Resonant two-color high-resolution spectroscopy of a negatively charged exciton in a self-assembled quantum dot

We report high-resolution resonant two-color laser spectroscopy on single self-assembled InGaAs quantum dots. The negatively charged exciton in a quantum dot can decay radiatively into both electron-spin ground states, the states with an electron in the quantum dot with its spin parallel or antiparallel to an applied magnetic field. The two decay paths can be used for optical spin alignment via optical pumping. We apply two laser fields at two different colors resonant with the two Zeeman split optical transitions to study the properties of the spin of the resident electron in the quantum dot in both the Faraday and Voigt geometries. The resonant two-color signal is monitored as a function of the laser power as well as the applied magnetic and electric field allowing us to determine spin decay and dephasing rates. We find the rate at which the optical spin alignment can be performed depending on the direction and the magnitude of the applied magnetic field. Finally we demonstrate the feasibility of performing coherent all-optical spin manipulation of an electron spin in a quantum dot.

Figure 1

(Color online) High-resolution resonant laser spectroscopy on a single QD in Faraday geometry. The data were recorded on dot A. In (a) the unpolarized resonance line of a QD charged with one electron is shown against energy detuning. The change in transmission through the sample of a narrow-band laser is measured, while the exciton resonance is detuned from the laser photon energy by the dc Stark shift. The transition is depicted in (b) on the right side with no applied magnetic field. The resonant laser drives the transition with a Rabi frequency $\Omega$, while the exciton decays spontaneously with rate $\Gamma$. On the right side of (b) the transitions of a negatively charged exciton are shown, with a magnetic field applied along the growth direction. The ground and excited states are split by the Zeeman energy. A transmission spectrum for a magnetic field of 0.5 T is shown in (c). The laser was linearly polarized to resolve both Zeeman split resonances. The energy difference between the two resonances corresponds to the sum of the Zeeman splittings of the excited and ground states. As depicted in (b) the Zeeman splitting of the exciton is determined by the hole and the Zeeman splitting of the ground state is determined by the electron. The lines are Lorentzian fits to the data, yielding a linewidth at half maximum of $\approx 3.5\mu \text{eV}$.

Figure 2

(Color online) Level schemes of a negatively charged QD in a magnetic field. In (a) the transitions responsible for the OSP are shown. A laser pumps resonantly the transition $|1⟩\leftrightarrow |3⟩$ with Rabi frequency ${\Omega }_1$ and creates an exciton. The exciton decays spontaneously into the ground state $|1⟩$ and $|2⟩$ at rate $\Gamma$ and ${\gamma }_s$, respectively, where transition rate ${\gamma }_{\text{s}}⪡\Gamma$ such that we label transition $|2⟩\leftrightarrow |3⟩$ as a forbidden transition. The spin-flip rate ${\gamma }_e$ leads to a relaxation of the spin polarization toward equal population of the two states $|1⟩$ and $|2⟩$. In (b) the two laser, repump scheme is shown. An additional laser pumps resonantly the forbidden transition $|2⟩\leftrightarrow |3⟩$, with Rabi frequency ${\Omega }_2$. This leads to a transfer of ground-state population from $|2⟩$ to $|1⟩$ and hence frustrates the OSP.

Figure 3

(Color online) Measured transmission contrast versus gate voltage, over the charging plateau, where the QD contains one resident electron, e. The data were recorded on Dot A. The transmission is normalized to the maximum contrast on the plateau. (a) The $X^{1-}$ plateau with no magnetic field applied. At the edges toward the empty (zero) and doubly negatively charged QD (2e), the transmission contrast decays following Eq. (4) (line), corresponding to a temperature $T=4.2\text{K}$. In (b) the transmission contrast is plotted in the same gate voltage range, with a magnetic field of 0.2 T applied. In the middle of the plateau the OSP leads to a drastic reduction in the transmission contrast. The line is a fit to the data of a product of Eq. (2) using Eq. (1) to describe $\vartheta$ and Eq. (5) to calculate the gate voltage dependent spin-flip rate ${\gamma }_e$ $\left(\hslash {\Gamma }_{\text{tun}}=0.05\mu \text{eV}\right)$.

Figure 4

(Color online) Transmission spectra in the OSP regime of the $X^{1-}$ gate voltage range, close to the cotunneling regime. The data were recorded on Dot B. The open circles are obtained from a one-laser transmission experiment at a magnetic field of 0.5 T with circular polarization. The weak remaining signal at $V_G=-1.065\text{V}$ arises because cotunneling suppresses OSP sufficiently. At the center of the charging plateau the signal is too weak to be resolved in the noise. According to the pump scheme in Fig. 2, a second linearly polarized laser is focused on the sample. The photon energy of this second laser is chosen such that at a gate voltage of $V_G=-1.065\text{V}$, the laser photons are resonant with the forbidden transition. The experiment is repeated: the transmission is monitored as the gate voltage is scanned while the photon energy of the two lasers remains constant. At the gate voltage where in the one-laser experiment a faint resonance is observed, an enhanced signal appears in the two-laser experiment. At an $\approx 25\text{mV}$ more negative gate voltage, a second resonance becomes visible, originating from the second laser, whose energy in this case matches the excitonic transition $|▲⟩\leftrightarrow |▲▼,△⟩$ at this gate voltage. The resonance condition of the second laser with the excitonic transition falls into the cotunneling regime, yielding a strong resonance. The gate voltage differs from that in Fig. 3 because the experiment was performed on a different QD. To check the symmetry of the pumping scheme, the experiment was repeated on the low-energy Zeeman branch, yielding the same behavior (data not shown).

Figure 5

(Color online) Two color, repump spectra of the $|▲⟩\leftrightarrow |▲▼,△⟩$ transition by pumping the $|▼⟩\leftrightarrow |▲▼,△⟩$ forbidden transition with differently polarized laser fields. The data were recorded on Dot B. In (a) the second laser was linearly polarized, according to the linear polarization basis defined by the neutral exciton fine structure splitting. In (b) and (c) the second laser field is left-handed and right-handed circularly polarized, respectively. The circular polarization basis was defined by one color spectroscopy on the $X^{1-}$ in a magnetic field (Ref. 21). No effect of the repumping signal on the polarization of the repump laser is observed. The laser pumping the $|▲⟩\leftrightarrow |▲▼,△⟩$ transition was ${\sigma }^{+}$ polarized. The lines are Lorentzian fits to the data.

Figure 6

(Color online) (a) Transmission contrast of a laser pumping the $|1⟩\leftrightarrow |3⟩$ transition with (open circles) and without (solid dots) a second laser pumping the $|2⟩\leftrightarrow |3⟩$ forbidden transition. All data were recorded on Dot B. A magnetic field of $B=0.5\text{T}$ was applied to obtain OSP. In (b) the spin-flip rate ${\gamma }_e$, obtained by fitting the quantum optical theory to the measured spectra, is plotted as a function of gate voltage. The line is a fit of Eq. (5) to the data points, with $\hslash {\Gamma }_{\text{tun}}=0.05\mu \text{eV}$. To account for the gate voltage independent hyperfine mediated spin-flip events, a constant term ${\gamma }_{\text{hf}}=0.42\text{MHz}$ was added to ${\gamma }_e$ [see Eq. (6)].

Figure 7

(Color online) Spin-flip rate ${\gamma }_e$ (a) and forbidden transition rate ${\gamma }_s$ (b) as a function of the applied magnetic field. The data was obtained by fitting the quantum optical theory to the spectra measured on Dot B in one-laser and two-laser spectroscopies. The evolution of the rates with magnetic field is explained by a simple model describing electron-nuclear spin flip-flop processes. The inset in (a) illustrates the model (Ref. 34), which assumes that the strength of the hyperfine interaction is linearly related to the overlap of the electronic states $|1⟩$ and $|2⟩$ each broadened by $\hslash \delta \omega$ and separated by the Zeeman energy $\left(2ϵ=\left|g_e{\mu }_{B}B\right|\right)$. The ratio of the overlap region (gray) of the two Lorentzians to the whole area of the Lorentzians gives the probability of equal energy of the two spin states. With this probability, an electron spin flip can occur where the spin is transferred to a nucleus. The lines in (a) and (b) are fits of Eq. (20) to the data points with the fitting parameter $\hslash \delta \omega =6.4\text{neV}$, which is of the same orders of magnitude as the dephasing rate ${\gamma }_2$ of an electron spin in this QD. In (b) an additional offset of 2.5 MHz is added, to account for the admixture of light holes to the excitonic state.

Figure 8

(Color online) The transmission contrast of the first laser, resonant with the $|1⟩\leftrightarrow |3⟩$ transition, is plotted against the Rabi frequency ${\Omega }_2$ of the second laser. The data were recorded on Dot B. The line is a fit of the contrast predicted by the quantum optical theory to the data points. The parameters are $\Gamma =1.2\text{GHz}$, ${\Omega }_1=2\text{GHz}$, ${\gamma }_s=35\text{MHz}$, and ${\gamma }_e=1.2\text{MHz}$, and the maximum transmission contrast ${\alpha }_0=1.93\times {10}^{-3}$. From the fit, an upper bound for the dephasing rate ${\gamma }_2=22\text{MHz}$ is obtained. The experiment was performed with an external magnetic field $B=0.5\text{T}$ applied.

Figure 9

(Color online) (a) One-laser transmission spectra of singly charged Dot C with an in-plane magnetic field applied $\left(B=0.7\text{T}\right)$. The three spectra were obtained for different polarizations of the laser field. For the upper curve, the laser was circularly polarized. The lower two curves were obtained for two orthogonal linear polarizations (see main text). The lines are Lorentzian fits to the data and an offset was added to the spectra for clarity. From the energy differences between the resonances, the Zeeman splitting of the electron and hole states can be calculated. In (b) these splittings are plotted as a function of the magnetic field. From a linear fit of the data points, the absolute values of the electron and hole $g$ factors are determined to be $\left|g_e\right|=0.47$ and $\left|g_h\right|=0.24$.

Figure 10

(Color online) Resonance energy of the $X^{1-}$ excitonic resonances of Dot C in Voigt geometry as a function of the magnetic field. The data points were obtained by fitting the spectra with Lorentz functions [see Fig. 9a]. The spectra were measured in the cotunneling regime to avoid OSP. The data points are fit by parabolas, using the $g$ factors from Fig. 9b and a diamagnetic shift of $-9.4\mu \text{eV}/{\text{T}}^2$. In the inset, the electronic and excitonic levels are sketched, together with the optical transitions between them.

Figure 11

OSP of the resident electron spin in a QD in Voigt geometry. The data were recorded on Dot C. (a) The transmission contrast (dark colors correspond to high transmission contrast) as a function of the gate voltage and the laser energy. The dashed lines denote the Coulomb blockade plateau, the voltage range in which the QD is charged with a single electron [compare to Fig. 3a]. The dotted lines mark the boundaries of the region in which the electron spin is optically pumped, leading to a much reduced transmission contrast. In between the dashed and dotted lines at the plateau edges the transmission contrast is high because of fast spin relaxation due to cotunneling [see also Fig. 3b]. The experiment was performed in a magnetic field of $B=0.7\text{T}$, with a circularly polarized laser field, yielding four distinct resonances [compare to Fig. 9a]. In (b), a second laser at a fixed energy 1.27997 eV was additionally used. This second laser was linearly ${\pi }_y$ polarized and the scanned laser was orthogonally polarized, along ${\pi }_x$. At the edges of the plateau again the signal due to cotunneling appears. In the middle of the plateau, the transmission signal of the scanned laser reappears due to optical repumping, according to the three repumping schemes (c), (d), and (e). In (c) the second, fixed laser is resonant with the $|\tilde{2}⟩\leftrightarrow |\tilde{3}⟩$ transition, while the scanned laser hits the $|\tilde{1}⟩\leftrightarrow |\tilde{3}⟩$ transition. In (d) the fixed and scanned lasers are resonant with the $|\tilde{1}⟩\leftrightarrow |\tilde{4}⟩$ and $|\tilde{2}⟩\leftrightarrow |\tilde{3}⟩$ transitions, respectively. Depending on which laser hits, which resonance at the given gate voltage, these repump signals appear as marked in (b). The recovery for the repump scheme (d) is weaker than for (c) and (e) because the polarization of the scanned laser does not match the polarization of the resonance it is pumping. In (e), similar to (c), the fixed laser is resonantly pumping the transition $|\tilde{1}⟩\leftrightarrow |\tilde{4}⟩$, while the scanned laser photon energy matches the $|\tilde{2}⟩\leftrightarrow |\tilde{4}⟩$ transition energy.

Figure 12

(Color online) Power dependence of the repump signal in a two-laser experiment. One laser resonantly pumps the $|\tilde{1}⟩\leftrightarrow |\tilde{3}⟩$ transition with a Rabi frequency ${\Omega }_1=2.5\text{GHz}$. The Rabi frequency of the second laser, resonant with the transition $|\tilde{2}⟩\leftrightarrow |\tilde{3}⟩$, is varied. The data were recorded on Dot C. The line represents a fit of the quantum optical theory to the data, with a dephasing rate ${\gamma }_2=6\text{GHz}$. The other parameters are $\Gamma ={\gamma }_s=4.2\text{GHz}$, ${\gamma }_e=1.2\text{MHz}$, and ${\alpha }_0=1.8\times {10}^{-3}$. The magnetic field was $B=0.7\text{T}$. The Rabi frequency of ${\Omega }_1=2.5\text{GHz}$, which leads to a power broadening and a reduced transmission contrast compared to Fig. 8, was chosen for optimum signal-to-noise ratio. The dashed line is a plot of the calculated transmission contrast expected for a dephasing rate ${\gamma }_2=22\text{MHz}$, the upper bound determined for Dot B in Sec. 5.