Nonclassical light from an incoherently pumped quantum dot in a microcavity

Semiconductor microcavities with artificial single-photon emitters have become one of the backbones of semiconductor quantum optics. In many cases, however, technical and physical issues limit the study of optical fields to incoherently excited systems. We analyze the model of an incoherently driven two-level system in a single-mode cavity. The specific structure of the applied master equation yields a recurrence relation for the steady-state values of correlations of the intracavity field and the emitter. We provide boundary conditions that permit a systematic solution which is numerically less demanding than standard methods. The method allows us to directly infer reasonable cutoff conditions from the system parameters. Different cavity systems from previous experiments are analyzed in terms of field correlation functions which can be measured via homodyne correlation measurements. We find that nonclassical correlations occur in systems of moderate quantum-dot–cavity coupling rather than strong coupling. Our boundary conditions also allow us to derive analytical results for the overall quantum state and its higher-order moments. We obtain very good approximations for the full quantum state of the field in terms of the characteristic functions. It turns out that for every physically reasonable set of system parameters, the state of the intracavity field is nonclassical.

Figure 1

Scheme of a quantum dot (two-level system) in a (semiconductor) microcavity. The left mirror is supposed to reflect perfectly, whereas the right one is partially transparent giving rise to a loss of the cavity mode.

Figure 2

$I_1=〈{\widehat{a}}^{†}\widehat{a}〉$ for resonance $\delta =0$ as a function of $g/\kappa$. The normalized quantum-dot decay is $\mathrm{\Gamma }/\kappa =1$ and the normalized pumping strength is varied from $p/\kappa =0.5$ to 2 in steps of $0.5$, where a longer dashing corresponds to a stronger pumping.

Figure 3

Renormalized correlation functions $I_n/(I_1^{n}n!)$ for parameters $\mathrm{\Gamma }/\kappa =1,\delta =0$, and $g/\kappa =1$ (blue/darker) or $g/\kappa =5$ (orange/lighter). Depicted are the orders $n=2,3,4$ with increasing dashing size, respectively.

Figure 4

Condition $I_{2n}/I_n^2$ plotted for orders $n=1,3,5$ and parameter sets A (blue/darker) and B (orange/lighter) for varied pumping strength $p$. A longer dashing corresponds to higher orders $n$. Note that the order of magnitude of $p$ is ${10}^3$ times higher compared to the other parameters; see beginning of Sec. 4.

Figure 5

$B_{2n}/B_n^2$ plotted for $n=1,3,5$ and parameter sets A (blue/darker) and B (orange/lighter) against the pumping strength $p$. A longer dashing corresponds to higher orders $n$. Note that the order of magnitude of $p$ is ${10}^3$ times higher than the other parameters.

Figure 6

Plot of $B_1-{\left|R_0\right|}^2$ for set A (blue/darker) and set B (orange/lighter). Negative values occur for entanglement between the intracavity field and the TLS.

Figure 7

Characteristic function for the parameter sets A (blue/darker) and B (orange/lighter) and pumping strength $p=1\times {10}^{11}{\mathrm{s}}^{-1}$.

Figure 8

Plot of the lower (blue/darker) and upper (orange/lighter) boundaries for the moments. Also seen are the approximated ratios ${\beta }_n/(-{\alpha }_{n+1})$ where the linear order is kept (green dashed line) and the asymptotic relations hold (red dot-dashed line). The parameters were taken from set B and the pumping strength is $p=1\times {10}^{11}{\mathrm{s}}^{-1}$.