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Abstract
The echo state property, which is related to the dynamics of a neural network excited by input driving signals, is one of the well-known fundamental properties of recurrent neural networks. During the echo state, the neural network reveals an internal memory function that enables it to remember past inputs. Due to the echo state property, the neural network will asymptotically update its condition from the initial condition and is expected to exhibit temporally nonlinear input/output. As a physical neural network, we fabricated a quantum-dot network that is driven by sequential optical-pulse inputs and reveals corresponding outputs, by random dispersion of quantum-dots as its components. In the network, the localized optical energy of excited quantum-dots is allowed to transfer to neighboring quantum-dots, and its stagnation time due to multi-step transfers corresponds to the hold time of the echo state of the network. From the experimental results of photon counting of the fluorescence outputs, we observed nonlinear optical input/output of the quantum-dot network due to its echo state property. Its nonlinearity was quantitatively verified by a correlation analysis. As a result, the relation between the nonlinear input/outputs and the individual compositions of the quantum-dot network was clarified.
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1. Introduction
The echo state property is one mandatory requirement of recurrent neural networks, meaning that effects from the initial conditions of the network should dissipate as time elapses, and the network's condition is updated by the next input not as a simple over-write but as a weighted-addition with the initial condition [1,2]. Generally, the nonlinearity of the system and the memory capacity are fundamental aspects of the property. By utilizing such a network, a machine learning system can effectively handle time-series data by taking past data into consideration with the memory function as actual reservoir computing [3,4], which is a framework for computation derived from recurrent neural network theory that maps input signals into higher dimensional computational spaces through the nonlinear dynamics of a neural network with a fixed composition.
A physical neural network is a type of artificial device in which a physical material and phenomena are used to emulate the function of a neural synapse. It is used to emphasize the reliance on physical hardware used to emulate the behaviors of neurons, as opposed to software-based approaches that simulate the input/output of neural networks. So far, various styles for physical implementation have been discussed, such as electrochemical cells [5], analog VLSI [6], memristive nanodevices [7], and so on. Here, we focus on the use of quantum-dots (QDs) as basic components of a physical neural network that is driven by optical pulse inputs and the outputs of which are fluorescence from the QDs. A QD is a semiconductor material, a few nanometers in size, with characteristic optical properties and behavior governed by the laws of quantum mechanics. To date, various networks of optical energy consisting of aligned QDs have been actively studied for implementing various optical functions, such as photonic wires [8,9], nanophotonic logic gates [10–13], and memory [14,15].
In this study, we employed randomly dispersed QDs in a cured-resin solution as a physical QD network. Such dispersed QDs are expected to behave as clusters of minute networks. By exciting the network with incident laser light, the network holds localized optical energies. The optical energies in the network are allowed to transfer from one QD to another [16,17]. As the energy is transferred, various paths in the QD network are expected to produce a cascade of fluorescence outputs from the network via various variations of the energy transfers. Additionally, because the QDs are randomly dispersed, various compositions of QD networks are necessarily included in a single QD sample; namely, various outputs can be obtained by spatially parallel optical processing in a QD sample. In this paper, we focus on nonlinearity of the system among the two aspects of the echo state property. We fabricated several samples containing various QD networks in which QDs were spatially dispersed in a cured-resin solution with several mixing conditions. Their echo state property and corresponding nonlinear optical input/output were verified quantitatively from the results of photon counting of the fluorescence outputs obtained in response to sequential short light pulses input to the QD networks from a laser. Finally, we discuss the relation between the verified input/outputs and the corresponding QD networks. As a result, the presence and behavior of the echo state property in each QD network were clarified by qualitative analysis of the nonlinearity of each input/output.
2. Echo state of QD network
In our scheme for realizing optical input/output of the QD network, the QD network is irradiated with a sequence of light pulses from a laser as optical inputs. The light pulses are incident on the QD network spatially in parallel and excite multiple QDs. The optical energy of the input light must be larger than the band-gap energy of the QDs. The excited QDs probabilistically emit fluorescence photons that have similar optical energy to the band-gap energy. On the other hand, part of the optical energy is allowed to transfer from an excited QD to another QD. After single or multiple steps of energy transference, some of the optical energy in the network is emitted from a destination QD. Generally, the timings of light emissions via multiple energy transfers are necessarily later than the emission involving no energy transfer. As a result, the fluorescence output from the QD network can be sequentially obtained by sparse counting the fluorescence photons.
As shown in Fig. 1, our QD network consists of two types of QDs: donor- and acceptor-QDs. As one of the typical phenomena of localized optical energy transfer, based on the mechanism of Förster-resonance energy transfer (FRET), a donor-QD, initially in its electronic excited state, may transfer energy to an acceptor-QD through nonradiative dipole–dipole coupling. Generally, the fluorescence spectrum of a donor-QD and the absorption spectrum of an acceptor-QD partially overlap, so that FRET from a donor-QD to a neighboring acceptor-QD (and also a neighboring donor-QD) is allowed probabilistically. Furthermore, based on the FRET mechanism, FRET from an acceptor-QD to a donor-QD is prohibited, and an acceptor-QD often acts as the destination in each network. During the energy transfer, optical energy is percolated in the QD network as the echo state, which can be regarded as the optical input being partially memorized, with subsequent forgetting.
Fig. 1. Schematic diagram of the echo state property of a QD network, which is due to localized transfer and temporal stagnation of optical energy during periodic irradiation of light pulses. Excited QDs are emphasized by yellow halos. Based on the FRET mechanism, in the case where all QDs consist of the same material, acceptor-QDs are larger than donor-QDs.
Additionally, as shown in the lower half of Fig. 1, when the next light pulse is input to the QD network during the holding time of the echo state, the transfer process and corresponding emission of fluorescence photons reveal a different tendency from the previous condition of the network, because the internal state of the QD network is different from the previous condition, namely, unlike the upper half of Fig. 1, some QDs are already excited. Here, saturated situation, which is all QDs are excited, is not anticipated. As a result, in such a case, the optical outputs in response to sequential optical inputs cannot be predicted by a simple linear sum of a single input/output. Therefore, and nonlinearity of the input/output can be quantitatively evaluated by comparison between the linear sum of single input/outputs.
3. Experimental QD network
In this paper, we used two types of QDs as components of the QD network: CdS dots (NN-labs, CS460; peak wavelength of the emission: 465-490 nm, 3.0 nmol/mg, represented in catalog) and CdS/ZnS dots (NN-labs, CZ600; peak wavelength of the emission: 600-620 nm, 1.0 nmol/mg, represented in catalog) with toluene solutions. The CdS and CdS/ZnS dots work as donor and accepter, respectively. Fluorescence and absorbance spectra of the QD solutions are shown in Fig. 2(a). Additionally, polydimethylsiloxanes resins (PDMS; Dow Corning, Sylgard184) were utilized as the base materials. The basic procedure to prepare a QD sample using these materials is shown in Fig. 2(b) and is as follows:
Fig. 2. (a) Fluorescence spectra (solid lines) and absorbance spectra (dashed lines)of donor and acceptor QDs. (b) Schematic diagram of experimental process. (c) Appearance of three QD samples under UV light illumination.
First, the two QD solutions are mixed with 1,000 µL of PDMS base solution. In order to control the configuration of the QD networks, the mixing ratio of the CdS dots was varied as shown in Table 1.
Table 1. Mixing ratio of QD solutions for our samples
After mixing, each mixture was heated to evaporate the toluene. Then, 100 µL of polymerization initiator PDMS was added to the mixture, and the resulting solution was dropped on cover glass. The mixture was spread on the cover glass using a spin coater (MIKASA, MS-B100, rotational speed: 3,000 rpm) for 100 seconds so that the QDs were randomly dispersed in the resin, and respective QD networks were expected to be formed. The assumed thickness of the samples was less than 1 µm. After the mixtures were thinned, in order to fix the alignments of the QDs in each mixture, the thinned samples were heated on a hot plate (EYELA, RCH-1000) at 150 °C for 600 s. The prepared samples appeared transparent under room light; however, they emitted fluorescence under UV light illumination, as shown in Fig. 2(c).
4. Time constants of the optical output
Here, we focus on light emission from acceptor-QDs to verify the echo state in the QD network. Here, photons emitted from each QD are necessarily output at various timings, whether the photons are emitted via energy transfers or not. That is, the results of photon counting, which is triggered by the timing of the optical input, indicate the characteristics of the echo state of the QD network.
For experimental verifications, we used a Ti:Al2O3 laser (Spectra Physics, MaiTai), which emitted optical pulses with a pulse length of 100 fs, with an optical parametric amplifier (Spectra Physics, TOPAS-prime) and a wavelength converter (Spectra Physics, NirUVis) as a light source for irradiating the QD samples. The oscillation frequency and wavelength were set to 1 kHz and 457 nm, respectively. The laser power and polarization were controlled appropriately to excite the QD samples and count the fluorescence photons effectively, as shown in Fig. 3.
The delay line generates a time lag Δt between the first and the second pulses incident on the QD sample. The range of Δt is set from 0.64 ns to 7.4 ns, and the corresponding optical length can be controlled by a stage controller driven by a stepping motor with a position resolution of 20 µm/step. Fluorescence photons induced by optical excitation of the QD samples went through the focusing lens again and were reflected to the detecting setup by a Glan-Thompson polarizer. After passing through a bandpass filter (UQG optics, 0G530, transmission wavelength: 620 ± 10 nm), the fluorescence photons propagated to a photon detector (Nippon Roper, NR-K-SPD-050-CTE). Because the irradiation light passed through polarizers, only the fluorescence photons from acceptor-QDs were selectively obtained by the single photon detector. Then, via synchronization with the trigger signal using a Si-PIN photo diode (EOT, ET-2030) on a time-to-amplitude converter (TAC; Becker & Hickl GMbH, SPC-130EMN), time-resolved intensities were obtained, and the results were collated using PC software (Becker & Hickl GMbH, SPC-130MN) as the lifetime of each fluorescence. Here, excitation power is set as 5.0 µW, which is surely enough suppressed not to induce saturated situation of QD excitations.
Before verifying the nonlinearity based on the echo state, fluorescence relaxation due to multiple incident laser pulses was experimentally verified in the setup as the basic specifications of our three samples, Samples A, B, and C.
The left hand side in Fig. 4(a) shows one example of the obtained photon counting result from acceptor-QDs in response to double incident laser pulses, which was obtained at a certain area in Sample A. As shown, two rising phases are recognized, which are due to the first and the second incident laser pulses. The time lag Δt between the two pulses was 7.4 ns. Under such a condition, the second pulse was irradiated before the induced optical energy is dissipated and corresponding photons being counted in response to the first incident pulse, which corresponds to the hold time of the echo state induced by the first incident pulse. Therefore, we focused on photon counting after the second incident pulse, which is extracted as the right-hand side of Fig. 4(a).
Fig. 4. (a) Experimentally obtained photon counting result of a QD sample under irradiation by two pulses with time difference Δt of 7.4 ns. (b) Comparison of time constant τ of fluorescence between three QD samples: Samples A, B, and C.
To discuss the spatial variation of the QD networks in each sample, three individual areas were irradiated in the three samples, and fluorescence photons were counted. Then, in order to quantitatively compare the relaxation of the fluorescence of each sample, the results were fitted by the exponential equation:
where A and B are individual positive constants, and τ denotes the time constant of the fluorescence lifetime. Values of these parameters were appropriately selected to be well-matched with experimental results. As the result, $\tau $ was calculated to be 0.1–0.3 ns for the three samples, as shown in Fig. 4(b). As shown, clear differences appeared between the results for each sample. Namely, in the case of Sample C, which contained no donor-QDs and a smaller total number of QDs, as shown in Table 1, energy transfer between QDs rarely occurred, and QDs excited by the laser pulse directly emitted photons without any energy transfers. Therefore, Sample C revealed the shortest τ. On the other hand, in the case of Sample A, which contained the largest number of donor-QDs, frequent energy transfers are expected to occur. As a result, excited optical energy is allowed to stagnate over a longer time in the QD network as the echo state and is obtained at various timings due to various energy transfers. Therefore, Sample A revealed the longest τ among the three. The results indicate a clear relation between the configuration of the QD network and the extent of the echo state due to FRET between QDs.5. Nonlinearity of the optical output
In order to quantitatively evaluate the nonlinearity of each photon counting result, we employed correlation analysis of the photon counting results in response to double incident pulses with a linear sum of single input/outputs, which are the photon counts due to a single pulse incident on each sample. As the correlation analysis, Pearson correlation coefficients, R, were calculated by
Fig. 5. Comparison of correlation coefficients corresponding to nonlinearity of fluorescence of three samples (Sample A, B, and C).
Overall, as shown in Fig. 5, with increasing Δt, higher and more-converged correlation coefficients were observed, which implies that the echo states induced by the first incident optical pulse are gradually dissipated. On the other hand, in the case of Δt shorter than 1.0 ns, the second pulse was incident before dissipation of the echo state excited by the first incident pulse. As a result, lower correlations and corresponding higher nonlinearity were successfully revealed.
Furthermore, the three lines in the results for Sample C revealed quite similar curves, and the correlation coefficients were more than 0.95, which corresponds to smaller nonlinearity. These findings are considered to be due to sparse alignment of the QDs, as schematically shown in the insets in Fig. 5. Specifically, the input/outputs varied from area to area, and FRET between QDs was rarely allowed. As a result, the echo state and corresponding nonlinear input/output were not sufficiently revealed in Sample C. On the other hand, in the case of Sample A, because the number of QDs was enough in all areas, a lot of paths for FRET are expected. As a result, similar tendencies of the input/outputs were obtained in each area, and higher nonlinearity than Sample C was revealed. In the case of Sample B, tendencies showing the most variation in each area were observed in, and higher nonlinearity was often revealed in some areas. As a result, we verified the echo state property of our QD samples and quantitatively measured hold times of less than 1.0 ns with our experimental conditions. As indicated in Fig. 5, the hold time and spatial variation of the echo state property clearly depended on the composition of the QD sample.
6. Conclusion
This paper shows results of investigation of the applicability of randomly-dispersed QDs as physical neural network with optical input/output. Based on the idea that QD network is expected to reveal nonlinear input/output due to short-time echo state of optical energy in the network, photon countings of the fluorescence outputs obtained in response to sequential short light pulses were carried out to verify the optical input/output of our original QD network. As the result, clear nonlinearity of the input/output, which is fundamental requirement for realization of effective machine learnings, was qualitatively verified. Remember, nonlinearity of the QD network is only one aspect of the echo state property. Namely, if the nonlinearity is emphasized, its memory capacity become decreased. On the other hand, if the nonlinearity is suppressed, its memory capacity become increased [18]. For further discussions on the echo state property, it is necessary to be additionally evaluated on memory capacity of the QD network.
In addition to the nonlinearity, larger spatial variation is also expected as one of the fundamental requirements in QD networks for physical implementations of reservoir computing [3], because varied nonlinear input/outputs in a single reservoir are useful for effective machine learning based on the QD reservoir, i.e., so-called nano-photonic reservoir computing. Optimization of the composition required for targeted processes for nano-photonic reservoir computing remains an open topic for further investigation.
Funding
Japan Science and Technology Agency (JPMJCR18K2).
Disclosures
JST CREST Grant JPMJCR18K2, Japan.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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