# Precise Photon Correlation Measurement of a Chaotic Laser
## 1 Introduction  
Semiconductor lasers subject to external optical feedback exhibit a rich variety of nonlinear dynamical behaviors and are used to generate high-dimensional chaotic lasers [1; 2]. This configuration has attracted great interest for a wide range of applications, like optical chaos communication [3; 4; 5; 6; 7], secure key distribution [8], high-speed physical random number generation [9; 10; 11; 12], chaos-based optical computing [13] and sensing [14; 15; 16]. It is fundamentally important to understand the underlying physical mechanisms of the chaotic laser, and practically useful to improve the laser performance and motivate its applications. Previous research mainly focused on clarifying intensity statistics and autocorrelation (AC) of chaotic lasers to characterize chaotic processes [17; 18; 19; 20; 21]. Intensity statistics are closely relevant to the extractable rate of random numbers [9; 10; 22] and the AC is a good indicator of a chaotic modulating bandwidth in optical chaos communications [4; 19]. However, the macro-scale intensity statistics and AC are not sufficient to reveal all properties of a given chaotic laser, and there is also a significant discrepancy between experimental and theoretical probability density distributions of the laser intensity [19]. Recent research reveals that quantum correlation is more accurate in assessing statistical properties and more sensitive to control parameters compared to the AC function [23; 24]. However, the previous research is concentrated on the properties of the quantum dot laser in the low-intensity (low-gain) situation, and the bunching effect of the chaotic laser, i.e., \(g^{(2)}(0)>1\) at \(\tau=0\), is only revealed in the fully developed chaotic (high-gain) regime. Studies on high order photoncorrelation of high-dimensional chaotic lasers are sparse, especially second order photon correlation \(g^{(2)}(\tau)\) at non-zero delay time. The landmark experiment on photon correlation was first conducted by [PERSON] (HBT), demonstrating spatial second order photon correlation \(g^{(2)}\) of a thermal light [25]. Soon afterwards, this experiment inspired [PERSON]'s seminal work on quantum optics theory, which described the photon correlation of different light fields by correlation functions of quantum statistics [26; 27; 28]. The photon correlation \(g^{(2)}\) is fundamentally different from the first order correlation and is harnessed in many applications, such as photon bunching and anti-bunching measurement [29; 30; 31; 32], spatial interference [33; 34], ghost imaging [35; 36; 37], the azimuthal HBT effect [38], single photon detection [39], etc. The \(g^{(2)}(\tau)\) also carries a wealth of information on the statistical probability of different photons arriving at time delay \(\tau\)[40]. Up until now, there are many approaches to obtain photon correlation \(g^{(2)}(\tau)\), typically including two-photon absorption (TPA) measurement [41; 42], photon coincidence counting [43], time interval measurement of photon pairs [44]. Recently, the HBT experiment was explored to observe chaos from quantum dot lasers with external feedback [23]. However, research on photon correlation \(g^{(2)}(\tau)\) of high-dimensional chaotic waveforms is rare and there is still an obvious disagreement between experimental and theoretical \(g^{(2)}(\tau)\). The calculation of \(g^{(2)}(\tau)\) from the photon pair time interval distribution provides a good way to measure the photon correlation of pseudo-thermal light with microsecond coherence time [44]. But for a chaotic laser, the coherence time is much shorter than that of pseudo-thermal light, and the resolution time must be shorter than the coherence time of the laser in the measurement. Although the shorter coherence time does not affect the bunching effect or \(g^{(2)}(0)\) of the chaotic laser [24], that makes the measurement of \(g^{(2)}(\tau)\) (\(\tau\
eq 0\)) at very short timescales using the HBT technique more difficult, owing to the limited response time of single-photon detectors [45].