Few-Photon Model of the Optical Emission of Semiconductor Quantum Dots

The Jaynes-Cummings model provides a well established theoretical framework for single electron two level systems in a radiation field. Similar exactly solvable models for semiconductor light emitters such as quantum dots dominated by many particle interactions are not known. We access these systems by a generalized cluster expansion, the photon-probability cluster expansion: a reliable approach for few-photon dynamics in many body electron systems. As a first application, we discuss vacuum Rabi oscillations and show that their amplitude determines the number of electrons in the quantum dot.

Figure 1

Scheme of the model system: (a) a QD is embedded inside a WL leading to confined QD states coupled to a WL continuum. (b) The emission of the confined states couples to a photon mode in a cavity.

Figure 2

The JCM (solid, red), standard cluster-expansion atom like using one-electron assumption (dashed or green), and standard cluster-expansion using $f_{\mathrm{sp}}^e=f^e{f}^h$ in the QD-WL system (dotted or blue) and PPCE using $f_{\mathrm{sp},n}^e=f_n^e{f}_n^h$ in the QD-WL system in a new scheme (dotted dashed, black). With the initial conditions: $f^e=f^h=0.95$, $p_0=0.8$, $p_1=0.2$, and a coupling constant $|g^{vc}|=0.51/\mathrm{fs}$ for a QD resonant with the cavity mode. [(a) ${\gamma }_p=1/(10\mathrm{ps})$ and (b) ${\gamma }_p=0/(\mathrm{ps})$].

Figure 3

The PPCE solution for semiconductor QDs. With the initial conditions: $f^h=0.3$ and $f^e=0.1$, $p_0=0.05$, $p_1=0.95$ and a coupling constant $g=0.5\mathrm{meV}$ for a QD resonant with the cavity mode and ${\gamma }_p=(10\mathrm{ps}{)}^{-1}$. In (a) the different lines are $f^e$ (solid, red), $f^h$ (green, long dashed), $g^1(0)$ (blue, short dashed). (c) The maximum value of $g^{(2)}$ for PPCE is plotted over the number of electrons in the QD, with $p_1=1.0$ (black, straight) and $p_2=1.0$ (red, dashed).