Time- and frequency-domain two-particle correlations of a driven dissipative Bose-Hubbard model

We theoretically investigate the time- and frequency-domain two-particle correlations of a driven dissipative Bose-Hubbard model (BHM) at and near a dissipative phase transition (DPT). We compute Hanbury Brown-Twiss (HBT) type two-particle temporal correlation function $g^2(\tau)$ which, as a function of time delay $\tau$, exhibits oscillations with frequencies determined by the imaginary part of Liouvillian gap. As the gap closes near a transition point, the oscillations at that point dies down. For parameters slightly away from the transition point, the HBT correlations show oscillations from super-bunching to anti-bunching regimes. We show that the Fourier transform of HBT correlations into frequency domain provide information about DPT and Liouvillian dynamics. We numerically solve the many-body Lindblad master equation and calculate Wigner distribution of the system in steady state to ascertain DPT. Below certain drive strength, the Fourier transform shows a two-peak structure while above that strength it exhibits either a Lorenzian-like single-peak structure or a structure with two-dips. The width of the single-peak structure is minimum at the phase transition point and the peak of this structure always lies at zero frequency. The positions of the two symmetrical peaks in case of two-peak structure are given by the imaginary parts of the Liouvillian gap while their half width at half maximum (HWHM) is given by the real part of the gap. The positions and the widths of the two dips are also related to low lying eigenvalues of the Liouvillian operator. We discuss quantum statistical properties of the model in terms of the HBT correlation function and its Fourier transform.