Luminescence spectra of quantum dots in microcavities. III. Multiple quantum dots

We discuss the spectral line shapes of $N$ quantum dots in strong coupling with the single mode of a microcavity in the presence of a continuous, incoherent pumping. Nontrivial features in the response of the system are induced by detuning the emitters or probing the direct exciton emission spectrum. We describe dark states, quantum nonlinearities, emission dips, and interferences and show how these various effects may coexist, giving rise to highly peculiar line shapes.

Figure 1

(Color online) Sketch of the two main channels of emission from a quantum dot in a microcavity, here illustrated with a micropillar. The cavity emission, or end emission ($S_a$, in blue), emits along the axis of the structure. The QD emission, or side emission ($S_{\sigma }$, in purple), emits through the leaky modes (pictured as a beam). The emission spectra can be decomposed into their polariton emission [dashed (red) curves], which is intrinsic to the coupling inside the structure and is therefore identical in both cases. The channel of detection superimposes to the polariton emission an interference part [dotted (orange) curve], which summed with the polariton contribution provides the observed spectrum, the filled line shapes. For the parameters chosen, the Rabi splitting between polaritons is observed in the QD emission but not in the cavity emission.

Figure 2

(Color online) One dot in a cavity. Emission spectrum from the radiation channel of the exciton $S_1\left(\omega \right)$, from the radiation channel of the cavity mode $S_a\left(\omega \right)$, and eigenstates of the system. Parameters: ${\gamma }_a/g=0.5$, ${\gamma }_1/g=0.1$, $P_1/g={10}^{-3}$, and $P_a=P_1$.

Figure 3

(Color online) Cavity [solid (blue) curve] and dot [dotted (purple) curve] emission spectra for one dot in a cavity as a function of pumping power. In the spontaneous emission regime (a), there are no qualitative differences between the two types of spectrum. Increasing pumping (b)–(d) shows markedly different behaviors in the two channels of emission. Whereas the cavity spectrum does not present a particularly rich phenomenology (collapse of the Rabi doublet), the dot emission displays more characteristic line shapes. Side elbows are formed [outlined with arrows in (b)] and strong deviations from Lorentzian lines are obtained even when the spectrum has only one peak. Parameters: ${\gamma }_a/g=0.5$, ${\gamma }_1/g=0.1$, $P_a=0$, and (a) $P_1/g={10}^{-3}$, (b) $P_1/g=0.5$, (c) $P_1/g=1$, and (d) $P_1/g=2$.

Figure 4

(Color online) Cavity spectrum of one dot in a cavity as a function of detuning. Lower panels, decomposition of the spectrum in its dressed states emission lines [$\left|1,\pm \right.〉\to \left|\mathrm{vacuum}\right.〉$, dashed (green) curve; and $\left|2,\pm \right.〉\to \left|1,\pm \right.〉$, dotted (red) curve], showing how (a) Jaynes-Cummings transitions are hindered at resonances but (b) are revealed out of resonance. Parameters are the same as in Fig. 3 but with the detunings $\Delta /g=0$, $0.6$, 1, and 2 and $P_1/g=0.4$.

Figure 5

(Color online) Dot emission spectrum of one dot in a cavity, out of resonance, as a function of pumping. The Rabi doublet in the spontaneous emission regime (a) gives rise, with increasing pumping (b)–(d), to an emission dip: the incoherent excitation is coherently scattered to the cavity which enters the lasing regime. This is better seen at nonzero detuning. Parameters are the same as in Fig. 3 but for $\Delta =2g$.

Figure 6

(Color online) Detail of Fig. 5d. The interference that arises as the system enters lasing is best seen in the detuned system because dressed states [solid (purple) curve] are then of a markedly different character; cf. the exciton line broadened by pumping (on the right), and the narrow line of the cavity that enters lasing (at the origin). The interferences between dressed states [dotted (brown) curve], which added to the dressed state emission provide the photoluminescence spectrum [filled (purple) area], strongly modify the results of kinetic theory applied to the dressed states.

Figure 7

(Color online) Emission spectrum from the radiation channel of the first exciton $S_1\left(\omega \right)$, the second exciton $S_2\left(\omega \right)$, the cavity mode $S_a\left(\omega \right)$, and the eigenstates for a strongly coupled system of two excitons and the cavity mode. The eigenvectors of the three eigenstates $\left|{\lambda }_2\right.〉$, $\left|{\lambda }_1\right.〉$, and $\left|{\lambda }_0\right.〉$ of the coupled system display the contributions of the individual quantum states $\left|1,0\right.〉$, $\left|0,X_1\right.〉$, and $\left|0,X_2\right.〉$ to the specific eigenstate. Parameters: same as in Fig. 2 with identical pumping of the dots.

Figure 8

(Color online) Emission spectrum from the radiation channel of the cavity mode $S_a\left(\omega \right)$, and eigenstates for a strongly coupled system of two excitons and the cavity mode where the two excitons cross out of resonance from the cavity mode. Parameters: same as in Fig. 7.

Figure 9

(Color online) Spectral shapes in the cavity emission when two dots are strongly coupled to the cavity mode. (a) With increasing excitation (solid) the emission dip effect becomes visible in the cavity emission. (b) It is better seen slightly out of resonance since it can now “feed” on the line previously canceled by subradiance. (c) Such interferences are strong even in systems that do not exhibit spontaneous emission features (such as Rabi splitting), although this still requires strong coupling. Parameters: $g_1=g_2=g$ (setting the unit), ${\gamma }_a=g$, ${\gamma }_1={\gamma }_2=0.1g$, $P_a=0$; then (a) ${\Delta }_1={\Delta }_2=0$, $P_1={10}^{-4}g$ with (dashed) $P_2={10}^{-4}g$ and (solid) $P_2=0.1g$. (b) Same as (a) but for ${\gamma }_a=2g$, $P_1={10}^{-3}g$, $P_2=0$, ${\Delta }_1=0$ and (solid) ${\Delta }_2=0.1g$ or (dotted) ${\Delta }_2=0$. (c) Same as (a) but for ${\gamma }_a=5g$, ${\Delta }_1=0$ with (dotted) ${\Delta }_2=0$ or (solid) ${\Delta }_2=0.1g$.

Figure 10

(Color online) Luminescence spectra of the cavity in strong coupling with one QD and in weak coupling with another QD, in the cases where (a) the strongly coupled dot or (b) the weakly coupled dot is excited. In the latter case, the effective cavity pumping from the weakly coupled, or spectator, QD allows to resolve a line splitting, which would not be observed if there would be only one dot. Parameters: ${\gamma }_a=3g$, ${\gamma }_1={\gamma }_2=g$, $g_1=g$, $g_2=g/10$, ${\Delta }_1={\Delta }_2=0$, with $P=0.2g$ on (a) the first or (b) the second dot.

Figure 11

(Color online) Emission spectrum from $N=2$ dots in strong coupling with the cavity mode in the detuning configuration where the dots drift with slopes proportional to the control parameter in the ratios 1 and $1/2$, respectively, both meeting the cavity at the same point. The left and right graphs, corresponding to standard and excellent strong coupling, respectively, are to be compared. The profusion of dressed states in the best system and the even greater number of possible transitions between them give rise to the crowded set of lines in the dot emission, featuring both crossing and anticrossing of the lines. The figure is also to be compared with Fig. 12 with a different detuning configuration, realizing in particular a qualitatively different profile of the cavity emission in the case ${\gamma }_a=0.5g$.

Figure 12

(Color online) Same as Fig. 11 but for a different configuration of detuning, with one QD always at resonance and the other tuning through with slope proportional to the control parameter. The profile varies topologically in the cavity emission of the less strongly coupled system, which is the case of easiest experimental access.

Figure 13

(Color online) Emission spectra from $N=2$ dots in strong coupling with the cavity mode in the case ${\gamma }_a=0.1g$, at resonance and as a function of increasing pumping. The nonlinear regime breaks the subradiance in the cavity emission and the central peak appears, followed by quantum nonlinear features, and ultimately the system goes toward lasing, with characteristic features such as the emergence of a single narrowing line in the cavity emission and of an emerging Mollow triplet in the dot emission.

Figure 14

(Color online) Same as Fig. 13 but for a lower $Q$ system, with ${\gamma }_a=0.5g$. Increasing pumping also breaks subradiance and lets the central peak appear in the cavity emission. In this case, however, quantized nonlinearities cannot be resolved and the system requires higher pumping to show signatures of lasing in the cavity emission. The transition is therefore merely that from a doublet to a triplet that ultimately collapses to a single line in the cavity, and of a singlet to a triplet in the dot emission.

Figure 15

(Color online) Emission spectrum from the radiation channel of the cavity mode $S_a\left(\omega \right)$ and the eigenstates of a strongly coupled system with $N=5$ QDs. Parameters are the same as in Fig. 7.

Figure 16

(Color online) Counterpart for $N=3$ of Fig. 11, with two dots drifting with slopes in the ratio 1 and 1/2 and a third dot pinned at resonance. The cavity emission displays both features of peak cancellation and quantum nonlinearities in the case ${\gamma }_a/g=0.5$, and a complex profile of crossing and anticrossing lines in the case ${\gamma }_a/g=0.1$, akin to dot emission with $N=2$. The dot emission becomes so crowded as to lose any recognizable pattern, already for $N$ as low as three.

Figure 17

(Color online) Cavity emission when three dots are strongly coupled to the cavity mode at low pumping with (a) resonance and (b) one case out of resonance shown explicitly. This case brings together most features described above: the Dicke nonlinearities (here with $N=3$), the subradiance with disappearance of the sharp central line at resonance, and the emission dip at the cavity mode. Note how the last feature results in the sharp horizontal line in the density plot.