Cavity quantum electrodynamics with semiconductor quantum dots: Role of phonon-assisted cavity feeding

For a semiconductor quantum dot strongly coupled to a microcavity, we theoretically investigate phonon-assisted transitions from the exciton to a cavity photon, where the energy mismatch is compensated by phonon emission or absorption. By means of a Schrieffer-Wolff transformation we derive an effective Hamiltonian, which describes the combined effect of exciton-cavity and exciton-phonon couplings, and compute the scattering rates within a Fermi-golden-rule approach. The results of this approach are compared with those of a recently reported description scheme based on the independent boson model [U. Hohenester et al., Phys. Rev. B 80, 201311(R) (2009)] and a numerical density-matrix approach. All description schemes are shown to give very similar results. This demonstrates that phonon-assisted cavity feeding can be described in terms of a simple scattering process and does not require a non-Markovian treatment as suggested elsewhere. We present results for the spontaneous emission lifetime of a quantum dot initially populated with a single exciton or biexciton and for the spectral properties of an optically driven dot-cavity system operating in the strong-coupling regime. Our results demonstrate that phonon-assisted feeding plays a dominant role for strongly coupled dot-cavity systems when the detuning is of the order of a few millielectron volts.

Figure 1

(Color online) Level schemes used in our simulations. (a) Ground and excited quantum-dot states $e$, $g$ and cavity. The exciton state is coupled to the cavity, with the cavity coupling $g$. The exciton and cavity decay with rates ${\gamma }_{\text{rad}}$ and $\kappa$. In case of laser pumping, we also consider transitions from $g$ to $e$. (b) Biexciton level scheme, consisting of the dot ground state $g$, the exciton states $x$ and $y$, with orthogonal polarizations, and the biexciton state $XX$. In our simulations we use a basis of the bare quantum-dot states and these states together with up to two photons in the cavity. When the cavity and the dot are not perfectly aligned, both exciton transitions can couple to the cavity. We assume that the $XX\to x$ and $x\to g$ transitions couple with strength $g\text{cos}\theta$ and the $XX\to y$ and $y\to g$ transitions with strength $g\text{sin}\theta$ to the cavity. Here $\theta$ is an angle that determines the degree of mixing. Pumping brings the system from the ground state via the exciton states $x$ and $y$ to the biexciton state $XX$.

Figure 2

(Color online) Cavity feeding rate for the phonon-assisted transition from the exciton to the cavity. We use the dot and material parameters listed in Table and an exciton-cavity coupling $g=150\mu \text{eV}$. The solid and dashed lines report results obtained from Eqs. (20, 13) for the independent boson model and the Schrieffer-Wolff perturbation approach, respectively.

Figure 3

(Color online) Spectra for driven dot-cavity system and for different quality factors. We use a level scheme consisting of the quantum-dot ground and excited states and the cavity. The panels report results for detunings of (a) $\Delta =-50\mu \text{eV}$, (b) $\Delta =0$, and (c) $\Delta =50\mu \text{eV}$. The temperature is set to $T=10\text{K}$.

Figure 4

(Color online) Exciton decay for a dot embedded in the cavity and in the presence of phonon-assisted transitions. The solid and dashed lines report results of our density-matrix and master-equation approaches, respectively. For clarity, the decay transients for different detunings are offset in time. Throughout, the decay for positive (blue) detuning (exciton energy larger than cavity energy ${\omega }_{\text{cav}}$) is faster than for negative (red) detuning. In all simulations we use $Q=5000$ and $T=10\text{K}$. The inset reports results for the correlation function $C\left(t\right)$ of the independent boson model (19). The solid lines correspond to the analytic expressions, and the symbols show the results of the density-matrix approach. Both results are in perfect agreement, thus demonstrating the accuracy of our numerical approach.

Figure 5

(Color online) SE decay time of initially excited quantum dot for different quality factors and temperatures (same line labeling as in Fig. 2). The gray lines report results where phonon scatterings have been artificially neglected. In the insets we compare, for a smaller range of detunings, the density-matrix approach (symbols) with the perturbation approach (solid lines), finding perfect agreement throughout.

Figure 6

(Color online) Spectrum of driven dot-cavity system, as depicted in Fig. 1. (b) Spectrum as a function of detuning between exciton and cavity. One observes the formation of polariton states, which anticross at zero detuning as a result of strong coupling. In our simulations we use a quality factor of $Q=15000$, a temperature of $T=10\text{K}$, a radiative dot lifetime of ${\tau }_{\text{rad}}=7\text{ns}$, and an exciton dephasing time of 50 ps. (a) Intensity of lower and upper polariton branches. The intensity is integrated over the regions indicated by the dashed lines in panel (b). The bright lines show results of simulations where phonon scatterings have been artificially neglected. The numbers in circles are used for the discussion in the text, and the dashed line indicates the position of zero detuning where the two polariton modes have equal intensity.

Figure 7

(Color online) Relative intensity of quantum-dot-like polariton branch as a function of (a) quality factor $Q$ (with $T=10\text{K}$) and (b) temperature (with $Q=15000$). The dashed lines indicate zero detuning. The relative intensity of the polariton modes is computed by integrating over the area for the lower (red) [upper (blue)] polariton for negative (positive) detunings, as indicated in Fig. 6b, and dividing by the sum of the upper and lower polariton intensities. For discussion see text.

Figure 8

(Color online) Time evolution of biexciton and exciton populations in the biexciton cascade decay. In the simulations, at time zero only the biexciton is populated, which decays via the exciton states to the ground state. The different rows report the population transients for the exciton $X_{\text{fast}}$, which couples with $g\text{cos}\theta$ strongly to the cavity, the exciton $X_{\text{slow}}$, which couples with $g\text{sin}\theta$ weakly to the cavity, and the biexciton $XX$. The different columns report results for different mixing angles $\theta$ [see Fig. 1b]. In our simulations we use $Q=15000$ and $T=10\text{K}$. The contour lines indicate where the population has decayed to 37% of its maximum value. For the fast exciton and the biexciton decay these contour lines are also magnified to make the features at early times better visible. The dashed lines indicate the position where the biexciton is in resonance with two cavity photons.

Figure 9

(Color online) Steady-state exciton and biexciton populations of a driven biexciton system. We use $Q=15000$ and $T=10\text{K}$. The labeling of the exciton and biexciton states is identical to Fig. 8. The different columns correspond to different pumping rates, indicated on top of the panels.

Figure 10

(Color online) Simulated spectra for driven biexciton system. We use $Q=15000$, $T=10\text{K}$, a mixing angle of $\theta =5°$, and a pump rate of 1/(5 ns). To make the biexcitonic features better visible, we use a logarithmic color scale. The intensities of the polariton modes are computed in accordance to the prescription given in the caption of Fig. 6. The numbers in circles are used in the discussion in the text.