Dynamics of the excitations of a quantum dot in a microcavity

We study the dynamics of a quantum dot embedded in a three-dimensional microcavity in the strong coupling regime in which the quantum dot exciton has an energy close to the frequency of a confined cavity mode. Under the continuous pumping of the system, the confined electron and hole can recombine either by spontaneous emission through a leaky mode or by stimulated emission of a cavity mode that can escape from the cavity. The numerical integration of a master equation including all these effects gives the dynamics of the density matrix. By using the quantum regression theorem, we compute the first- and second-order coherence functions required to calculate the photon statistics and the spectrum of the emitted light. Our main result is the determination of a range of parameters in which a state of cavity modes with Poissonian or sub-Poissonian (nonclassical) statistics can be built up within the microcavity. Depending on the relative values of pumping and rate of stimulated emission, either one or two peaks close to the excitation energy of the dot and/or to the natural frequency of the cavity are observed in the emission spectrum. The physics behind these results is discussed.

Figure 1

Ladder of levels (continuous lines) for a two-state QD coupled to a single optical mode of a microcavity. States labeled as $Gn$ with $n=0,1,2,\dots$ correspond to having the QD in its GS coexisting with $n$ cavity photons. The same for states $Xn$ with the QD in its excited state $X$. Double continuous lines depict the coupling $g$, dashed lines the pumping with rate $P$, dotted lines the leaky modes emission with rate $\gamma$, and dash-dotted lines the emission of cavity modes with rate $\kappa$.

Figure 2

Density matrix diagonal elements ${\rho }_{in,in}$ for exciton $(i\equiv X)$ states (dashed lines), and ground $(i\equiv G)$ states (continuous lines), from $n=0$ up to $n=35$ cavity photons. $g=1$, $\Delta =5$, $\gamma =0.1$, $\kappa =0.1$, and $P=15$. The inset shows a zoom out of the graphic to observe high occupations at very short times.

Figure 3

Density matrix diagonal elements ${\rho }_{in,in}$ for both the exciton $(i\equiv X)$ and the ground $(i\equiv G)$ states, from $n=0$ up to $n=3$ cavity photons. $g=1$, $\Delta =5$, $\gamma =0.1$, $\kappa =0.5$, and $P=1$. The inset shows a zoom of the vertical axis of the graphic.

Figure 4

Density matrix off diagonal element ${\rho }_{X0,G1}$ with detunning $\Delta =5$ for $g=1$, $\gamma =0.1$, and two different values of $\kappa$ and $P$.

Figure 5

Imaginary part of the density matrix off diagonal element ${\rho }_{X0,G1}$ at perfect resonance $(\Delta =0)$ for $g=1$, $\gamma =0.1$, and different values of $\kappa$ and $P$. All the real parts of the same matrix elements are zero. The inset shows a zoom of the graphic for small values of $t$.

Figure 6

$\overline{N_{\mathit{ph}}}$, in gray scale, as a function of $\kappa$ and $P$ for perfect resonance $\Delta =0$ with $g=1$ and $\gamma =0.1$. Some contours of constant $\overline{N_{\mathit{ph}}}$ are drawn as continuous lines to clarify the figure.

Figure 7

$\overline{N_{\mathit{ph}}}$, in gray scale, as a function of $\kappa$ and $P$ for a detuning $\Delta =5$ with $g=1$ and $\gamma =0.1$. Some contours of constant $\overline{N_{\mathit{ph}}}$ are drawn as continuous lines to clarify the figure.

Figure 8

Second-order coherence function $g^{\left(2\right)}(\tau =0)$, in gray scale, as function of $\kappa$ and $P$ for perfect resonance $\Delta =0$. $g=1$ and $\gamma =0.1$. The dashed area shows the region supporting a Poisson distribution of cavity photons in which $g^{\left(2\right)}(\tau =0)$ close to 1. Some contours of constant $g^{\left(2\right)}$ are drawn as continuous lines to clarify the figure.

Figure 9

Second-order coherence function $g^{\left(2\right)}(\tau =0)$, in gray scale, as function of $\kappa$ and $P$ for a detuning $\Delta =5$. $g=1$ and $\gamma =0.1$. The dashed area shows the region supporting a Poisson distribution of cavity photons in which $g^{\left(2\right)}(\tau =0)$ close to 1. Some contours of constant $g^{\left(2\right)}$ are drawn as continuous lines to clarify the figure.

Figure 10

Spectrum of emission for $g=1$, ${\omega }_X=1000$, $\Delta =5$, $\gamma =0.1$, $\kappa =0.1$, and $P=15$.

Figure 11

Spectrum of emission for $g=1$, ${\omega }_X=1000$, $\Delta =5$, $\gamma =0.1$, $\kappa =5$, and $P=1$.

Figure 12

Spectrum of emission for $g=1$, ${\omega }_X=1000$, $\Delta =5$, $\gamma =0.1$, $\kappa =5$, and $P=15$.

Figure 13

Spectrum, in gray scale, of stimulated emission (cavity modes escaping from the cavity) as a function of frequency and $P$. $g=1$, ${\omega }_X=1000$, $\Delta =5$, $\gamma =0.1$, and $\kappa =0.1$. Some contours of constant intensity are drawn as continuous lines to clarify the figure.

Figure 14

Spectrum, in gray scale, of spontaneous emission (emission of leaky modes) as a function of frequency and $P$. $g=1$, ${\omega }_X=1000$, $\Delta =5$, $\gamma =0.1$, and $\kappa =0.1$. Some contours of constant intensity are drawn as continuous lines to clarify the figure.