Phase randomness in a semiconductor laser: Issue of quantum random-number generation

Gain-switched lasers are in demand in numerous quantum applications, particularly in systems of quantum key distribution and in various optical quantum random number generators. The reason for this popularity is natural phase randomization between gain-switched laser pulses. The idea of such randomization has become so familiar that most authors use it without regard to the features of the laser operation mode they use. However, at high repetition rates of laser pulses or when pulses are generated at a bias current close to the threshold, the phase randomization condition may be violated. This paper describes theoretical and experimental methods for estimating the degree of phase randomization in a gain-switched laser. We consider in detail different situations of laser pulse interference and show that the interference signal remains quantum in nature even in the presence of classical phase drift in the interferometer provided that the phase diffusion in a laser is efficient enough. Moreover, we formulate the relationship between the previously introduced quantum reduction factor and the leftover hash lemma. Using this relationship, we develop a method to estimate the quantum noise contribution to the interference signal in the presence of phase correlations. Finally, we introduce a simple experimental method based on the analysis of statistical interference fringes, providing more detailed information about the probabilistic properties of laser pulse interference.

Figure 1

Probability density functions ${\tilde{f}}_{\tilde{S}}^Q$ defined by Eq. (29) at different values of ${\sigma }_{\phi }$ and $\mathrm{\Delta }\theta$.

Figure 2

Calculated dependences ${\tilde{S}}_{\mathrm{th}}\left({\sigma }_{\phi }\right)$ for the three values of $\mathrm{\Delta }\theta$: 0, $\pi /2$, and $\pi$.

Figure 3

The dependence of the statistical distance $d$ defined by Eq. (35) on ${\sigma }_{\phi }$ and $\mathrm{\Delta }\theta$ in the form of a color map (a). The slices of the map at different values of $\mathrm{\Delta }\theta$ are shown in (b).

Figure 4

Theoretical dependences of the phase diffusion standard deviation (${\sigma }_{\phi }$) on the bias current $I_b$ and modulation current amplitude $I_p$ at different pulse repetition rates.

Figure 5

Schematic representation of the experimental setup to measure phase diffusion in a gain-switched laser.

Figure 6

Theoretical dependence of the probability density function of the normalized interference signal $\tilde{S}$ [Eq. (16)] on the standard deviation of phase fluctuations ${\sigma }_{\phi }$ (a) and the corresponding dependence of its standard deviation ${\sigma }_{\tilde{S}}$ (b).

Figure 7

Experimental dependence of the probability density function of the normalized interference signal $\tilde{S}$ on the bias current $I_b^{\exp }$ (on the left) and the corresponding dependence of its standard deviation ${\sigma }_{\tilde{S}}$ (on the right) at pulse repetition frequency $f_p=1.25\mathrm{GHz}$.

Figure 8

Experimental dependence of the probability density function of the normalized interference signal $\tilde{S}$ on the bias current $I_b^{\exp }$ (a) and the corresponding dependence of its standard deviation ${\sigma }_{\tilde{S}}$ (b) at pulse repetition frequency $f_p=2.5\mathrm{GHz}$.

Figure 9

An example of measured probability density function of the laser pulse interference as a function of the interferometer temperature. Pulse repetition rate was 1.25 GHz; the bias current was $I_b^{\exp }=54$ mA. The temperature shift $\mathrm{\Delta }\tau$ was measured with respect to 300 K.

Figure 10

Experimental dependences of the mean value $\langle \tilde{S}\rangle$ of the interference signal on the phase shift $\mathrm{\Delta }\theta$ (filled circles) and corresponding statistical interference fringes ${\sigma }_{\tilde{S}}\left(\mathrm{\Delta }\theta \right)$ (empty circles) at pulse repetition frequency of 1.25 GHz. In all panels one division on the $\mathrm{\Delta }\theta$ axis equals to $\pi$.

Figure 11

Experimental dependences of the mean value $\langle \tilde{S}\rangle$ of the interference signal on the phase shift $\mathrm{\Delta }\theta$ (filled circles) and corresponding statistical interference fringes ${\sigma }_{\tilde{S}}\left(\mathrm{\Delta }\theta \right)$ (empty circles) at pulse repetition frequency of 2.5 GHz. In all panels one division on the $\mathrm{\Delta }\theta$ axis equals to $\pi$.

Figure 12

Experimentally measured standard deviations of phase diffusion (${\sigma }_{\phi }$) and normalized pulse intensity fluctuations (${\sigma }_s$) as functions of the bias current at 1.25 and 2.5 GHz of pulse repetition rate. Orange asterisks denote values of ${\sigma }_s$ that were used when fitting ${\sigma }_{\tilde{S}}$ fringes at $I_b^{\exp }<47$ mA (for 1.25 GHz) and $I_b^{\exp }<33$ mA (for 2.5 GHz).

Figure 13

Schematic representation of a cascade of interferometers on the chip we used in the experiment (on the top) and the output of the system for various configurations of phases ${\phi }_c^k$ in assumption that a single laser pulse of the Gaussian shape with the rms width of 20 ps arrives at the input of the interferometer.

Figure 14

Theoretical statistical fringes at various values of ${\sigma }_{\phi }$ and ${\sigma }_s$.