Luminescence spectra of quantum dots in microcavities. I. Bosons

We provide a unified theory of luminescence spectra of coupled light-matter systems realized with semiconductor heterostructures in microcavities, encompassing (i) the spontaneous emission case, where the system decays from a prepared (typically pure) initial state, and (ii) luminescence in the presence of a continuous incoherent pump. We show how, by provoking a self-consistent quantum state, the pump considerably alters the emission spectra, even at vanishing intensities. The main outcome of our analysis is to unambiguously identify strong coupling in situations where it appears in disguise or only seems to appear. Here, we consider bosonic matter fields, in which case fully analytical solutions can be obtained. This describes the case of quantum wells or large quantum dots, or the limit of low excitation where the probability to have particles in the system remains much smaller than one.

Figure 1

(Color online) Solid black: bare energies of the cavity photon (horizontal line) and of the exciton (tilted) as a function of detuning $\Delta$. Dashed black: eigenenergies of the system Hamiltonian, without dissipation nor pumping [Eq. (2)]. The excitonlike state at large negative $\Delta$ has become a photonlike state at large positive $\Delta$, and vice versa. Around $\Delta =0$, both modes are an admixture of exciton and photon. Dotted blue: Correction of the eigenenergies when pump and decay are taken into account for parameters of point (c) in Fig. 8. Solid red: actual position of the observed peaks in the photoluminescence (PL) spectra for the same parameters. For these parameters, the three descriptions of SC give the same qualitative results.

Figure 2

(Color online) Schema of our system for the SS case: self-assembled QDs in a semiconductor microcavity. The QD sketched on the right is in SC with the cavity mode with coupling strength $g$, while the one of the left is in WC. The electron-hole pairs created by the incoherent pumping of the structure provide an effective electronic pumping, $P_b$, of the SC dot, while the pumping of the WC dot results in an effective cavity pumping $P_a$, through rapid conversion of the excitons into cavity photons.

Figure 3

(Color online) Complex Rabi $R/g$, separated in its real (a) and imaginary (b) parts, as a function of the decoherence parameter ${\Gamma }_{-}/g$ for various detunings ($\Delta /g$ from $-1.6$, up, to 0, bottom, by steps of 0.4). Solid black lines correspond to resonance.

Figure 4

(Color online) Strong-coupling steady-state spectra (blue solid line) and their decomposition into Lorentzian (dotted purple) and dispersive parts (dashed green) for various detunings $\left(\Delta /g=0,1,3\right)$ with parameters of point (c) of Fig. 8: ${\gamma }_a=3.8g$, ${\gamma }_b=0.1g$, $P_a=0.5g$, $P_b=0.1g$. The vertical black lines mark the positions of the bare modes (cavity at ${\omega }_a=0$ and exciton at ${\omega }_b=-\Delta$), showing the “level repulsion” of SC.

Figure 5

(Color online) Time dynamics of the correlator $\mathfrak{R}{⟨a^{†}\left(t\right)a\left(t+\tau \right)⟩}^{\text{SE}}$, cf. Eq. (33). Only the pattern of oscillations is of interest here (lighter blues correspond to higher values). In all cases, both the $t$ and $\tau$ dynamics tend to zero. Figures (a) and (b) show the SE of an exciton and of an upper polariton, respectively, in a very strongly coupled system (${\gamma }_a=0.2g$ and ${\gamma }_b=0.1g$). Figure (c) shows the SE of an exciton in weak coupling $\left({\gamma }_a=5.9g\right)$. The oscillations in $\tau$, rather than in $t$, are the mark of strong coupling.

Figure 6

(Color online) (a) Strong-coupling spectra $S_0^{\text{SE}}\left(\omega \right)$ and (b) its corresponding mean number dynamics $n_a^{\text{SE}}\left(t\right)$ for the spontaneous emission of three different initial states. In blue solid, one exciton; in purple dashed, one photon; and in brown dotted, one upper polariton. Parameters are ${\gamma }_a=1.9g$ and ${\gamma }_b=0.1g$. Inset of (b) is the same in log scale.

Figure 7

(Color online) Dynamics of ${\text{lim}}_{t\to \infty }⟨a^{†}\left(t\right)a\left(t+\tau \right)⟩/n_a^{\text{SS}}$, Eqs. (14a, 33), for the steady state corresponding to points (a)–(e) in Fig. 8. In the inset, the same in log scale. Solid lines [(b) blue, (c) purple, and (d) brown] of SC feature oscillations of the correlator, as the mark of SC. Dashed lines [(a) green and (e) blue] correspond to weak coupling. Note that although the blue dashed line (e) appears to be similar to other SC lines, it does not oscillate in the log scale, where it only features a single local minimum. In the same way, the brown line (d) that seems not to oscillate actually features an infinite set of local minima, as is revealed in the log scale.

Figure 8

(Color online) Phase space of the steady-state strong/weak coupling as a function of $P_b/g$ and ${\gamma }_a/g$ for the parameters ${\gamma }_b=0.1g$ and $P_a=0.5g$. The red lines delimit the region where there is a steady state [Eqs. (62, 63)]. The blue line, Eq. (64), separates the strong (in shades of blue) from the weak (shades of red) coupling regions. The dotted black line, Eq. (65), separates SC and WC regions in the absence of pumping. The brown line, Eq. (66), separates the regions where one (dark blue) or two (light blue) peaks can be resolved in the luminescence spectra. This defines three areas in the SC region: (1) two peaks are resolved in the spectra, (2) the two peaks cannot be resolved and effectively merge into one, albeit in SC, and (3) SC is achieved thanks to the pump $P_a$ (with one or two peaks visible depending of the overlap with the light or dark area) despite the large dissipation that predicts WC according to Eq. (59). In the same way we can distinguish three regions in weak coupling: (I) standard WC, (II) SC with a two peaked spectrum, and (III) WC due to pumping $P_b$. The surrounding figures (a)–(e) show spectra (filled) from these regions and their decomposition into Lorentzian (green) and dispersive (brown) parts. Parameters correspond to the points in the inset: (a) ${\gamma }_a=3.8g$ and $P_b=g$, (b) ${\gamma }_a=3.8g$ and $P_b=0.5g$, (c) ${\gamma }_a=3.8g$ and $P_b=0.1g$, (d) ${\gamma }_a=4.49g$ and $P_b=0.1g$, (e) ${\gamma }_a=4.8g$ and $P_b=0.1g$. Observe how, in SC, two eigenstates have emerged, even in the cases—like in (b)—where they are not seen in the total spectrum. In the same way, in WC, all the emission emanates from the origin, although a two-peak structure can arise as a result of a resonance, also centered at the origin.

Figure 9

(Color online) Rabi splitting at resonance (dotted blue) given by $\pm \mathfrak{R}\left(R_0\right)$, Eq. (35), and the observed position of the peaks in the PL spectra (solid red) as a function of $P_b/g$. Parameters are those of the line of points (a)–(c) of Fig. 8: ${\gamma }_a=3.8g$, ${\gamma }_b=0.1g$, $P_a=0.5g$. The corresponding $P_b$ are marked for those points.

Figure 10

(Color online) Phase space of SC/WC as a function of the pumps $P_b/g$ and $P_a/g$ for fixed decay parameters ${\gamma }_a=3.8g$ and ${\gamma }_b=0.1g$. As in Fig. 8, the red lines mark the physical regions and the blue one the SC (blue shades)/WC (red shades) transition, with the same regions 1 and 2 of SC and III of WC, also with the points (a)–(c), of Fig. 8. In the inset, zoom of the low-pump region, showing the importance of both the angle, $\text{arctan}\left(1/\alpha \right)$, and the magnitude, $\sqrt{P_a^2+P_b^2}$, of a given point in the diagram.