Abstract
In optical materials energy is usually extracted only from the lowest excited state, resulting in fundamental energyefficiency limits such as the Shockley–Queisser limit for singlejunction solar cells. Photoncutting materials provide a way around such limits by absorbing highenergy photons and ‘cutting’ them into multiple lowenergy excitations that can subsequently be extracted. The occurrence of photon cutting or quantum cutting has been demonstrated in a variety of materials, including semiconductor quantum dots, lanthanides and organic dyes. Here we show that photon cutting results in bunched photon emission on the timescale of the excitedstate lifetime, even when observing a macroscopic number of optical centres. Our theoretical derivation matches well with experimental data on NaLaF_{4}:Pr^{3+}, a material that can cut deepultraviolet photons into two visible photons. This signature of photon cutting can be used to identify and characterize new photoncutting materials unambiguously.
Introduction
Optical materials are used to convert the energy of photons into other useful forms. Examples include photovoltaic materials converting light into electrical energy, phosphors transforming one colour of light into another and photocatalysts using photon energy to make or break chemical bonds. In most materials the conversion process generates no more than one quantum of output energy (for example, an energetic electron or a colourconverted photon) per one photon absorbed. The energy efficiency of such processes has a fundamental limit that is approximately inversely proportional to the energy of the incoming photon. This is, for example, a major factor determining the Shockley–Queisser efficiency limit for singlejunction solar cells^{1}.
The process of photon cutting or quantum cutting can substantially improve the energy conversion efficiency for highenergy photons by ‘cutting’ them into multiple lowerenergy excitations. This possibility was first hypothesized in 1957 (ref. 2) and has since been demonstrated experimentally in various materials, including semiconductor quantum dots (multiexciton generation)^{3,4,5,6,7}, organic dyes (singlet fission)^{8,9,10,11,12,13,14} and lanthanide ions^{15,16,17,18}.
Most experiments for the characterization of photoncutting materials rely on (timeresolved) photoluminescence or transient absorption measurements as a function of excitation wavelength^{3,6} or density of optical centres^{12,16,17}. Usually, the proof that photons are ‘cut’ into multiple excitations is indirect, except in rare cases where the excitations can be extracted with very high efficiency^{7,19}. This leads to ambiguities in the identification and characterizations of new photoncutting materials. For example, the occurrence of multiexciton generation in semiconductor quantum dots is usually concluded from fast decay components in transient absorption data^{3,20,21,22}, but these can also originate from trapping of charge carriers or charging of the quantum dots^{23}. Triplet states in dye molecules can be generated by the photoncutting process of singlet fission^{11,14}, but also by regular intersystem crossing^{9}. Similarly, nonradiative energy transfer from a highly excited lanthanide ion can result in photon cutting through distribution of the energy over multiple excited centres^{15,16}, but it is not trivial to distinguish this from processes generating only one excitation, while excess energy is lost as heat^{24}. In view of this, it is not surprising that previous studies have sometimes reported contradictory conclusions on the occurrence or efficiency of photon cutting^{20,21,24}.
Here we propose that direct proof of photon cutting in a material is possible by the observation of nonPoissonian photon emission statistics. Bunched emission has been reported from multiexciton states in single CdSe quantum dots^{25}. However, analysing photon cutting on a single optical centre is challenging at best and impossible for many photon cutters that rely on energy transfer between centres. We derive here that a photoncutting material exhibits photon bunching even if it contains a macroscopic number of optical centres. Photon bunching should therefore be observable from any photon cutter where the excitations can be extracted as light^{6,15,16,23,26,27}. We demonstrate this phenomenon experimentally on the photoncutting phosphor NaLaF_{4} doped with Pr^{3+} (ref. 28).
Results
Derivation of photon bunching from a photoncutting material
We start from the general energy level scheme of a photon cutter (Fig. 1a). A highenergy photon (purple) excites the system to a high excited state Y. This is followed by a cascade of transitions: first to an intermediate excited state X (blue) and then further to the ground state G (red). In both steps of the cascade, a photon is emitted. Figure 1b is an exemplary photon detection trace of a single optical centre exhibiting such cascade emission. All photons from the first transition (blue) are followed by a photon from the second transition (red), with as average time interval the decay time τ_{X} of the intermediate state X. For a material with a large number N of optical centres the emitted photons are also bunched in pairs, but now the pairs overlap in time (Fig. 1c). The statistics of bunched emission are different from the statistics of regular (that is, nonphoton cutting) photoluminescence. The peculiar photon statistics of a photon cutting material can be described mathematically and investigated experimentally using the normalized photon–photon crosscorrelation function.
where I_{1} is the intensity of the first emission step in the cascade and I_{2} is the intensity of the second step. The crosscorrelation function describes how likely it is to detect a photon from the second transition at time τ after detection of a photon from the first transition. In the Methods section, we derive the crosscorrelation function analytically for an experiment on a macroscopic photon cutter (as in Fig. 1c) in which the first photon (blue in Fig. 1a) and the second photon (red in Fig. 1a) are spectrally separated and directed to two independent detectors with negligible dark count rates:
with the steadystate population and k_{XG} the decay rate of the intermediate level X. Bunching is observed as an additional signal decaying with rate k_{XG} on top of a constant unity background caused by Poissonian photons statistics.
In Fig. 1g, we plot the analytical correlation function (lines) for an ideal photon cutting material together with the correlation function from a Monte Carlo simulation (dots), for different numbers of optical centres (N) in the material. The excitation rate is set at Φ=2k_{XG}/N, which corresponds to a constant steadystate population of =2 in the limit of large N. The crosscorrelation function shows an increased likelihood of detecting a (red) photon from the second transition after detection of a (blue) photon from the first transition for any number N of optical centres. The analytical model (equation (2)) matches well with the Monte Carlo results for an ensemble of emitters with N>10, including macroscopic materials containing a number of optical centres on the order of Avogadro’s constant (Fig. 1g, green and blue), clearly revealing the occurrence of photonpair emission in the photon statistics. The analytical model is less accurate for a small N (yellow and red), because the approximation of no groundstate depletion is justified only for large N and low excitation rates. In agreement with equation (2), Fig. 1h shows that an increasing excitation rate Φ (and therefore increasing steadystate population ) results in a smaller bunching amplitude. At the same time a higher excitation rate Φ results in a higher photon count rate and therefore in decreased statistical noise on the correlation function. Interestingly, the bunching signal and the statistical noise both scale with 1/Φ, so that the excitation rate has no net effect on the signaltonoise ratio in a photonbunching experiment. See Supplementary Fig. 3 for an analysis of what this means for the fit uncertainties.
Figure 1c–f illustrate complications that may arise in real experiments on photoncutting materials. Ideally, the collection and detection of photons would have unity efficiency (Fig. 1c), but in practice this efficiency is finite so that many photon emission events go unnoticed (Fig. 1d). Moreover, detectors have a finite dark count rate leading to random background counts (Fig. 1e). Finally, for many photoncutting materials it is not possible to spectrally distinguish the two photons emitted in the cascade process (Fig. 1f). The correlation functions for increasing experimental complexity are plotted in Fig. 1i. A nonunity detection efficiency does not lower the bunching amplitude, but only lowers the count rates and therefore increases the noise on the data (compare blue and green). Detector dark counts lower the bunching amplitude (yellow), and should therefore be minimal. (In Supplementary Note 1 we investigate the effect of dark counts analytically.) The bunching amplitude decreases further if the two emitted photons cannot be separated spectrally (red). The bunching signal now becomes symmetric about τ=0, because the order of detector clicks is no longer sensitive to order of the emitted photons (Fig. 3d).
Experimental demonstration of photon cutting in NaLaF_{4}:Pr^{3+}
To experimentally test the occurrence of photon bunching in the emission from a macroscopic photoncutting material, we measure the phosphor NaLaF_{4} doped with 1% Pr^{3+} (ref. 28). We use ∼10 mg of the material, containing N=10^{17}–10^{18} optical centres that in the experiment are excited more or less homogeneously. Figure 2a shows the mechanism of photon cutting in Pr^{3+}: ultraviolet light excites the ion to the ^{1}S_{0} level (level Y in Fig. 1a), after which a radiative transition to the ^{1}I_{6} level, rapid nonradiative relaxation to the ^{3}P_{0} level (level X in Fig. 1a) and a radiative transition to one of the ^{3}H_{J} levels (level G in Fig. 1a) follow^{29,30}. In Supplementary Fig. 2 and Supplementary Note 2, we discuss how other types of photon cutters, such as quantum dots exhibiting multipleexciton generation, can be mapped on the model of Fig. 1a.
Our experimental setup is sketched in Fig. 2b. The blue photons from the first radiative transition and the red–green photons from the second radiative transition are separated (Fig. 2c,d) with a dichroic filter and sent to two independent detectors. Crosscorrelating the two detector signals yields Fig. 2e. We observe an increased likelihood of detecting a redgreen photon after detection of a blue photon. The inset of Fig. 2e shows that temporal decay of the bunching signal in the crosscorrelation function (green) matches the photoluminescence decay of the intermediate ^{3}P_{0} level (red), in agreement with equation (2). This is a direct proof of photon cutting in NaLaF_{4}:Pr^{3+}, with the ^{3}P_{0} level of Pr^{3+} as the intermediate state.
The magnitude of photon bunching
In Fig. 3a–c we show how the bunching signal changes with excitation density Φ. The amplitude of the bunching signal (Fig. 3a) is inversely proportional to the excitation density, while the decay time is constant at 18.4 μs (Fig. 3b). This decay time is in good agreement with the reported 18 μs^{28} of the intermediate state ^{3}P_{0} (=X in Fig. 1a). Meanwhile, the ratio of count rates on the two detectors is constant at 0.33 red photons per 1 blue photon, in good agreement with the reported value of 0.40 (ref. 28).
With the aim to investigate the effect of spectral separation, we have performed the experiment both with and without spectrally separating the two photons emitted in the cascade process. Spectral separation was achieved by using a dichroic mirror in the emission path (as in Fig. 2e), while the experiment without spectral separation used a 50/50 beamsplitter. In Fig. 3d we show the two resulting crosscorrelation functions. The crosscorrelation function with spectral separation (green) is similar to the result in Fig. 2e. The crosscorrelation function of the experiment with the 50/50 beamsplitter (grey) shows symmetric bunching with a lower amplitude, in agreement with the simulations in Fig. 1i (red). We derive in Supplementary Note 1 that the ratio between the bunching amplitudes of the experiments with and without spectral separation is related to the radiative efficiencies of the two transitions in the photoncutting material. Based on this we can estimate that NaLaF_{4}:Pr^{3+} emits 0.35 red photons per 1 blue photon. This agrees well with the value of 0.40 reported by Herden et al.^{28} and with the value of 0.33 obtained from the ratio of the count rates (Fig. 3c).
We have shown that the bunching amplitude equals 1/, where is the steadystate population in intermediate state X. Throughout this work, we used low excitation densities, so that =5–100 (Figs 2e and 3) in our bulk powder. Such excitation is strong enough that the photon count rate is reasonable (10^{3}–10^{4} s^{−1}), but weak enough for a significant bunching amplitude. In Fig. 3e,f we investigate how long a measurement using weak excitation (=5; total count rate 1,500 s^{−1}) must last to clearly observe bunching over the Poissonian background. Figure 3e shows correlation functions for experiments with different durations T. In Fig. 3f we plot the fitted bunching amplitude as a function of experiment duration, including the 2σ confidence interval on the fit (blue) and the standard deviation of the noise (red). Already after 5 min, the bunching amplitude exceeds the noise by 2σ and after 15 min by 6σ, although photon bunching is not yet clear by visual inspection of the crosscorrelation function (Fig. 3e). The time T_{s} at which the bunching amplitude and noise level are equal (green arrow in Fig. 3f) can be used to calculate the detection efficiency η if either the efficiency of the first or second radiative transition is known (see Supplementary Information equation (16)). Based on the efficiencies reported by Herden et al.^{28}, we estimate that our detection efficiency is η=0.4%.
Discussion
Our work has demonstrated theoretically and experimentally that a photoncutting material exhibits bunched emission, even if it contains a macroscopic number of optical centres. In fact, the magnitude of bunching does not depend on the total number of optical centres, but only on the steadystate population of centres in the excited state. We predict that similar photon statistics should be observable for many other photoncutters recently reported^{3,4,6,15,16,17,20,21,22,23,26,27}. An interesting analogy exists with bunched cathodoluminescence, which has recently provided evidence that the impact of an individual electron on a semiconductor material can generate multiple excitations^{31,32}.
Photonbunching experiments will provide definite proof of photon cutting for materials where controversy exists^{20,24}. The magnitude of the bunching signal depends on the photoncutting efficiency of the material, as well as on the detection efficiencies (see Supplementary Note 1). With careful calibration of the experimental setup and reference measurements on one of the steps of the cascade, it is possible to determine the absolute photoncutting efficiency of materials. We envision that experiments as presented here will contribute to the development and optimization of existing and new photoncutting materials.
Methods
Photon correlation
We used a microcrystalline NaLaF_{4}:Pr^{3+} 1% powder provided by Jüstel and colleagues^{28}. Approximately 10 mg of powder was glued as a thin layer of a few mm^{2} to a nonluminescent background using SPI silver paint. The spectral output of a Micropack DH2000 deuterium lamp filtered using an Acton Optics & Coatings 180N 180±10 nm bandpass filter illuminated the sample homogeneously, exciting Pr^{3+} to a 4f^{1}5d^{1} level, from which rapid nonradiative relaxation to the ^{1}S_{0} level takes place. The excitation intensity was controlled with two apertures of adjustable size. A Zeiss LD ECEpiplanNEOFLUAR 100 × numerical aperture 0.75 objective collected the emission light and emission not originating from the cascade pathway was filtered out with a 350 nm longpass filter. Two nominally identical Hamamatsu R10699 photomultiplier tubes, operated at 1,000 V and cooled with Peltier elements, were used as detectors. They have dark count rates of 10–20 cps. Becker & Hickl GmbH HFAC26dB amplifiers were used to amplify the signals, which were recorded with a PicoQuant TimeHarp 260 photon counting module, operated in timetagged timeresolved mode and set at a discriminator level of −100 mV. In Supplementary Fig. 4 and Supplementary Note 4 we show that an experiment with one detector is not possible, because detector afterpulsing interferes with the detection of photon pairs. For the experiment with spectral separation (Figs 2e and 3), the emitted light was split with a Thorlabs DMLP425 dichroic mirror and sent to the two separate detectors. For the experiment without spectral separation (Fig. 3d, grey), the emission was divided equally over the two detectors using a Thorlabs BSW26 50/50 nonpolarizing beamsplitter. Supplementary Note 3 describes how the raw list of photon arrival times is converted into the normalized crosscorrelation function.
Spectra
Emission spectra were recorded on an Edinburgh Instruments FLS920 spectrofluorometer, using a Micropack DH2000 deuterium lamp plus Acton Optics & Coatings 180N 180±10 nm bandpass filter for excitation and scanning a single monochromator after a 350 nm longpass filter to direct the cascade emission to a Peltiercooled Hamamatsu R928P photomultiplier tube. The spectrum of the redgreen photons was measured with the Thorlabs DMLP425 dichroic mirror in the emission path and the spectrum of the blue photons was obtained by subtracting the redgreen spectrum from the spectrum in the absence of the dichroic mirror. The luminescence decay curve of the ^{3}P_{0} was obtained by pulsed excitation with an Ekspla NT 342B laser at 446 nm and detection at 609 nm using a Triax 550 monochromator and Hamamatsu H742202 photomultiplier tube, coupled to a PicoQuant TimeHarp 260 photo counting module set at a discriminator level of −100 mV.
Derivation of bunching strength
We derive the crosscorrelation function for photons emitted by a macroscopic photoncutting material with N optical centres. Each centre has the general energylevel structure as in Fig. 1a: a highly excited state Y can decay to an intermediate excited state X and then further to the ground state G. The two steps have total transition rates k_{YX} and k_{XG}, respectively, and radiative efficiencies and with the superscript ‘r’ denoting the radiative part of the transition rate. A continuouswave light source pumps the centres from state G to state Y at an excitation rate Φ=σI/ħω, with σ the absorption crosssection, I the pump intensity and ℏω the photon energy. The two photons emitted in the cascade process are separated spectrally and sent to two detectors (1 and 2, respectively), each with zero dark count rate. This model describes our experiment on Pr^{3+} well, as explained in Supplementary Note 1. There we also discuss situations where spectral separation of the two emitted photons is not possible (such as for multiexciton generation in colloidal quantum dots) or where dark counts are not negligible.
A photon count on detector 1 signifies that one of the optical centres in the material just underwent the transition Y→X. This particular centre is therefore in the intermediate excited state X directly after the detection event and can subsequently decay further to the ground state at a rate k_{XG}. The expectation value for the population of state X as a function of delay time τ after the click on detector 1 is
with =ΦN/k_{XG} the steadystate population of state X. The decay of this one centre has a negligible effect on the excitedstate populations and =ΦN/k_{YX} in the rest of the material, because we consider a macroscopic number of optical centres . Realizing that the detector count rates (I_{1} and I_{2}) are proportional to the excitedstate populations, we can express the normalized crosscorrelation of the signals on detectors 1 and 2 as:
Monte Carlo simulation
We performed rejectionfree kinetic Monte Carlo simulations of an ensemble of photoncutting optical centres, each with an energylevel structure as in Fig. 1a. The simulation keeps track of the populations N_{G}, N_{X} and N_{Y} in the ground state G, the intermediate state X and the highestenergy excited state Y. Three processes can occur in the ensemble of optical centres: (1) absorption at rate ΦN_{G}, (2) decay of a centre in state Y at rate k_{YX}N_{Y} and (3) decay of a centre in state X at rate k_{XG}N_{X}. A simulation step consists of (A) randomly selecting one of the processes (1–3) to occur, taking into account the relative probabilities, (B) drawing a residence time Δt from an exponential distribution with average (ΦN_{G}+k_{XG}N_{X}+k_{YX}N_{Y})^{−1}, (C) updating the simulation time to t→t+Δt, (D) storing time t in case of photon emission and (E) updating the populations of the three states. In the simulation of experiments with spectral separation, photon emission events from states X and Y are stored in separate channels. In the simulation of experiments with finite detector efficiency, photon emission events are randomly excluded from the storage. Dark count events were added separately by drawing the intervals between events from an exponential distribution with average D^{−1}, where D is the dark count rate.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: de Jong, M. et al. NonPoissonian photon statistics from macroscopic photon cutting materials. Nat. Commun. 8, 15537 doi: 10.1038/ncomms15537 (2017).
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Acknowledgements
We are grateful to Thomas Jüstel and coworkers for supplying us with the NaLaF_{4}:Pr^{3+} 1% sample. The work was supported by the EU Marie Curie Initial Training Network LUMINET (316906) and by the Netherlands Center for Multiscale Catalytic Energy Conversion (MCEC), an NWO Gravitation programme funded by the Ministry of Education, Culture and Science of the government of the Netherlands.
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F.T.R. conceived the idea of the experiment. M.d.J. designed, performed and analysed the experiment. F.T.R. derived the analytical correlation functions. M.d.J. performed the Monte Carlo simulations. M.d.J., A.M. and F.T.R. discussed the results and wrote the manuscript.
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de Jong, M., Meijerink, A. & Rabouw, F. NonPoissonian photon statistics from macroscopic photon cutting materials. Nat Commun 8, 15537 (2017). https://doi.org/10.1038/ncomms15537
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DOI: https://doi.org/10.1038/ncomms15537
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