^{1,a)}, Takahisa Harayama

^{1,2}, Peter Davis

^{1}, Ken Tsuzuki

^{3}, Ken-ichi Arai

^{1}, Kazuyuki Yoshimura

^{1}and Atsushi Uchida

^{4}

### Abstract

We present an experimental method for directly observing the amplification of microscopic intrinsic noise in a high-dimensional chaotic laser system, a laser with delayed feedback. In the experiment, the chaotic laser system is repeatedly switched from a stable lasing state to a chaotic state, and the time evolution of an ensemble of chaotic states starting from the same initial state is measured. It is experimentally demonstrated that intrinsic noises amplified by the chaotic dynamics are transformed into macroscopic fluctuating signals, and the probability density of the output light intensity actually converges to a natural invariant probability density in a strongly chaotic regime. Moreover, with the experimental method, we discuss the application of the chaotic laser systems to physical random bit generators. It is experimentally shown that the convergence to the invariant density plays an important role in nondeterministic random bit generation, which could be desirable for future ultimate secure communication systems.

In real physical systems, there always exists microscopic noise, and the system is randomly perturbed. If the deterministic part of its system is strongly chaotic, an initial uncertainty due to the noise is rapidly amplified by the dynamics so that macroscopic observables will be unpredictable after long time. In such dynamical systems, an invariant probability distribution is fundamental and it plays an important role in characterizing the long time behaviors and the statistical properties of the systems.

^{1}The convergence rate to the invariant distribution is also an important feature, and it is deeply related to statistical properties of nonequilibrium systems.

^{2–5}So far, there have been many mathematical and numerical studies on the fundamental statistical properties of strongly chaotic dynamical systems. However, the experimental investigation of the statistical properties has not yet been done in detail on the basis of the direct observation of a noise amplification effect in strongly chaotic systems. In this article, we experimentally study the statistical properties of the noise amplification in a laser system with delayed feedback, by actually observing the convergence to an invariant probability distribution of the observable. Moreover, the application of the noise amplification effect by the chaotic dynamics to nondeterministic random bit generation is discussed. It is experimentally shown that nondeterministic random bit generation is possible if the sampling time interval of the observable is much longer than the time needed for the convergence to the natural invariant probability distribution.

I. INTRODUCTION

II. CHAOTIC LASERS WITH DELAYED FEEDBACK

III. EXPERIMENTAL METHOD

IV. EXPERIMENTAL RESULTS

V. APPLICATION TO NONDETERMINISTIC RANDOM BIT GENERATION

VI. SUMMARY

### Key Topics

- Probability theory
- 34.0
- Chaotic systems
- 24.0
- Chaos
- 22.0
- Chaotic dynamics
- 11.0
- Entropy
- 11.0

##### H01S5/00

## Figures

A schematic of a laser with delayed optical feedback.

A schematic of a laser with delayed optical feedback.

A schematic of the photonic integrated device that we study in this work.^{19} It consists of a single frequency DFB, two semiconductor optical amplifiers (SOA1 and SOA2), a photodiode (PD) and a passive waveguide. HR-coating: high reflection coating for optical feedback, AR-coating; anti-reflection coating for avoiding extra optical feedback.

A schematic of the photonic integrated device that we study in this work.^{19} It consists of a single frequency DFB, two semiconductor optical amplifiers (SOA1 and SOA2), a photodiode (PD) and a passive waveguide. HR-coating: high reflection coating for optical feedback, AR-coating; anti-reflection coating for avoiding extra optical feedback.

The power spectra and the absolute values of the autocorrelation for the injection current to SOA1 mA (a,b), 10 mA (c,d), and 15 mA (e,f). In these examples, the injection currents to SOA2 and DFB are fixed at 5 mA and 35 mA, respectively. For a comparison of the signal amplitudes, the power spectrum of the laser noise observed for mA is shown in figure (a). Dotted lines represent (b), (d), and (f).

The power spectra and the absolute values of the autocorrelation for the injection current to SOA1 mA (a,b), 10 mA (c,d), and 15 mA (e,f). In these examples, the injection currents to SOA2 and DFB are fixed at 5 mA and 35 mA, respectively. For a comparison of the signal amplitudes, the power spectrum of the laser noise observed for mA is shown in figure (a). Dotted lines represent (b), (d), and (f).

A schematic of experimental setup for measuring the effect of noise amplification in a laser device with delayed optical feedback. OSC; digital oscilloscope (50 GS/s, bandwidth 12.5 GHz).

A schematic of experimental setup for measuring the effect of noise amplification in a laser device with delayed optical feedback. OSC; digital oscilloscope (50 GS/s, bandwidth 12.5 GHz).

Time trace of the chaotic light intensity. The upper panel represents the pulse current. In this example, the low level and high level of the pulse are respectively 0 mA and 5 mA.

Time trace of the chaotic light intensity. The upper panel represents the pulse current. In this example, the low level and high level of the pulse are respectively 0 mA and 5 mA.

Four temporal waveforms of chaotic intensity starting from an initial state. The injection current is applied to SOA1 at time . The values of the injection current are 5 mA (a), 10 mA (b), 15 mA (c).

Four temporal waveforms of chaotic intensity starting from an initial state. The injection current is applied to SOA1 at time . The values of the injection current are 5 mA (a), 10 mA (b), 15 mA (c).

Time dependence of the probability density of the chaotic light intensity signals for . The dotted curve represents the probability density obtained from a single long chaotic time series in the case of the same injection current value.

Time dependence of the probability density of the chaotic light intensity signals for . The dotted curve represents the probability density obtained from a single long chaotic time series in the case of the same injection current value.

The time dependence of the deviation from the invariant probability density for *J* = 5 mA (a), 10 mA (b), 15 mA (c). Dotted lines represent (a), (b), and (c).

The time dependence of the deviation from the invariant probability density for *J* = 5 mA (a), 10 mA (b), 15 mA (c). Dotted lines represent (a), (b), and (c).

Time dependence of the Shannon's entropy.

Time dependence of the Shannon's entropy.

Time dependence of the Shannon's entropy of the XOR-ed bit sequences.

Time dependence of the Shannon's entropy of the XOR-ed bit sequences.

## Tables

Results of NIST Special Publication 800-22(rev. 1a) statistical tests. For each test, 100 samples of 1 Mbit sequences and significance level were used. “Success” denotes that the P-value of the uniformity of p-value is larger than 0.0001, and that the proportion of the bit sequences satisfying p-value for 100 samples is larger than 0.960150377. For the tests which produce multiple P-values and proportions, the worst case is shown.

Results of NIST Special Publication 800-22(rev. 1a) statistical tests. For each test, 100 samples of 1 Mbit sequences and significance level were used. “Success” denotes that the P-value of the uniformity of p-value is larger than 0.0001, and that the proportion of the bit sequences satisfying p-value for 100 samples is larger than 0.960150377. For the tests which produce multiple P-values and proportions, the worst case is shown.

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