Figure 1

(Color online) Upper panel: The valence and conduction band states of a prototypical semiconductor are charcterized according to the projection of the total angular momentum onto the positive $z$ axis, $m_j$. Solid lines depict valence to conduction band excitations of an electron that are promoted by absorption of left-hand circularly polarized light $[{\sigma }^{(-)}={\widehat{e}}_L=({\widehat{e}}_x-i{\widehat{e}}_y)∕\sqrt{2}]$, while dashed lines indicate the absorption of right-hand circularly polarized light $[{\sigma }^{(+)}={\widehat{e}}_R=({\widehat{e}}_x+i{\widehat{e}}_y)∕\sqrt{2}]$. Lower panel: Two relative orientations of the QD local reference frame $(x^{\prime },y^{\prime },z^{\prime })$ relative to the fixed (laboratory) axis system $(x,y,z)$ are shown. In each case the incoming photon propagates along $\widehat{z}$. Considering the selection rules for one-photon absorption, it can be shown that in (a) left-hand circularly polarized light excites the ${\Psi }_{-1}$ QD exciton. However, when the QD is rotated 180° right-hand circularly polarized light excites that same exciton.Reuse & Permissions

Figure 2

Kinetic scheme used for simulations of the 3-TG signals. The exciton states interconvert according to the rate constants $k_{pm}$ and $k_{mp}$.Reuse & Permissions

Figure 3

(Color online) Calculated imaginary (absorptive) contribution to the heterodyne-detected VHVH 3-TG signals for cases of (i) no spin flips, where the decay of the signal follows exciton population recombination, and (ii) a spin flip time $1∕k_s$ of $1\mathrm{ps}$. The dashed line (iii) shows the homodyne-detected signal calculated for the same conditions as (ii).Reuse & Permissions

Figure 4

Synopsis of the generation of a 3-TG signal for the VHVH experiment, showing the signal evolution for two population times. The initial pump pulse sequence (VH) creates equal numbers of $F=+1$ and $F=-1$ excitons, with corresponding densities ${\rho }_{pp}$ and ${\rho }_{mm}$, respectively. If those population densities are probed at some time delay $t_p$ later, then the corresponding third-order polarizations $P_1^{\left(3\right)}$ and $P_3^{\left(3\right)}$ are induced and radiate from the sample to be detected as a signal. That signal intensity decays with $t_p$ according to the usual kinetics of exciton trapping and recombination. Exciton spin flips are monitored by the population densities ${\rho }_{m^{\prime }{m}^{\prime }}$ and ${\rho }_{p^{\prime }{p}^{\prime }}$. The state $m^{\prime }$ is physically indistinguishable from $m$, but is differentiated because while $m$ was directly excited by the pump sequence, $m^{\prime }$ was formed by a spin flip from the state $p$. The VHVH 3-TG experiment can monitor the history of the exciton populations because the third-order polarizations radiated by ${\rho }_{m^{\prime }{m}^{\prime }}$ and ${\rho }_{p^{\prime }{p}^{\prime }}$, $P_2^{\left(3\right)}$ and $P_4^{\left(3\right)}$, are phase shifted by $\pi$ from $P_1^{\left(3\right)}$ and $P_3^{\left(3\right)}$. The four radiated polarizations thus interfere destructively and the total signal intensity decays according to the increasing ratio of $({\rho }_{pp}+{\rho }_{mm})$ to $({\rho }_{m^{\prime }{m}^{\prime }}+{\rho }_{p^{\prime }{p}^{\prime }})$ To illustrate that, the time-resolved polarizations radiated at $t_p=500\mathrm{fs}$ and $t_p=3500\mathrm{fs}$ are compared in (a) and (b) respectively. Those conditions correspond to the circled points on the time-integrated signal shown in Fig. 3. It is clear that even though the total time-integrated signal is zero at $t_p=3500\mathrm{fs}$, the exciton populations are still significant. The total signal is zero because of interference between the the radiated polarizations ${\mathbf{P}}_1^{\left(3\right)}(0,t_p,t)$ to ${\mathbf{P}}_4^{\left(3\right)}(0,t_p,t)$.Reuse & Permissions

Figure 5

(a) Simulated VHVH 3-TG decays for various spin flip rates: (i) no spin flips, (ii) $k_s^{-1}=5\mathrm{ps}$, (iv) $k_s^{-1}=1\mathrm{ps}$, (v) $k_s^{-1}=0.5\mathrm{ps}$. For the simulation result (iii) the forward and reverse spin flip rates differ: $k_{pm}^{-1}=1\mathrm{ps}$ and $k_{mp}^{-1}=2.5\mathrm{ps}$. (b) Plot of the exciton population densities ${\rho }_{pp}$ (solid line) and ${\rho }_{p^{\prime }{p}^{\prime }}$ (dashed line) for the simulation with $k_s^{-1}=1\mathrm{ps}$. (c) Similar to (b), but for simulation (iii).Reuse & Permissions

Figure 6

Experimental setup for optical heterodyne detected transient grating (OHD-TG) measurements. $\lambda ∕2$: half-wave plate, $P$: polarizer, $L$: lens, $\mathit{DO}$: diffractive optics, $M$: $\mathit{PM}$: parabolic mirror, $\mathit{CS}$: cover slip, $G$: glass plate, $S$: sample, $\mathit{PD}$: photodiode, $C$: chopper.Reuse & Permissions

Figure 7

(Color online) (a) Imaginary part of the heterodyne-detected 3-TG signals for CdSe quantum dots with average radius $R=1.72\mathrm{nm}$. The VVVV, VHVH, and VHHV signals are compared. The intensity scale for this plot is linear. (b) Comparison of (i) the VHVH signal for the $R=1.72\mathrm{nm}$ CdSe sample shown in (a) with (ii) the homodyne-detected VHVH 3-TG data for CdSe with $R=1.70\mathrm{nm}$. Curve (iii) is the homodyne signal reconstructed from the real and imaginary parts of the heterodyne-detected 3-TG data, which is evidently qualitatively similar to curve (ii).Reuse & Permissions

Figure 8

(Color online) Results of a control experiment showing the heterodyne-detected 3-TG signals for VVVV, VHVH, and VHHV polarization sequences. These signals all contain the same information for this laser dye (rhodamine 6G) because its excited states obey selection rules for linearly-polarized light.Reuse & Permissions

Figure 9

Simulated RRRR and RRLL 3-TG decays for a spin flip rate $k_s^{-1}=1\mathrm{ps}$. The inset compares the difference between the RRRR and RRLL signal intensity to a comparable simulation of the VHVH 3-TG signal.Reuse & Permissions