The continuous-wave (CW) optically injected semiconductor laser is a typical configuration to generate chaotic signals. However, such a configuration only yields bandwidth-limited chaos within a narrow range of injection parameters [see Supplementary Note I, Figure S1]. Herein, we consider using the on-chip DFB laser array subject to intensity-modulated optical injection to generate wideband chaotic signals. To begin with, the input carrier at (ξ_i, f_i) = (0.7, 39.5 GHz) is sent into the DFB laser to excite the period-one (P1) dynamics. The injection ratio ξ_i is defined as the square root of the power ratio between the optical input and the free-running DFB laser. The frequency detuning f_i between the TLs and DFB lasers can be adjusted by changing the emitting wavelength of the TLs. The optical spectrum of CH4 with P1 oscillation as an illustrative example is shown in Figure 2a, where the spectrum of the free-running DFB laser (gray curve) is also plotted for comparison. A frequency difference of 43.28 GHz between the red-shifted cavity frequency and the regeneration frequency from the optical injection is observed. After the frequency beating in PD, a microwave jittering around 43.28 GHz is obtained, as shown in the inset of Figure 2b. To gain the chaotic dynamics, a microwave signal is modulated into an optical carrier from TLs and then injected into each DFB laser. The optical modulation depth is described by a sideband-to-carrier ratio (SCR). (24) One can observe a broadband optical spectrum in Figure 2c when sending an intensity-modulated optical input with (SCR, f_m) = (−14.5 dB, 35.4 GHz) into a DFB laser [the optical spectrum of intensity-modulated input is shown in Supplementary Note I, Figure S2d]. The low modulation sideband (which is 7.88 GHz away from the lower oscillation sideband of the P1 dynamics) strongly perturbs the low oscillation sidebands so that the laser destabilizes into a chaotic dynamical state. (25) Further, we observe a wide microwave power spectrum (blue curve) in Figure 2d after the optical-to-electrical conversion. To quantitatively estimate the chaotic bandwidth, we turn to the 80% standard bandwidth definition. (26) One can see from Figure 2d that the high-frequency component can be significantly raised and the corresponding chaotic bandwidth is estimated to be 30.01 GHz. For comparison, the chaos induced by CW optical injection into the DFB laser at (ξ_i, f_i) = (0.23, 9 GHz) is displayed as the red curve in Figure 2d, where the relaxation resonance frequency of the free-running DFB laser dominates the component and rapidly decreases to the noise floor (gray curve) at around 25 GHz. The chaotic bandwidth is estimated to be 11.7 GHz, which is the broadest one achieved in our experiment by using the CW optical injection. In fact, similar results can also be obtained in the other three channels (see Supplementary Note I, Figure S2). It should be noted that such broadband optical chaos can be achieved in wide parameter regions by using the intensity-modulated optical injection scheme as demonstrated in our previous work; (27) moreover, the chaotic bandwidth achieved in the current work can be further promoted by properly manipulating the injection parameters and utilizing PD and ESA devices with a wider bandwidth.

Subsequently, the noise-like output signals are gathered using an 8-bit real-time OSC, and the intensity series (blue curve) with asymmetry probability density distribution is shown in Figure 2f. To confirm the dynamic state of the laser, we calculate the largest Lyapunov exponent (28) using the experimental data, and the results are tabulated in Supplementary Note I, Table S2. The generation of chaotic signals in our system is confirmed by the strictly positive LLEs observed across all channel outputs. Furthermore, the electronic noise (red curve) shown in Figure 2f has an amplitude at least 2 orders of magnitude smaller than the optical chaotic waveform. The complexity/dimension of chaotic output is evaluated using the correlation dimension (CD) via the Grassberger–Procaccia algorithm. (29−32) This algorithm calculates the correlation integral C(r), representing the likelihood of point pairs with Euclidean distances less than or equal to r in the phase space of delayed embedding. The CD is estimated from the slope of convergence in the logarithmic plot of C(r) against r (more details are presented in Supplementary Note I). It can be seen from Figure 3h,i that the intensity-modulated optical injection system can produce a higher chaotic dimension (CD = 4.1826) compared to the CW optical injection system (CD = 3.3567). We conclude that the periodic intensity modulation introduces additional nonlinearities, which enhance the dimension of the generated optical chaos.

The randomness of chaotic lasers stems from their extreme sensitivity to initial conditions and noise, which renders the system evolution unpredictable. (34−37) In this study, the modulation of an external optical signal induces complex dynamical behaviors in the laser output, thereby amplifying the inherent system noise and enhancing the randomness and unpredictability of the output. This characteristic randomness guarantees chaotic lasers as a desired source for RBG, suitable for various application domains. Benefiting from the large chaotic bandwidth, the intensity time series can support a sampling rate of 80 GS/s. As shown in Figure 2f, the intensity distribution of laser output (CH4) displays an asymmetric profile. We know that any RBG based on a realistic physical process inevitably suffers from technical imperfections, resulting in a nonideal distribution of sampled data. This, in turn, can give rise to statistical biases and correlations in the generated bit sequence. To realize faster RBG, multibit extraction methods have been proposed and demonstrated. (38) The physical entropy source is sampled through multibit analog-to-digital conversion (ADC) and the RBG rate can be promoted to the order of magnitude of Tb/s, (38−41) after adopting various advanced postprocessing methods, which can be summarized into two categories. One is guided by information-theoretic considerations. The RBG rate within the information limit L is described by the following relation (38) where S is the sampling rate of the ADC, V is the number of significant bits (e.g., quantified by Shannon entropy and min-entropy), and D is the channel number of chaotic signals. The other one is based on pragmatic considerations, where V usually exceeds the Shannon entropy and min-entropy using several postprocessing methods, such as high-order derivatives, (42) high-order finite differences, (41,43,44) time-shifted bit-order reverse exclusive-OR (XOR) operation. (40)

In this work, we adopt two methods to process the raw data in Figure 3a. For method-i, the number of retained bits is selected conservatively based on the constraints imposed by information theory. (41) The min-entropy of our experimental data is estimated to be 5.2, which indicates that extracted random bits from each raw 8-bit resolution sample cannot be larger than 5.2 (see Supplementary Note II). Specifically, a delay difference is employed to achieve a symmetric distribution as illustrated in Figure 3b, which is beneficial for unbiased bit extraction. The differential data is then digitalized to 8 bits and the 5 LSBs are retained for RBG. A uniform and flat probability distribution is observed in Figure 3c, which indicates that the generated bit sequence contains different bit patterns with nearly equal frequencies. Last but not least, the bit-wise XOR operation is used to enhance the inherent randomness and thus further eliminate residual correlations. (42,45)

To evaluate the randomness of the long random-bit sequences, the statistical bias B and the serial autocorrelation coefficients C_k are calculated as shown in Figure 3d,e. B and C_k are defined as (41,43) where b_i (i = 1,2, . . .) denotes the values “0” or “1”, k is the delay in bits, and the averaging ⟨·⟩ is executed over the index i. In a random bit sequence with a high degree of quality and a length of N, both measurement of statistical deviation B and the magnitude of the serial autocorrelation coefficients |C_k|, should remain below the threshold defined by three standard deviations: 3σ_B = 1.5N^–0.5 for B and 3σ_B = 3N^–0.5 for C_k with a probability of 99.7%. In Figure 3d, we plot the dependence of the statistical bias B on the bit sequence length N in a log–log scale. It is evident that the statistical bias consistently remains beneath the threshold of the 3σ_B criterion (indicated by the red dashed line). Figure 3e displays the 200 autocorrelation coefficients for a bit sequence with a length of 10 G bits. One can find all values C_k are within the range of the 3σ_C-criterion (indicated by red dashed lines). Consequently, the random bit sequence generated from method-i well satisfies both criteria.

The randomness of the bit sequence for both method-i and method-ii is tested using the benchmark National Institute of Standards and Technology (NIST) 800-22, (46) and the results are shown in Figure 3j,k. Here, each test is evaluated using 1000 samples of 1 Mbit sequences with a significance level of α_H = 0.01. One can see all 15 NIST tests fulfill a pass rate greater than 98.05608% and a P-value larger than 0.0001 (red), which indicates good statistical randomness of the generated bits from the parallel chaos by using the method-i.

Let us now consider the randomness extraction using method-ii, in which the number of extracted random bits is merely chosen to pass standard randomness tests. First, the calculation of high-order finite differences is performed, where all sampled data are transformed from an 8-bit resolution into a 52-bit resolution integer data type (41,43) (more details are described in Supplementary Note II). The 52nd-order finite differences are used to symmetrize the initial probability density distribution [see Figure 3f]. Next, the randomness extraction is examined.

Although all numbers consist of 52 bits, it is not possible to extract all significant bits from each number because retaining all bits leads to the emergence of strong correlations in the generated bit sequences. To eliminate these correlations, several most significant bits (MSBs) need to be discarded. We find that the extraction of 47 LSBs from each sample offers the whole bits with equal frequency [see Figure 3g] and the resultant bit sequence can pass all the NIST statistical tests for randomness [see Supplementary Note II, Figure S4b]. The B and the C_k of the random bit sequences extracted from 47 LSBs are tested, and the results are shown in Figure 3h,i. It can be seen that the B of the random bit sequence always keeps below the significance level of 3σ_B up to 50 G bits, while all C_k are kept within the range of 3σ_C-criterion. Moreover, the ACF of the generated bit sequence for both postprocessing methods shows correlations below the lower limit $1/\sqrt{N}$ (see Supplementary Note II, Figure S5).

To investigate the effect of photodetection noise on RBG, we have also disconnected the light source and collected the noise signal at a sampling rate of 80 GS/s, and the corresponding time series are plotted in Figure 2f (red). After undergoing identical postprocessing procedures, the generated bit sequence fails to pass the NIST test even for only extracting 1 LSB from each noise sample sequences. This indicates that the level of the detection noise does not affect the random bit quality. Eventually, we obtain a physical random bit sequence with verified randomness of 1.6 Tb/s (80 GS/s × 5 bit × 4 channels) by extracting 5 LSBs at a sampling rate of 80 GS/s. By extracting 47 LSBs using the method-ii, the RBG rate can be promoted to 15.04 Tb/s (80 GS/s × 47 bit × 4 channels). It should be noted that since the extracted bits from each sample exceed the minimal entropy threshold, i.e., deviating from information theory, the random bits generated by method-ii are termed physical-based pseudo RBG (a notion proposed in our previous work (41) and supported in a review (34)).

In Table 1, we compare our work with other parallel RBGs using optical chaos. In the context of bandwidth calculations, superscript “a” denotes the utilization of the 80% standard bandwidth definition, while superscript “b” signifies the application of the 10 dB bandwidth method. The sampling rate of the recorded data is a primary factor affecting the rate of RBG. Large chaotic bandwidth supports higher sampling rates while maintaining a low correlation between adjacent samples. (38) For example, a low correlation is ensured by down-sampling the raw data to 6.67 GS/s due to a limited chaotic bandwidth of a few GHz, (13) markedly reducing the RBG rate by over an order of magnitude. On the contrary, our scheme provides a viable solution to generate a chaotic signal with a continuous spectral distribution of more than 44 GHz. This ensures a high sampling rate while keeping uncorrelation between adjacent samples. Limited the bandwidth of detection devices, we demonstrate a chaotic signal with a bandwidth of only 30.01 GHz, which is slightly lower than that of the three-cascaded semiconductor laser scheme. (38) The integrated laser array enables our scheme to be a promising miniaturization physical entropy source.

The MAB problem involves optimizing decisions in uncertain environments, balancing exploration of new options with exploitation of known rewards to maximize cumulative gains. (53) Optical chaos, characterized by its inherent randomness and unpredictability, offers a powerful tool for this challenge. By leveraging optical systems to generate random-like signals, it is possible to enhance decision-making processes in MAB problems, effectively navigating the exploration-exploitation trade-off. Here, we try to address the intricate MAB problem by utilizing a specially fabricated laser array to concurrently generate parallel wideband chaotic signals. Figure 4 illustrates a schematic diagram of the optical decision-making process. For an M-armed bandit problem, where M = 2^k with k being a natural number. By binary encoding the slot machines, each binary sequence corresponds to a specific slot machine, producing outputs [D₁, D₂, D₃, D₄]. We adopt a modified strategy to perform the parallel optical decision-making process illustrated as follows: (20) A laser array chip simultaneously produces four chaotic signals I_i(t) (i = 1 to 4) and is compared with the threshold values TH₁, TH₂, TH₃, and TH₄ of each channel. A decision is derived from the comparison results, i.e. if I_i ≤ TH_i, D_i = 0, else D_i = 1. Specifically, assume that the four-channel chaotic signals sampling at t₁ are I₁(t₁), I₂(t₁), I₃(t₁), and I₄(t₁), and then they are separately compared with the threshold values TH₁, TH₂, TH₃, and TH₄. If I₁ ≤ TH₁, the MSB is confirmed as D₁ = 0; if I₂ ≤ TH₂, the second-most significant bit is D₂ = 0; if I₃ ≤ TH₃, the third-most significant bit is determined as D₃ = 0; if I₄ ≤ TH₄, the LSB is D₄ = 0. Consequently, slot machine 1, indicated by D = [0, 0, 0, 0], is selected. If a reward is achieved from slot machine 1, the threshold values are adjusted to make a similar decision more likely in the next cycle. Conversely, if no reward is received, the threshold values are modified to decrease the likelihood of making the same choice again (see Supplementary Note III).

In the experiment, we address the 16-arm bandit problem by utilizing four-channel chaotic signals generated by the on-chip DFB laser array, which are standardized and normalized. The probabilities of winning for each of the 16 arms are set as follows: P = [0.2, 0.1, 0.4, 0.5, 0.1, 0.4, 0.4, 0.9, 0.3, 0.4, 0.2, 0.4, 0.2, 0.1, 0.4, 0.2]. As depicted in Figure 5a, the decision maker begins by exploring a wide range of options to identify the slot machine with the highest probability of success. After sufficient exploration, slot machine 8 consistently emerges as the preferred choice with the highest likelihood of winning prizes, indicating highly successful decision-making. To provide a more intuitive assessment of decision performance, we define the total number of selections as T, the number of correct selections as T_hit, and the correct decision ratio (CDR) as T_hit/T. In this work, a CDR exceeding 0.9 is regarded as successful decision-making. The period during which CDR first exceeds 0.9 is termed the convergence cycle y. To demonstrate the superior performance of parallel chaotic signals in optical decision-making, we compare the processing outcomes between four-channel and single-channel chaotic signals in Figure 5b. Within the same processing system, the four-channel chaotic signals required significantly fewer iterations to reach the convergence cycle [pink dashed line in Figure 5b] compared to single-channel chaotic signals. This observation highlights the potential for widespread utilization of parallel chaotic signals in solving the MAB problem.

Scalability is a pivotal aspect when evaluating decision performance. The study on the capacity of parallel chaotic signal processing to handle large-scale slot machines begins with a comparison of the MAB problems for M = 2, 4, 8, and 16, as depicted in Figure 5c. The CDR grows exponentially as the number of slot machines increases. However, successful decisions can still be made irrespective of the size increment, indicating the suitability of parallel chaotic signals for solving large-scale MAB problems.

We explore the accuracy of decision-making in a dynamically changing environment. That is, the higher reward probability of the slot machine changes over time. Figure 5d displays the evolution of the CDR in a variable environment, where the target slot machine varies randomly from 3 to 10 at the 1000th cycle. For the cases of four-channel and single-channel, the probabilities of winning for each of the 16 arms are set as P = [0.8, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2]. From this figure, one can see that after the abrupt change in the target machine, the CDR first drops to zero and then rapidly increases. Moreover, four-channel parallel decision-making also demonstrates a shorter time to reach 0.9 compared to single-channel decision-making even if the reward probability is suddenly converted. To reveal the learning process behind the decision-making, we examine the changes in threshold values corresponding to environmental variations, as shown in Figure 5e. In the first 1000 cycles, the threshold values TH₁ and TH₂ rise until they eventually fluctuate around a maximum value of 0.5, while the threshold values TH₃ and TH₄ decrease until they eventually stabilize, fluctuating around a minimum value of −0.5. Following the rules mentioned above, the target slot machine is determined as [0, 0, 1, 1]. When the target slot machine changes unexpectedly at 1001 cycle, after a temporary fluctuation around 0, the values of TH₁ and TH₃ are decreased to about −0.5, and the values of TH₂ and TH₄ are increased to about 0.5. Under this scenario, the I₁(t) and I₃(t) are larger than the TH₁ and TH₃, while the I₂(t) and I₄(t) are smaller than the TH₂ and TH₄. Consequently, the slot machine [1, 0, 1, 0] is most likely to be rewarded, which is consistent with the preoptimized slot machine settings.

Considering the on-chip laser array can be further expanded, (48) we numerically simulate the 8-channel laser array to solve the MAB problem. Figure 5f presents a comparison of the convergence cycle between parallel chaotic decision-making and other methods. (13) By fitting a power function, we derive the relationship between the convergence cycle y and the number of slot machines M as y = 19.48M^0.96. In comparison to the upper confidence bound 1 (UCB1)-tuned algorithm and the Thompson sampling algorithm, the method based on optical chaos demonstrates a smaller scaling index, making it well-suited for addressing large-scale MAB problems. Moreover, our approach, by utilizing the calculation of Shannon entropy, demonstrates a faster convergence rate compared to UCB1-tuned (see Supplementary Note III). We speculate that our proposed scheme can generate broadband optical chaos (i.e., chaotic bandwidth is greater than 30 GHz) characterized by rapid fluctuations and high-frequency oscillations in each channel, and these features enable the system to conduct extensive exploration and selection in an extremely short period, thereby accelerating the process of identifying the optimal solution compared with other methods. Additionally, compared to traditional chaotic systems, integrated laser chip technology offers a compact, cost-effective, and scalable mode of decision-making. The ongoing expansion of chip capacity also opens up new possibilities for the scalability of decision-making systems.