Luminescence spectra of quantum dots in microcavities. II. Fermions

We discuss the luminescence spectra of coupled light-matter systems realized with semiconductor heterostructures in microcavities in the presence of a continuous, incoherent pumping, when the matter field is fermionic. The linear regime—which has been the main topic of investigation both experimentally and theoretically—converges to the case of coupling to a bosonic material field, and has been amply discussed in the first part of this work. We address here the nonlinear regime, and argue that, counter to intuition, it is better observed at low pumping intensities. We support our discussion with particular cases representative of, and beyond, the experimental state of the art. We explore the transition from the quantum to the classical regime, by decomposing the total spectrum into individual transitions between the dressed states of the light-matter coupling Hamiltonian, reducing the problem to the positions and broadenings of all possible transitions. As the system crosses to the classical limit, rich multiplet structures mapping the quantized energy levels melt and turn to cavity lasing and to an incoherent Mollow triplet in the direct exciton emission for very good structure. Less ideal figures of merit can still betray the quantum regime, with a proper balance of cavity versus electronic pumping.

Figure 1

(Color online) Chain of correlators—indexed by $\left\{\eta \right\}=\left(m,n,\mu ,\nu \right)$—linked by the dissipative Jaynes-Cummings dynamics. On the left (right), the set ${\cup }_k{\mathcal{N}}_k$ (${\cup }_k{\tilde{\mathcal{N}}}_k$) involved in the equations of the two-time (single-time) correlators. In green are shown the first manifolds ${\mathcal{N}}_1$ and ${\tilde{\mathcal{N}}}_1$ that correspond to the Boson case, and in increasingly lighter shades of blues, the higher manifolds ${\mathcal{N}}_k$ and ${\tilde{\mathcal{N}}}_k$. The equation of motion $⟨a^{†}\left(t\right)C_{\left\{\eta \right\}}\left(t+\tau \right)⟩$ [$⟨{\sigma }^{†}\left(t\right)C_{\left\{\eta \right\}}\left(t+\tau \right)⟩$] with $\eta ∊{\mathcal{N}}_k$ requires for its initial value the correlator $⟨C_{\left\{\tilde{\eta }\right\}}⟩$ with $\left\{\tilde{\eta }\right\}∊{\tilde{\mathcal{N}}}_k$ defined from $\left\{\eta \right\}=\left(m,n,\mu ,\nu \right)$ by $\left\{\tilde{\eta }\right\}=\left(m+1,n,\mu ,\nu \right)$ [$\left(m,n,\mu +1,\nu \right)$], as seen on the diagram. The red arrows indicate which elements are linked by the coherent (strong coupling) dynamics, through the coupling strength $g$, while the green/blue arrows show the connections due to the incoherent cavity/exciton pumpings, respectively. The self-coupling of each node to itself is not shown. This is where ${\omega }_{a,\sigma }$ and ${\gamma }_{a,\sigma }$ enter.

Figure 2

(Color online) Spectral structure of the Jaynes-Cummings model at resonance and without pumping. (a) Positions ${\omega }_p$ of the lines in the luminescence spectrum. Only energies higher than ${\omega }_a$ are shown (not their symmetric below ${\omega }_a$). We take ${\omega }_a=0$ as the reference energy. In green (thick), the first manifold, and in increasing shades of blue, the successive higher manifolds which form a pattern of branch-coupling curves that define different orders of SC. (b) half-width at half-maximum ${\gamma }_p$ of the lines (with ${\gamma }_{\sigma }=0$). In both (a) and (b), the blue filled region results from the accumulation of the countable-infinite vanishing lines. (c) Eigenenergies of the Jaynes-Cummings Hamiltonian with decay as an imaginary part of the bare energies (the Jaynes-Cummings ladder). This provides a clear physical picture of panel (a) where the peaks positions arise from the difference of energy between lines of two successive manifolds. Lines $A_k$ of (a) stem from the emission from manifold $k$ in strong coupling into manifold $k-1$ in weak coupling or vacuum. Lines $B_k$ and $C_k$ stem from the emission between the two manifolds in SC. Solid lines are those plotted in (a), dotted lines produce the symmetric lines, not shown. The horizontal line $W$ at 0 in (a) arises from decay between two manifolds in WC. Scheme (c) also reproduces the broadening of the lines (b) with the sum of the imaginary parts of the eigenenergies involved in the transition.

Figure 3

(Color online) Blue squares give the decay rates for the cavity and quantum dot estimated by Khitrova et al. in Ref. 43 for four references systems having achieved SC at this time: photonic crystals and pillar microcavities nearby point 3, microdisks, and atomic systems nearby point 2. With green circles, the three sets of parameters used in this text. Points 2 and 3 average over their two nearest neighbors and represent these systems. Point 1 represents a very good system in very strong coupling that might be realizable in the near future. Parameters are fractions because numerical computations have been done with arbitrary precision arithmetic (with the values given).

Figure 4

(Color online) Populations and statistics of the points marked 1, 2, and 3 in Fig. 3. Each row shows the triplet $n_a$ (first column), $n_{\sigma }$ (second) and $g^{\left(2\right)}\left(0\right)$ (third) for a given point ($n\text{th}$ row corresponds to point $n$). All plots share the same $x$-axis in logarithmic scale of $P_{\sigma }/g$ ranging from ${10}^{-3}$ to ${10}^3$. All $y$ axes are rescaled to its specific graph, at the exception of $n_{\sigma }$ which is always between 0 and 1. The color code corresponds to different values of $P_a$. Each color code applies to its row and is given in the last column. The qualitative behavior is roughly the same for all points: there is a peak in $n_a$ that is subsequently quenched as the dot gets saturated. In $g^{\left(2\right)}$, there is on the other hand, a local minimum of fluctuations that can be brought to the Poissonian limit of 1 (allowing for a lasing region) and maintained over a large plateau in good cavities.

Figure 5

(Color online) Spectral structure in the cavity emission of the Jaynes-Cummings model as a function of ${\gamma }_a/g$, with some electronic pumping ($\Delta =0$, ${\gamma }_{\sigma }=0$, $P_a=0$, ${\omega }_a=0$). Panel (a) is for $P_{\sigma }=g/50$ and (b) and (c) for $P_{\sigma }=g/10$. Panel (c) is a zoom on the central peaks of the entire picture (b). In blue (red) are the peaks with $L_p^a>0$ ($L_p^a<0$).

Figure 6

(Color online) Spectral structure in the cavity emission at resonance as a function of $P_{\sigma }/g$ for point 2 ($\Delta =0$, $P_a=0$, ${\omega }_a=0$). In insets, we zoom over the central peaks and the region $0\le P_{\sigma }/g\le 5$ (upper, in linear scale) and ${10}^{-3}\le P_{\sigma }/g\le 3$ (lower, logarithmic scale), showing the complex structures that arise. In blue (red) are the peaks with $L_p^a>0$ ($L_p^a<0$). At sufficiently high pumping, all eigenvalues have collapsed to zero, defining an extreme case of weak coupling. Blue lines here come on top.

Figure 7

(Color online) Jaynes-Cummings forks as they appear in the luminescence spectrum of a QD in a microcavity with system parameters given by point 1 of Fig. 3 and for pumping rates $\left(P_a,P_{\sigma }\right)/g$ given by (a), (0,0.057); (b), (0.225,0.087), and (c), (0.001,0.27). The two outer peaks at $\pm 1$ correspond to the vacuum Rabi doublet of the linear limit. Inner peaks correspond to transitions with states of more than one excitation. Although the underlying structure is the same, many variations of the actual line shapes can be obtained.

Figure 8

(Color online) Expanded view in logarithmic scale of a spectrum similar to those of Fig. 7, this time with $\left(P_a,P_{\sigma }\right)/g=\left(0.002,0.076\right)$. Transitions up to the third manifold (shown in insets) are resolvable. Others are lost in the broadening. The transition energies of the Jaynes-Cummings ladder are shown by vertical lines (up to the third manifold). The Rabi peaks that correspond to transitions from the first manifold to vacuum (line 1) are in this case dominated by higher transitions that accumulate close to the center (line 0).

Figure 9

(Color online) Point 1 of Fig. 3. Spectral emission over a wide range of electronic pumping $P_{\sigma }/g$ from ${10}^{-3}$ to ${10}^3$ showing the three main regimes: multiplet emission, lasing, and quenching. Cavity pumping only affects quantitatively the main features of the emission in this case of very strong coupling, so a small set is shown as representative enough: two upper rows (blue) correspond to cavity emission for no and large cavity pumping, respectively, and lower row (violet) to the direct exciton emission for no-cavity pumping. The Jaynes-Cummings fork is clearly resolved at small $P_{\sigma }$ $\left(<0.2g\right)$ and is enhanced by the cavity pumping. At higher electronic pumping $\left(P_{\sigma }\approx 0.5g\right)$, the multiplet structure collapses into a dominant doublet of inner peaks while the vacuum Rabi peaks melt into its shoulders. Then the system is brought into lasing $\left(g<P_{\sigma }<30g\right)$ and is finally quenched $\left(P_{\sigma }>30g\right)$.

Figure 10

(Color online) Point 1 of Fig. 3. Details of the loss of the multiplet structure with increasing exciton pumping and zero cavity pumping. The two upper rows (blue) correspond to the cavity emission $S_a\left(\omega \right)$ and the two lower (violet) to the exciton direct emission $S_{\sigma }\left(\omega \right)$. The spectral structure is richer in the exciton spectra that develop a Mollow-triplet-like emission.

Figure 11

(Color online) Point 1 of Fig. 3. Incoherent Mollow triplets are observed in the exciton direct emission with broad satellite peaks at approximately $\pm 2\sqrt{n_a}$ and a strong narrow central peak taking over a narrow resonance. Three upper rows show the spectra over the interval $\left|\omega \right|\le 15g$ allowing to see the satellites. Three lower rows are the same in the window $\left|\omega \right|\le 3g$, allowing to see the narrow resonance and peak that sit at the origin. Values of the electronic pumping are given in the frame of the first three rows. Cavity pumping is zero but influences very little the Mollow triplets. Rightmost figure superposes various spectra at increasing electronic pumping, showing the drift and broadening of the satellites, and putting to scale the very strong coherent feature at the origin. The incoherent Mollow triplet appears thus very differently from its counterpart under coherent excitation.

Figure 12

(Color online) Same as Fig. 9 over the same range of $P_{\sigma }$, but for Point 2 of Fig. 3. In this case, cavity pumping has a strong influence on the cavity luminescence spectra, so we show more cases, namely, $P_a/g=0$ (upper row), $\approx 0.08$ (second), $\approx 0.11$ (third) and $\approx 0.15$ (fourth) as well as the exciton direct emission spectra $S_{\sigma }$ in the fifth columns with two cases of cavity pumping, $P_a/g=0$ (outer peaks) and $\approx 0.15$, corresponding to first and fourth rows of the cavity emission. Exciton spectra are less qualitatively affected by the cavity pumping. With electronic pumping only, no particular feature is observed in the cavity emission. In this case, cavity pumping makes a huge difference by revealing the underlying Jaynes-Cummings ladder.

Figure 13

(Color online) Point 3 of Fig. 3. Spectral emission for the indicated electronic pumping $P_{\sigma }/g$: ${10}^{-3}$ (first column), $\approx 0.23$ (second), $\approx 7.56$ (third, lasing) and 1000 (fourth, quenching), for $P_a/g=0$ (thick line with no coloring), $\approx 0.20$, $\approx 0.81$, and $\approx 1.42$ as indicated in the top left panel, and similarly for others (apart from the case $P_a=0$, inner peaks corresponds to higher pumpings). In this system, broadening is always too high to allow any manifestation of the underlying Jaynes-Cummings structure. The structure could be mistaken for a bosonic system (or the other way around).

Figure 14

(Color online) Positions ${\omega }_p/g$ of the lines around ${\omega }_a=0$ in the luminescence spectrum with detuning and in the absence of pump. Columns correspond to various detunings, first column being the case of resonance (cf. Fig. 2). First three rows show in isolation the first, second, and third manifold, respectively. First manifold corresponds to the Boson or linear case (Ref. 1). Fourth row shows all manifolds together. Left-bottom panel is detailed for positive ${\omega }_p$ in Fig. 2.

Figure 15

(Color online) Anticrossing of the luminescence lines as detuning $\Delta ={\omega }_a-{\omega }_{\sigma }$ is varied. Here, ${\omega }_a=0$ is fixed and the QD bare energy is tuned from below the cavity (positive detuning) to above (negative detuning). Panels (a)–(d) correspond to point 1 and panel (e) to point 2. (a)–(d) are at zero cavity pumping, $P_a=0$. (a) and (d) are for $P_{\sigma }=0.03g$ [(d) is a zoom of (a)] and (b) and (c) for $P_{\sigma }=0.3g$. (a), (b), and (d) are the cavity emission $S_a\left(\omega \right)$ and (c) the direct exciton emission $S_{\sigma }\left(\omega \right)$. (e) is for $P_{\sigma }={10}^{-3}g$ and $P_a=g/5$ (cf., seventh row, first column of Fig. 12). The nonlinear central peaks give rise to very characteristic anticrossing profiles.