Experimental Upper Bounds for Resonance-Enhanced Entangled Two-Photon Absorption Cross Section of Indocyanine Green

Research on molecular entangled two-photon absorption (ETPA) has gained momentum over the past decade due to its potential application in nonlinear spectroscopy and bioimaging. While multiple theoretical predictions on ETPA cross sections have been made Landes et al. (2021a); Schlawin (2022); Kang et al. (2020); Burdick et al. (2018); Wittkop, Marmolejo-Tejada, and Mosquera (2023), the predicted values vary by orders of magnitude, and the formulations emphasize different parameters of the excitation flux. Although reported ETPA cross sections could be as high as $1\times 10^{-17}$ cm²/molecule Harpham et al. (2009); Guzman et al. (2010); Upton et al. (2013), recent measurements of ETPA cross sections reported values lower than $1\times 10^{-21}$ cm²/molecule for organic molecules with virtual intermediate states, even in dyes with near-unity quantum yields which facilitate fluorescence detection such as rhodamine 6G (R6G) Tabakaev et al. (2022); Parzuchowski et al. (2021); Landes et al. (2021b); Mikhaylov et al. (2022); Hickam et al. (2022), and quantum dot systems with large classical two-photon absorption (TPA) cross sections Gäbler et al. (2023). The measured fluorescence signals from ETPA are usually in the tens per second to tens per hour count rate range, which hinders many potential applications in imaging and sensing.

For classical TPA, it is known that a real intermediate state increases the cross section by 1-2 orders of magnitude Terenziani et al. (2008); Kobayashi, Mutoh, and Abe (2018); Drobizhev et al. (2002); Makarov et al. (2007); Hahn et al. (2021). Resonance-enhanced ETPA (r-ETPA) is predicted to be enhanced by similar orders of magnitude in atomic systems Oka (2018); Kang et al. (2020); Schlawin and Buchleitner (2017); Li and Hofmann (2023). While r-ETPA in atomic cesium Georgiades et al. (1995) and atomic rubidium Dayan et al. (2004) have been observed when excited by bright squeezed vacuum, the quantum enhancement on the resonance-enhanced TPA (r-TPA) cross sections was not quantified. Moreover, atoms generally have classical TPA cross sections in the range of $10^{-40}-10^{-35}$ cm⁴ s, which are 7-15 orders of magnitude larger than those of organic molecules ($10^{-50}-10^{-47}$ cm⁴ s) Sulham, Pitz, and Perram (2010); Davila (2018); Bjorkholm and Liao (1974); Xu et al. (2020); Terenziani et al. (2008); Nociarová et al. (2021). Atoms also have longer-lived excited state coherences than molecules Deutsch et al. (2010); Boyd et al. (2006); Young et al. (2020); Gustin et al. (2023); Hildner, Brinks, and van Hulst (2011); Paulus et al. (2020) which could facilitate coherent biphoton processes. To date, experimental studies in organic molecules have only focused on virtual state-mediated ETPA (v-ETPA) instead of r-ETPA processes Lee and Goodson (2006); Harpham et al. (2009); Guzman et al. (2010); Upton et al. (2013); Villabona-Monsalve et al. (2017, 2018); Eshun et al. (2018); Villabona-Monsalve, Burdick, and Goodson (2020); Tabakaev et al. (2021); Varnavski and Goodson (2020); Parzuchowski et al. (2021); Landes et al. (2021a, c); Mikhaylov et al. (2022); Burdick, Schatz, and Goodson (2021); Corona-Aquino et al. (2022); Tabakaev et al. (2022); Villabona-Monsalve et al. (2022); Triana-Arango, Ramos-Ortiz, and Ramírez-Alarcón (2023); Varnavski et al. (2023).

The experimental challenge in measuring molecular r-ETPA is that signals from the first singlet excited state (S1) can mask signals from the subsequent, weaker excitation to the second singlet excited state (S2). Traditional TPA measurement approaches such as transmission measurements, pump versus pairs power attenuation test Tabakaev et al. (2022); Mikhaylov et al. (2022); Dayan et al. (2005), and z-scans Sengupta et al. (2000); Ajami et al. (2010); Nag, De, and Goswami (2009) therefore become more difficult. Indocyanine green (ICG), however, is a unique case for r-ETPA because it has both an S1-S0 emission at $\sim$850 nm and an S2-S0 emission at $\sim$550 nm Kumari and Gupta (2019); Das et al. (2013); Pu et al. (2014), as shown in Figure 1. The S2:S1 fluorescence ratio allows quantitative comparison between the r-ETPA cross section and the classical one-photon absorption cross section. ICG is also important in medical imaging as the first FDA-approved near-IR contrast agent noa (2023a). Although traditionally used as a one-photon dye, ICG will have further medical applications in deep tissue imaging if it exhibits significantly enhanced r-ETPA.

In this work, spectrotemporally resolved emission signatures of ICG are measured when photoexcited by broadband entangled photons in the near-IR region. No r-ETPA signals are measured within the detection limits of the measurements, allowing us to place an experimental upper bound on the r-ETPA cross section of ICG at $6\times 10^{-23}$ cm²/molecule. For comparison, the classical r-TPA cross section is also measured to be $20(\pm 13)$ GM where 1 GM equals $10^{-50}$ cm⁴ s/molecule. The results suggest that having a resonant intermediate state does not significantly enhance the ETPA cross section of ICG in this case. The findings indicate that current measurement schemes with non-diffraction-limited focusing conditions, $<1\%$ fluorescence collection efficiencies, and $<10$ photons/s dark counts are not sufficiently sensitive for detecting r-ETPA signals in organic molecules with $<$ 20 GM classical TPA cross sections. The measured sequential absorption cross section for ICG is reported for the first time, and this may still be useful for future imaging and sensing applications.

Figure 3 compares the emission spectrum of ICG excited by the SPDC flux (red) with ICG’s classical r-TPA S2-S0 fluorescence spectrum (light blue) and classical S1-S0 fluorescence spectrum (purple). The S2-S0 and S1-S0 emission peaks by 403 nm excitation (deep blue) are spectrally separated so their peak intensity ratio is used to estimate the S2-S0 fluorescence quantum yield of ICG for later calculation of the classical r-TPA cross section. Specifically, the intensity values at 555 nm (S2-S0 fluorescence peak) and 850 nm (S1-S0 fluorescence peak) are scaled by the quantum efficiencies (QEs) of the spectrometer grating noa (2023b) and CCD camera noa (2023c). The ratio of the QE-corrected intensities is then multiplied by ICG S1-S0 fluorescence quantum yield of 0.12 in DMSO Berezin et al. (2007), resulting in an estimated ICG S2-S0 fluorescence quantum yield of 0.02. The emission spectrum from SPDC excitation (red) at 20 s integration time does not have an apparent S2-S0 emission peak. Therefore, the more spectrally sensitive Michelson interference scheme (Figure 2a) is employed to examine if one- versus two-photon signals can be separated.

Figure 4 compares the CCD interferograms of the ICG S1-S0 (purple) and S2-S0 (red) emission regions as a function of time delay. Coincidence rates from the SPDC flux, measured by a pair of SPADs and a timing circuit, are also shown in orange. Each data point at a certain time delay $\Delta\tau$ in the S1-S0 emission region (820-920 nm) represents a summed signal count accumulated over 10 s, and each data point in the S2-S0 emission region (450-650 nm) represents a summed signal count accumulated over 100 s. In the two-photon coincidence interferogram, for an entangled state, the peak amplitudes are larger than the valley amplitudes near time zero. In a TPA event, the interferogram of the fluorescence will follow that of the SPDC coincidences. For a one-photon absorption event, the interferogram’s peak and valley components have equal amplitudes, which is the case for the ICG S1-S0 emission pattern Figure 4a. On the other hand, the interferogram for the S2-S0 emission region is at or near noise level, indicating that no one- or two-photon signals exist within the spectral region despite this approach’s higher signal-to-noise ratio compared with the spectral measurement of Figure 3. The conclusions are further reinforced by the Fourier transforms of the interferograms Figure 4b, where only the SPDC coincidences have a strong 2f peak at $\sim$403 nm, and no peaks are measured in the S2-S0 emission region.

Finally, an epifluorescence and SPAD measurement scheme (Figure 2b) is used to confirm the lack of a two-photon induced signal at even longer integration times than possible with the Michelson scheme. The approach was previously used to successfully measure v-ETPA signals of R6G with 5+ hours of integration time Tabakaev et al. (2022). Error analysis is first performed using 1 mM R6G to determine the optimal integration time that stabilizes the standard deviation of the SPAD measurements. Three hundred sets of repeated measurements are used. Within each set, SPAD counts are measured for 1 minute in the laser-off configuration, followed by 1 minute of measurement in the laser-on configuration. A mean counts/s value is then calculated from each 1-min bin. The 150 repeated laser-off and 150 laser-on measurements are then analyzed separately to confirm that each accumulated standard deviation of the mean is proportional to $1/\sqrt{N}$ where $N$ is the number of repeated 1-min measurements Skoog, Holler, and Crouch . The data indicates that 50 min of integration time stabilizes time-dependent fluctuations and is still practical for performing multiple ETPA measurements.

Figure 5a shows a comparison between the SPAD counts from 1 mM R6G, 1 mM ICG, and solvent DMSO when excited by entangled photons. A wavelength range of 500-650 nm is selected by 8 OD of interference filters at the SPAD entrance, which encompasses both ICG’s S2-S0 r-TPA emission region and R6G’s S1-S0 v-TPA emission region.Although each signal in Figure 5a is already dark-count-subtracted, the solvent DMSO still shows positive counts, possibly indicating molecular Rayleigh scattering Miles, Lempert, and Forkey (2001) of residual CW pump or SPDC. The ICG signal is statistically indistinguishable from the DMSO signal, confirming that r-ETPA is undetectable with an upper bound of $6\times 10^{-23}$ cm²/molecule for the cross section as later discussed. The R6G signal has a statistically significant signal above the solvent scatter. A pump versus pairs attenuation power study Tabakaev et al. (2022); Mikhaylov et al. (2022); Dayan et al. (2005) using an ND filter wheel is performed to determine whether the signal is from one- or two-photon events. The measurement approach verifies ETPA when the fluorescence signal scales linearly with varying pump power but quadratically with pairs attenuation because the loss of one photon within an entangled pair destroys the entanglement. However, Figure 5b shows that the R6G signal counts scale linearly with both pump attenuation and pairs attenuation, indicating the presence of one-photon processes such as hot-band absorption Mikhaylov et al. (2022) instead of ETPA. While the pairs attenuation curve is shifted up with respect to the pump attenuation curve, we attribute this discrepancy as an artifact from the placement of the ND filter wheel in the beam path between pump and pair measurements. When moved, the ND wheel can alter background scatter at few photons per second levels. The results are, therefore, too close to the instrument noise floor to draw statistically sound conclusions about ETPA fluorescence in either ICG or R6G.

Experimental upper bounds can be placed on the r-ETPA cross section ($\sigma_{ETPA}$) of ICG based on the two measurement schemes, as summarized in Table 1. Each scheme’s excitation flux, focusing conditions, collection efficiency, and the optical properties of ICG are used in the calculation Hickam et al. (2022). Additionally, the classical r-TPA cross section of ICG ($\delta_{TPA}$) is calculated to be $20(\pm 13)$ GM according to Equation 1, and is included in the table for comparison. The measured $F(\textrm{800 nm})_{ICG}$ value is $1899(\pm 836)$ Hz, and the $F(\textrm{800 nm})_{R6G}$ value is $22365(\pm 429)$ Hz. To arrive at the $\delta(\textrm{800 nm})_{ICG}$ value, the following approximations are made: 1) because reported TPA cross sections of R6G vary by up to 1 order of magnitude, regardless of solvent and excitation conditions Makarov, Drobizhev, and Rebane (2008); Oulianov et al. (2001), $20(\pm 10)$ GM is used as an estimate for $\delta(\textrm{800 nm})_{R6G}$; 2) $\eta_{ICG}$ is estimated as 0.02 as explained earlier; 3) $\eta_{R6G}$ is approximated to be 0.95 Kubin and Fletcher (1982); Magde, Wong, and Seybold (2002). Because of the approximations, the calculated r-TPA cross section of ICG is emphasized as an order-of-magnitude estimation.

Using the excitation fluxes and focusing areas of the two measurement schemes, the r-ETPA cross section detection limits can be converted noa (2023d) to equivalent $\delta_{TPA}$ values of $2(\pm 2)\times 10^{13}$ GM and $3(\pm 2)\times 10^{12}$ GM, as shown in Table 1. Comparing these values with ICG’s classical r-TPA cross section of $20(\pm 13)$ GM, we conclude that having a resonant intermediate state must enhance ICG’s ETPA cross section by less than 11 orders of magnitude. The conclusion is supported by previous theories Oka (2018); Li and Hofmann (2023) suggesting that ETPA would only offer up to 4 orders of magnitude enhancement on the excited state population in atomic systems, and r-TPA may only be $\sim$1.7 times more prone to quantum enhancement than v-TPA when excited by 100 nm bandwidth of SPDC photons as used in our work Oka (2018). The determined cross section upper limits for R6G are also listed in Table 1. These detection limits agree with theoretical predictions Landes et al. (2021a) and previous experiments Parzuchowski et al. (2021); Hickam et al. (2022), suggesting that the measured upper bounds on ICG in this work are reasonable. The ICG detection limits differ from the R6G detection limits by $\sim$1 order of magnitude mainly because ICG’s absorption bands are broad and heavily overlap with emission bands, leading to a higher chance of fluorescence reabsorption. ICG also has a lower quantum yield than R6G. Note that the correlation time of the entangled photons is $<$ 50 fs (Figure 4a), which is much shorter than the sub-ns lifetime of ICG Berezin et al. (2007). Therefore, if present, ICG’s S1-S2 transition is not significantly hindered by the decay of the intermediate state.

The results are intended as upper bounds specific to the described experimental configurations, which have proven helpful in other studies of ETPA Hickam et al. (2022); Parzuchowski et al. (2021). In the future, if near UV to deep UV entangled photons can be created, molecules with higher S2-S0 quantum yields can be studied, such as azulenes, aromatic acenes, polyenes, and metalloporphyrins Itoh (2012). Earlier in the field, theories have predicted that maximizing the bandwidth of the entangled photon source, which in turn maximizes the brightness and minimizes the entangled photon correlation time, would be the best approach to increasing ETPA efficiency Oka (2018); Dayan et al. (2005); Landes et al. (2021a). However, experimentally, the broad momentum matching cone is difficult to collimate, leading to focused spot sizes that are still 1-2 orders of magnitude away from the diffraction limit Szoke et al. (2021). Furthermore, maintaining the single-photon-per-mode limit in nonlinear light-matter interactions may not even be critical. Bright squeezed vacuum, a class of SPDC fluxes that saturate the single-photon-per-mode limit, also exhibits many quantum-enhanced properties in two-photon absorption Georgiades et al. (1995); Dayan et al. (2004), optical harmonics generation Spasibko et al. (2017), and ultrafast spectroscopy Cutipa et al. (2022). Future experiments using bright squeezed vacuum sources generated in a single, waveguided spatial mode could further improve the detection limits for entangled multiphoton processes.

We investigated the validity of r-ETPA in ICG and placed upper bounds on the possible enhancement. We measured the spectrotemporally resolved emission signatures of ICG excited by broadband entangled photons in near-IR. Similar to many reported v-ETPA measurements, no r-ETPA signals are measured, with an upper bound for the cross section placed at $6\times 10^{-23}$ cm²/molecule. In addition, the classical r-TPA cross section of ICG at 800 nm is measured to be $20(\pm 13)$ GM. The same measurements are performed on R6G for bench-marking. The results suggest that having a resonant intermediate state does not significantly enhance the ETPA cross section of ICG beyond the detection limits of this study. The findings indicate that current measurement schemes with non-diffraction-limited focusing conditions, $<1\%$ fluorescence collection efficiencies, and $<10$ photons/s dark counts are not sufficiently sensitive for detecting r-ETPA signals in organic molecules with small classical TPA cross sections. Further improvements to lower the instrument detection limits will be the key to the successful detection of r-ETPA as well as v-ETPA signatures.

The authors have no conflicts to disclose.

Manni He: Conceptualization (equal), Data Curation (lead), Formal Analysis (lead), Methodology (equal), Resources (lead), Software (equal), Writing/Original Draft Preparation (equal), Writing/Review & Editing (equal).

Bryce P. Hickam: Conceptualization (supporting), Data Curation (supporting), Formal Analysis (supporting), Methodology (equal), Resources (supporting), Software (equal).

Nathan Harper: Formal Analysis (supporting), Methodology (supporting), Resources (supporting), Writing/Review & Editing (equal).

Scott K. Cushing: Conceptualization (equal), Funding Acquisition (lead), Methodology (supporting), Writing/Original Draft Preparation (equal), Writing/Review & Editing (equal).

The data that support the findings of this study are available within the article. Additional data are available from the corresponding author upon reasonable request.