Chaotic lasers have been widely used in high-speed secure communication, physical random key generation and distribution, sensing measurement, optical computation, and so on because of their wide bandwidth, large noise-like fluctuation, super sensitive dependence on initial values, and long-term unpredictability. With rapid development of these applications, a more accurate characterization of chaotic lasers is needed. At present, chaotic lasers are mainly evaluated through an analysis of their dynamic properties in time and frequency domains. The quantum statistics of chaotic lasers are also investigated through high-order coherence and photon number distribution. However, the phase-space quasi-probability distributions of chaotic lasers remain to be further studied. The phase-space Wigner quasi-probability distributions of quantum states, such as the squeezed state, Schr?dinger cat state, single-photon state, and multi-photon Fock state, have been reconstructed experimentally. However, system losses and the influence of noise in a practical experiment should be considered, and the high-fidelity measurement of the Wigner quasi-probability distribution in a phase space of chaotic lasers still needs to be further studied.

In this work, the phase-space Wigner quasi-probability distributions of chaotic lasers in the quasiperiodic, moderate, and coherent collapse states are reconstructed experimentally through balanced homodyne quantum tomography and the maximum likelihood method. First, by controlling the bias current and optical feedback strength, the quasiperiodic, moderate, and coherence collapse chaotic lasers with different bandwidths are prepared. Then, the chaotic lasers in these three different states are used as signal light and allowed to interfere with a local oscillator beam. Finally, the beams enter the balanced homodyne detector after 50:50 beam splitting. The local oscillator phase is scanned by piezoelectric ceramic transducer to obtain the amplitude quadrature at all phase angles. Based on the measured quadrature results of the chaotic lasers, the Wigner quasi-probability distributions in phase space and the density matrices of the chaotic lasers are reconstructed using the maximum likelihood method.

From the quasiperiodic state to coherence collapse, the auto-correlation of chaotic lasers changes from multi-periodic oscillation to periodic weakening chaos. The peak values of the time-delay signature decrease from 0.567 to 0.213, and the chaotic intensity and bandwidth increase continuously (Fig. 2). With an increase in the chaotic bandwidth and intensity, the measured phase-space Wigner quasi-probability distributions of the chaotic lasers are magnified by 1.5-3.0 times compared to the shot noise limit, and this chaotic amplification effect is continuously enhanced (Figs. 3 and 4). After removing the -44 dBm background noise, the fidelity of the Wigner quasi-probability distributions of the chaotic lasers is improved from 74.9%, 81.2%, and 84.8% to 79.9%, 86.8%, and 90.4%, respectively (Fig. 5). Then, after compensating for the 8.9% loss of the experimental system, the optimal fidelity of the reconstructed Wigner quasi-probability distribution function reaches up to 97.6%, realizing the high-fidelity phase-space reconstruction of chaotic lasers (Fig. 6).

In summary, the phase-space Wigner quasi-probability distributions of chaotic lasers in different states are reconstructed accurately. Three chaotic lasers, i.e. quasi-periodic, moderate, and coherent collapse chaotic lasers, are prepared experimentally, and their bandwidths are 3.2 GHz, 7.3 GHz, and 11.5 GHz, respectively. With an increase in bandwidth and a decrease in the time-delay period, the complexity of the chaotic laser increases. The quadrature signals of the chaotic lasers in the three states are measured, and the Wigner quasi-probability distributions in phase space and the density matrices of the chaotic lasers are reconstructed using balanced homodyne detection and the maximum likelihood method. Compared to the shot noise limit, the measured phase-space Wigner quasi-probability distributions of the chaotic lasers are magnified by 1.5-3.0 times. Meanwhile, the randomness of chaos increases, and the effect of chaotic amplification increases gradually. Finally, the high-fidelity phase-space reconstruction of the Wigner quasi-probability distributions of chaotic lasers with the fidelities of 95.5%, 97.0%, and 97.6% is achieved after removing the -44 dBm background noise and compensating for the 8.9% loss of the experimental system. Therefore, this method can enable the precise characterization of entropy sources in chaos-based secure communication.