Phase locking and quantum statistics in a parametrically driven nonlinear resonator

We discuss phase-locking phenomenon at low-level of quanta and quantum statistics for parametrically driven nonlinear Kerr resonator (PDNR). Oscillatory mode of PDNR is created in the process of a degenerate down-conversion of photons under interaction with a train of external Gaussian pulses. We calculate the distribution of photon-number states, the second-order correlation function of photons, the Wigner functions of cavity mode showing two-fold symmetry in phase space, and we analyze formation of phase-locked states in the regular as well as the quantum chaotic regime of the PDNR.

Figure 1

Excitation number against time intervals for the negative and positive detuning. The parameters used are $\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1},\tau =4\pi {\gamma }^{-1}/5$, and (a) $\mathrm{\Delta }/\gamma =-20$ and (b) $\mathrm{\Delta }/\gamma =20$.

Figure 2

Wigner functions for the positive and negative detunings at time instants corresponding to (a) and (c) the minimum and (b) and (d) the maximum of the excitation number observed in Fig. 1. The parameters employed are (a) and (b) $\mathrm{\Delta }/\gamma =-20,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1}$, and $\tau =4\pi {\gamma }^{-1}/5$ and (c) and (d) $\mathrm{\Delta }/\gamma =20,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1}$, and $\tau =4\pi /5{\gamma }^{-1}$. Note that (a) and (c) relate to the minimal value of the photon number, while (b) and (d) relate to the maximal value. The Wigner functions of (a) and (b) are for the chaotic regime, while (c) and (d) are for the regular regime of the PDNR.

Figure 3

Wigner function for the case of positive and negative detuning and for a time instant of mean excitation number that lies between the minimal and maximal values. The parameters used are $\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1},\tau =4\pi {\gamma }^{-1}/5$, and (a) $\mathrm{\Delta }/\gamma =20$ and (b) $\mathrm{\Delta }/\gamma =-20$.

Figure 4

Wigner functions for negative detuning for time instants that correspond to (a) the minimal and (b) the maximal values of the mean excitation number. The parameters used are $\mathrm{\Delta }/\gamma =-7.5,\chi /\gamma =0.5,\mathrm{\Omega }/\gamma =10,T=0.25{\gamma }^{-1}$, and $\tau =2\pi /5{\gamma }^{-1}$. Also shown is (c) the Wigner function for the regular regime, with a time instant at the maximal value of the mean excitation number. The parameters employed are $\mathrm{\Delta }/\gamma =15,\chi /\gamma =1,\mathrm{\Omega }/\gamma =10,T=0.5{\gamma }^{-1}$, and $\tau =2\pi {\gamma }^{-1}$.

Figure 5

Semiclassical results for the PDNR: (a) the mean excitation numbers (photon numbers) versus dimensionless scaled time and (b) the Poincaré section. The parameters are $\mathrm{\Delta }/\gamma =-10,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1}$, and $\tau =4\pi /5{\gamma }^{-1}$.

Figure 6

Averaged photon numbers and the second-order correlation function for both chaotic and regular regimes of the PDNR. The parameters are (a) $\mathrm{\Delta }/\gamma =-20,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1}$, and $\tau =4\pi {\gamma }^{-1}/5$ and (b) $\mathrm{\Delta }/\gamma =20,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1}$, and $\tau =4\pi /5{\gamma }^{-1}$.

Figure 7

Time dependence of the quantum purity and distribution of oscillatory excitation numbers for (a) and (b) negative and (c) and (d) positive detunings. The parameters are as follows: (a), (b) $\mathrm{\Delta }/\gamma =-20,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1}$, and $\tau =4/5\pi {\gamma }^{-1}$, (c), (d) $\mathrm{\Delta }/\gamma =20,\chi /\gamma =1,\mathrm{\Omega }/\gamma =20,T=0.5{\gamma }^{-1},\tau =4/5\pi {\gamma }^{-1}$.