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Randomness is essential ingredient for applications in cryptography, stochastic simulation, and fundamental science experiments [American Scientist 89, 4 (2001); PRL 115, 25 (2015)]. Common practice is to generate random numbers from mathematical algorithm using Pseudo-random number generators. We are interested in tackling this problem with quantum technology. Phase diffusion in spontaneous emission events is a quantum phenomena with inherent randomness [Opt. Express 19, 21 (2011)]. Implementations of this scheme using pulsed lasers can yield high-speed quantum random number generation (QRNG) [Opt. Express 22, 2 (2014)]. The general interest in the laser phase diffusion QRNG setup has been mainly focused on and motivated by the speed of the random number generations. Little has been stated about the performance of quantum phase noise as a randomness source in QRNG, from the perspective of the physics involved.
We reanalyze the process of phase diffusion based QRNG and give an intuitive explaining picture of the underlying physics. Our findings show that a pulsed process is beneficial over the continuous-wave approach and give a upper bound of maximum random bit rate for a given experimental setting. In detail, the output of a semiconductor laser contains fluctuations in intensity and phase. This was originally studied first by Charles Henry in 1982 [J. Lightwave Techn. 4, 3 (1986)]. Based on that one can identify two phenomena contributing to the phase noise, which takes the form of a random walk. First, there will be phase changes due to the carrier-induced change in refractive index in the semiconductor laser. Second, spontaneous emission events take place in the active medium [Opt. Express 19, 21 (2011)]. Naturally, taking spontaneous emission events into account necessitates the application of stochastic laser rate equations for theoretical description. We solve these equations numerically be a Monte-Carlo simulation and investigate several random walk scenarios. One of our essential findings is that the phase fluctuation becomes much larger in the pulsed laser regime than in the continuous-wave mode. Additionally, there is a natural limit on the maximum pulse repetition rate for a given pulse width, since the phase jumps become smaller for shorter pulse distances.
Since phase measurement is not a feasible procedure for optical signals, the phase fluctuation needs to be converted to an observable macroscopic parameter such as the optical intensity. This can be done by using an interferometer setup. The whole experimental setting is shown in Fig. 2 (left). An DFB laser diode is modulated using gain-switching technique to generate uniform amplitude signals with random phases. Each signal interferes with neighboring pulses through an unbalanced Mach-Zehnder interferometer. The delay in one arm was set to equal to the pulse repetition rate.
With this setup we investigate the influence of several parameters on the shape of the probability distribution determining the quality of QRNG. The theoretical as well as experimental findings can help to find physical standards for QRNG verification rather than the ones based on classical statistical information theory.
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