cross-check to results obtained from single-molecule studies.11,31 In summary, we have demonstrated that, while giving qualitatively accurate results for nonresonant HB spectra, the simple convolution methods previously employed16 19 for modeling nonresonant HB spectra in excitonically coupled systems give poor agreement with experimental results in simulations of resonant HB spectra. As an alternative, we present here a new method for calculating resonant HB spectra that explicitly includes the effects of excitonic line broadening and delocalization via the line shape theory of Renger and Marcus.20 The similarities and key differences between these approaches are explored through calculated resonant HB spectra for a variety of excitonically coupled dimers. It is demonstrated that the new method gives results that are in excellent qualitative agreement with literature experimental data. It is anticipated that the method will find immediate application in more accurate modeling of resonant HB spectra in excitonically coupled systems, leading to both a significant improvement in the accuracy of extracted physical parameters and a new means of detailed comparison between excitonic line shape theories and experimental data.
’ ASSOCIATED CONTENT
bS
Supporting Information. A more complete description of the line shape functions used in these simulations and additional comments on their numerical evaluation. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The author thanks Ryszard Jankowiak (Kansas State University, Department of Chemistry) and Virginia Naibo (Kansas State University, Department of Mathematics) for many insightful discussions. He also acknowledges support from a Fulbright U.S. Student Fellowship (2009 2010) for research in Warsaw, Poland, during which time some of these calculations were performed. ’ REFERENCES (1) Kharlamov, B. M.; Personov, R. I.; Bykovskaya, L. A. Stable ‘Gap’ in Absorption Spectra of Solid Solutions of Organic Molecules by Laser Irradiation. Opt. Commun. 1974, 12, 191. (2) Gorokhovskii, A. A.; Kaarli, R. K.; Rebane, L. A. Hole Burning in the Contour of a Pure Electronic Line in a Shpol’skii System. JETP Lett. 1974, 20, 216. (3) Denisov, Y. V.; Kizel, V. A. Energy Migration in EuropiumActived Borate Glasses and Relative Location of Energy Levels. Opt. Spectrosc. 1967, 23, 251. (4) Szabo, A. Laser-Induced Fluorescence-Line Narrowing in Ruby. Phys. Rev. Lett. 1970, 25, 924. 2720
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