Quantum statistics in a time-modulated exciton-photon system

We consider a system consisting of a large individual quantum dot with excitonic resonance coupled to a single-mode photonic cavity in the nonlinear regime when exciton-exciton interaction becomes important. Quantum statistics of coupled exciton-photon modes is studied for two regimes of driving: a monochromatic input field and a field with periodically time-modulated amplitude. We show that sub-Poissonian statistics for both modes are realized in the case of monochromatic driving for transient and steady-state regimes in the presence of decoherence and cavity-induced feedback. We also demonstrate that variances of quantum fluctuations of photon and exciton numbers display oscillations in the case of modulated input. In this case, we show an improvement of the degree of sub-Poissonian statistics and antibunching for both modes at periodic sequence of definite time intervals in comparison with the case of the steady-state regime for monochromatic driving. We also observe the Wigner functions with negative values in phase space for a time-modulated exciton-photon system.

Figure 1

Time evolution of the mean excitation numbers of photonic (1) and excitonic (2) modes for the nondissipative case, without any driving. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma ={\mathrm{\Delta }}_{ex}/\gamma =7.12$, $g/\gamma =5$, $\chi /\gamma =4$.

Figure 2

The results for the case of single-mode driving, ${\mathrm{\Omega }}_2=0$. (a) Temporal dependence of the mean excitation numbers of photonic (1) and excitonic (2) modes; (b) temporal dependence of the Mandel parameters for photonic (1) and excitonic (2) modes. The contour plots of the Wigner functions of (c) the photonic mode and (d) excitonic mode for time intervals corresponding to the maximal values of excitation numbers. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $g/\gamma$ = 5, $\chi /\gamma$ = 1, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 0.

Figure 3

The results for the case of single-mode driving, ${\mathrm{\Omega }}_2=0$, where curves (1) correspond to the photonic mode and curves (2) correspond to the excitonic mode. (a) and (c) Temporal dependence of the mean excitation numbers of modes; (b) and (d) temporal dependence of the Mandel parameters for excitonic and photonic modes. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $g/\gamma$ = 5, $\chi /\gamma$ = 1, ${\mathrm{\Omega }}_1/\gamma$ = 10, ${\mathrm{\Omega }}_2/\gamma$ = 0, $\delta /\gamma =0$ in (a) and (b); ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $g/\gamma$ = 7, $\chi /\gamma$ = 3, ${\mathrm{\Omega }}_1/\gamma$ = 10, ${\mathrm{\Omega }}_2/\gamma$ = 0 in (c) and (d).

Figure 4

The results for the case of a periodically driven photon-exciton system. Curves (1) correspond to the photonic mode, and curves (2) correspond to the excitonic mode. (a) Temporal dependence of the mean excitation numbers for both modes, (b) temporal dependence of the Mandel parameters for both modes, (c) the Wigner function of the photonic mode corresponding to the maximal occupancy of the mode, and (d) the Wigner function of the photonic mode corresponding to the minimal occupancy of the mode. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $g/\gamma$ = 5, $\chi /\gamma$ = 1, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 5, $\delta /\gamma =2$.

Figure 5

(a) The mean excitation number and (b) $Q$ factor depending on time intervals for various values of the modulation frequency: $\delta /\gamma =1.5$, curve (1), $\delta /\gamma =2.5$, curve (2), $\delta /\gamma =3$, curve (3). The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $g/\gamma$ = 5, $\chi /\gamma$ = 1, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 5.

Figure 6

The results for the case of the periodically driven photon-exciton system in the deeply sub-Poissonian regime. Curves (1) correspond to the photonic mode, and curves (2) correspond to the excitonic mode. (a) Time dependence of the mean excitation numbers, (b) time dependence of the Mandel parameters, (c) the contour plots of the Wigner function corresponding to the maximal occupancy of the photonic mode, and (d) time-dependent normalized second-order correlation functions for photonic and exciton modes. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $\chi /\gamma$ = 3, $g/\gamma$ = 7, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 5, $\delta /\gamma =2$.

Figure 7

The photon number distributions (a) at the minimum level of quanta and (b) at the maximum level of quanta. The exciton number distributions (c) at the minimum level of quanta and (d) at the maximum level of quanta. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $\chi /\gamma$ = 3, $g/\gamma$ = 7, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 5, $\delta /\gamma =2$.

Figure 8

The minimal values of the Mandel parameter depending on the exciton-photon coupling strength for several values of the exciton-exciton interaction: $\chi /\gamma$ = 1, dashed curve; $\chi /\gamma$ = 3, solid black curve; $\chi /\gamma$ = 5, solid pink curve. The parameters are ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 5, $\delta /\gamma =2$.

Figure 9

(a) Time evolution of the Mandel factor for three values of thermal photon numbers $n_{th}$; (b) the second-order correlation function versus time intervals for several numbers of thermal photons. The parameters are as follows: ${\mathrm{\Delta }}_{ph}/\gamma =7.12$, ${\mathrm{\Delta }}_{ex}/\gamma =7.12$, $\chi /\gamma$ = 1, $g/\gamma$ = 5, ${\mathrm{\Omega }}_1/\gamma$ = 5, ${\mathrm{\Omega }}_2/\gamma$ = 5, $\delta /\gamma =2$.