Extended Data Fig. 4: Measurements and simulations of the 800 nm/700 nm -emission as a function of Tm³⁺ concentration.

a. Single particle spectra, measured for different Tm³⁺ concentration pre-ANPs/ANPs, at their respective saturation regimes (excitation wavelength: 1064 nm; excitation intensity: 376 kW cm⁻² (4% Tm³⁺), 1,849 kW cm⁻² (5% Tm³⁺), 517 kW cm⁻² (15% Tm³⁺)). b. 800 nm / 700 nm -emission ratio as a function of Tm³⁺ concentration for different Tm³⁺ concentration pre-ANPs/ANPs, measured at their respective saturation regimes (excitation wavelength: 1064 nm; excitation intensity: 376 kW cm⁻² (4% Tm³⁺), 1,849 kW cm⁻² (5% Tm³⁺), 517 kW cm⁻² (15% Tm³⁺)). Error bars are the standard deviations of three single-NP measurements. c. Photon avalanche differential rate equation (DRE) model simulations of the 800 nm / 700 nm -emission ratio as a function of Tm³⁺ concentration. The DRE simulations for the different Tm³⁺ concentrations were performed with the respective parameters (per Tm³⁺ concentration) extracted from fitting the emission versus excitation graphs measured at zero applied force (Supplementary Note 6, Table S3, and Fig. S5). Note that the DRE model predicts Tm³⁺ emissions from these levels up to a certain excitation threshold, above which other, higher-level, excited states are populated and thus must be taken into account. Any discrepancies between the simulated values here and the measured ones in a-b may be accounted for by adding DREs for higher levels (above ³F₂). Other factors – such as determination of exact concentrations and hence energy transfer rates, or discernment of the precise 700 nm emission at lower concentrations and hence experimental 800 nm / 700 nm -emission ratios – may also reduce discrepancies between simulated and measured values. The DRE simulations show that higher cross-relaxation rates, attained at higher Tm³⁺ concentrations, decrease the 800 nm / 700 nm -emission ratio by: 1. Populating ³F₄ at a faster rate, which, in turn, populates ³F₃ at a faster rate by direct excited state absorption. 2. Depopulating ³H₄ at a faster rate.