Transiently changing shape of the photon number distribution in a quantum-dot–cavity system driven by chirped laser pulses

We have simulated the time evolution of the photon number distribution in a semiconductor quantum-dot–microcavity system driven by chirped laser pulses and compare with unchirped results. When phonon interactions with the dot are disregarded—thus corresponding to the limit of atomic cavity systems—chirped pulses generate photon number distributions that change their shape drastically in the course of time. Phonons have a strong and qualitative impact on the photon statistics. The asymmetry between phonon absorption and emission destroys the symmetry of the photon distributions obtained for positive and negative chirps. While for negative chirps transient distributions resembling thermal ones are observed, for positive chirps the photon number distribution still resembles its phonon-free counterpart but with overall smoother shapes. In sharp contrast, using unchirped pulses of the same pulse area and duration wave packets are found that move up and down the Jaynes-Cummings ladder with a bell shape that changes little in time. For shorter pulses and lower driving strength Rabi-like oscillations occur between low photon number states. For all considered excitation conditions transitions between sub- and super-Poissonian statistics are found at certain times. For resonant driving with low intensity the Mandel parameter oscillates and is mostly negative, which indicates a nonclassical state in the cavity field. Finally, we show that it is possible that the Mandel parameter dynamically approaches zero and still the photon distribution exhibits two maxima and thus is far from being a Poissonian.

Figure 1

Transient photon number distributions for laser excitations with unchirped pulses with (a) and (c) pulse area $\mathrm{\Theta }=5\pi$ and duration FWHM=2.4 ps, (b) and (d) pulse area $\mathrm{\Theta }=31.63\pi$ and duration $\mathrm{FWHM}=94.22\mathrm{ps}$. Panels (c) and (d) display results accounting for phonons that are initially at equilibrium at a temperature of $T=4\mathrm{K}$ while the corresponding phonon-free results are shown in (a) and (b). The pulse has its maximum at $t=t_0$. Black markers indicate the FWHM of the pulse.

Figure 2

Transient photon number distributions for laser excitations with chirped pulses with pulse area and FWHM before the chirp filter of $\mathrm{\Theta }=5\pi$ and FWHM=2.4 ps, i.e., ${\mathrm{\Theta }}_{\mathrm{chirp}}=31.63\pi$ and duration ${\mathrm{FWHM}}_{\mathrm{chirp}}=94.22\mathrm{ps}$ for $\left|\alpha \right|=40{\mathrm{ps}}^2$. (a) and (c) Calculated with chirp parameter $\alpha =-40{\mathrm{ps}}^2$, (b) and (d) $\alpha =+40{\mathrm{ps}}^2$. (c) and (d) Displayed are results accounting for phonons that are initially at equilibrium at a temperature of $T=4\mathrm{K}$ while the corresponding phonon-free results are shown in (a) and (b). The pulse has its maximum at $t=t_0$. Black markers indicate the FWHM of the pulse after the chirp filter.

Figure 3

The time-dependent (a) mean photon number and (b) Mandel parameter $Q\left(t\right)=(\langle \mathrm{\Delta }{n}^2\rangle -\langle n\rangle )/\langle n\rangle$ for the cases indicated by the labels. All curves are calculated with phonons initially at $T=4$ K, except for the gray curves which correspond to the phonon-free case. The blue curve is scaled down by a factor of 5 for better visibility. The inset in (b) corresponds to a zoomed-in scale.

Figure 4

Time evolution of the upper and lower laser-dressed state energies with respect to the excitation pulse maximum at $t=t_0$. While for negative chirps (a) phonon emission is probable (represented by black arrows), for positive chirps (b) phonon absorption is suppressed at low temperatures, which is indicated by the dashed arrows. Green curly arrows indicate transitions between laser-dressed states due to the QD cavity feeding.

Figure 5

(a) Linear absorption spectrum of the QDC system. (b) Time-dependent instantaneous frequency, blue (red) for positive (negative) chirp. $\mathrm{\Delta }t$ marks the time elapsed between the crossing of the two resonances. (c) and (d) Time evolution of the occupations of the lowest excited eigenstates of the QDC system [(c) for positive and (d) for negative chirp]. (e) and (f) Photon number distribution at $t-t_0=20\mathrm{ps}$ (gray); red, accounting only for $|n,+\rangle$ (e) or $|n,-\rangle$ (f) states. Here, only phonon-free results are shown.