Theory of fast nondeterministic physical random-bit generation with chaotic lasers

We theoretically show that completely stochastic fast physical random bit generation at a rate of more than one gigabit per second can be realized by using lasers with optical delayed feedback which creates high-dimensional chaos of laser light outputs. The theory is based on the mixing property of chaos, which transduces microscopic quantum noise of spontaneous emission in lasers into random transitions between discrete macroscopic states.

Figure 1

Schematic illustration of a physical device producing sequences of discrete macroscopic states a, b, c, and d appearing randomly. Microscopic noises act as entropy sources.

Figure 2

Schematic illustration of the growth of entropy during the convergence of an initial microscopic noise distribution to the natural invariant density, due to the mixing property Eq. (4).

Figure 3

Numerical calculation of the time dependence of the entropy of the bit obtained by using the $r$-adic map with $r=2$. The entropy actually approaches 1 much faster than the theoretical prediction.

Figure 4

The time evolution of the probability density distribution of the light intensity and the carrier density at times (a) 0.1 ns, (b) 0.13 ns, (c) 0.16 ns, (d) 0.22 ns, (e) 0.40 ns, (f) 0.86 ns, (g) 1.00 ns, and (h) 1.25 ns.

Figure 5

The invariant probability density of the light intensity (solid curve) and the time-evolving probability density of the light intensity after 1.25 ns (crosses).

Figure 6

The difference between the average transient probability density and the invariant probability density (solid curve). The average transient probability density was averaged over the transient probability densities obtained for 10 different randomly chosen initial states of $N$ and $I$ on the chaotic attractor. The transient probability density for each initial condition was obtained using ${10}^5$ different microscopic noise trajectories. The dotted line represents $\exp (-5.3t)$.

Figure 7

The autocorrelation function of the time evolution of the light intensity. The dotted line represents $\exp (-5.3t)$.

Figure 8

The “vanishing time” of the autocorrelation functions. The colors indicate the time (ns) required for the exponentially decaying envelope of the autocorrelation function to become less than 0.1 calculated numerically with the spontaneous emission factor $C_s={10}^{-5}$. (a) Phase diagram of time delay and feedback strength for a fixed phase $\theta =0$. (b) Phase diagram of the delay phase shift and feedback strength for a fixed time delay ${\tau }_D=0.182$ ns.

Figure 9

The rf spectra of the chaos laser with delayed optical feedback (red curve) and a solitary laser (green curve). The spontaneous emission factor $C_s$ is ${10}^{-3}$ for the chaos laser and ${10}^{-5}$ for the solitary laser in correspondence with laser experiments (see text).

Figure 10

An example of the time series and the bit sequences obtained by numerically solving the Lang-Kobayashi Eqs. (10) and (11). Temporal waveforms of the light intensity, and corresponding binary digitized signals 10110001$\cdots$. The dots mark points sampled with a 0.8 GHz sampling rate. The broken line represents the threshold value for digitizing the light intensity. The random bit sequences are generated after the XOR operation combining two bit sequences extracted from two different time series starting from different initial states.

Figure 11

The entropy averaged over ${10}^5$ random bit sequences generated by sampling the time series numerically produced by the Lang-Kobayashi model with the clock times corresponding to the horizontal axis.

Figure 12

The entropy averaged over ${10}^5$ random bit sequences obtained by a logical XOR operation on pairs of bits generated by sampling the different time series produced numerically from the Lang-Kobayashi model with clock times corresponding to the horizontal axis.