Optical Stark shift to control the dark exciton occupation of a quantum dot in a tilted magnetic field

When a detuned and strong laser pulse acts on an optical transition, a Stark shift of the corresponding energies occurs. We analyze how this optical Stark effect can be used to prepare and control the dark exciton occupation in a semiconductor quantum dot. The coupling between the bright and dark exciton states is facilitated by an external magnetic field. Using sequences of laser pulses, we show how the dark exciton and different superposition states can be prepared. We give simple analytic formulas, which yield a good estimate for optimal preparation parameters. The preparation scheme is quite robust against the influence of acoustic phonons. We further discuss the experimental feasibility of the used Stark pulses. Giving a clear physical picture, our results will stimulate the usage of dark excitons in schemes to generate photons from quantum dots.

Figure 1

Sketch of the system in the ground-state manifold of a QD including energy shifts and coupling to the light and magnetic fields. We only consider the “$-$” subsystem marked by color. The arrows indicate the spin configuration of the participating excitons.

Figure 2

(a) Dynamics of the occupations $g, b$, and $d$ of the states in the three-level system, (b) evolution of the dressed-state energies $E_i$ and (c) sequence of the laser fields. Parameters for this example are $\hslash \mathrm{\Delta }=15\mathrm{meV}, {\delta }_{bd}=1.5\mathrm{meV}$ and $\hslash {\mathrm{\Omega }}_0=9.95\mathrm{meV}$.

Figure 3

(a) Mean occupation of the dark exciton state after the action of the Stark pulse as function of detuning and strength of the Stark pulse $\hslash {\mathrm{\Omega }}_0$ for a bright-dark splitting of ${\delta }_{bd}=1.5\mathrm{meV}$. The red dashed line indicates Eq. (11). (b) Final occupation as function of detuning and effective bright-dark splitting. The colored dots indicate the parameters shown in the examples of the dynamics shown on the right (red and black) and in Fig. 2 (blue). (c), (d) Occupations $g, b$, and $d$ in the three-level system (c) for detuning $\hslash \mathrm{\Delta }=1.5\mathrm{meV}$ bright dark-splitting ${\delta }_{bd}=1.5\mathrm{meV}$ and $\hslash {\mathrm{\Omega }}_0=4.24\mathrm{meV}$ [red dot in (b)] and (d) $\hslash \mathrm{\Delta }=15\mathrm{meV}, {\delta }_{bd}=0.15\mathrm{meV}$ and $\hslash {\mathrm{\Omega }}_0=3.01\mathrm{meV}$ [black dot in (b)].

Figure 4

Dynamics of (a) occupations $g, b, d$ of the states in the three-level system and (b) the coherences ${\rho }_{ij}$ with $i,j\in \{g,b,d\}$ under (c) the laser sequence. Same parameters as in Fig. 2.

Figure 5

Dynamics of the occupation including the exciton-phonon interaction for $T=1\mathrm{K}$ for (a) the dark-state preparation scheme [cf. Fig. 2] and (c) the preparation of the superposition [cf. Fig. 4]. (d) Dynamics of the coherences including the exciton-phonon interaction for the preparation into the superposition state in (c). (b) Difference of the final occupation without phonons [cf. Fig. 3] to the case including phonons as a function of detuning and bright-dark splitting. Note that a blue hue indicates a phonon enhancement of the preparation fidelity.

Figure 6

(a) Classification of the electric driving signal as a function of $1/\alpha$ and ${\tau }_{\text{S}}$. Colors indicate the frequency where 90% of the electrical driving signal is present (f90). Orange dashed line: 45 GHz line for modern EOM systems. The dots indicate the value used for the dark state preparation (red: Fig. 2; orange: this figure). (b) Dynamics of the occupation of the three-level system without phonons for the pulse sequence shown in (c). Parameters for this example are $\hslash \mathrm{\Delta }=15\mathrm{meV}, {\delta }_{bd}=1.5\mathrm{meV}$, and $\hslash {\mathrm{\Omega }}_0=10.1\mathrm{meV}$.