Signatures of long-range spin-spin interactions in an (In,Ga)As quantum dot ensemble

We present an investigation of the electron spin dynamics in an ensemble of singly charged semiconductor quantum dots subject to an external magnetic field and laser pumping with circularly polarized light. The spectral laser width is tailored such that ensembles with an increasing number of quantum dots are coherently pumped. Surprisingly, the dephasing time $T^{*}$ of the electron spin polarization depends only weakly on the laser spectral width. These findings can be consistently explained by a cluster theory of coupled quantum dots with a long-range electronic spin-spin interaction. We present a numerical simulation of the spin dynamics based on the central spin model that includes a quantum mechanical description of the laser pulses as well as a time-independent Heisenberg interaction between each pair of electron spins. We discuss the individual dephasing contributions stemming from the Overhauser field, the distribution of the electron $g$ factors, and the electronic spin-spin interaction as well as the spectral width of the laser pulse. This analysis reveals counterbalancing effects on the total dephasing time when increasing the spectral laser width. On one hand, the increasing deviations of the electron $g$ factors reduce the dephasing time. On the other hand, more electron spins are coherently pumped and synchronize due to the electronic spin-spin interaction which extends the dephasing time. We find an excellent agreement between the experimental data and the dephasing time in the simulation using an exponential distribution of Heisenberg couplings with a mean value $\overline{J}\approx 0.26\mu \mathrm{eV}$.

Figure 1

Laser pulse spectra with varying spectral widths $\mathrm{\Delta }E$ and corresponding temporal lengths $\mathrm{\Delta }\tau$. The uppermost spectrum is the photoluminescence (PL) emission of the QD ensemble.

Figure 2

Time-resolved ellipticity traces showing the spin precession and dephasing of the QD ensemble at (a) $B_{\mathrm{ext}}=1\mathrm{T}$ and (b) $B_{\mathrm{ext}}=6\mathrm{T}$, both taken at $T=6$ K for the different pulse widths given in Fig. 1.

Figure 3

Measured (red, blue) signal decay times in dependence of the spectral pulse width $\mathrm{\Delta }E$ at $B_{\mathrm{ext}}=0.25\mathrm{T}$ (a), $B_{\mathrm{ext}}=1\mathrm{T}$ (b), and $B_{\mathrm{ext}}=6\mathrm{T}$ (c). The data with constant power (blue, open circles) are measured by keeping the total average laser power constant for all spectral widths. For adjusted power (red, full circles), we used a fixed spectral excitation power density such that all QDs are pulsed by a $\pi$ pulse. Hence the total average power linearly depends on the spectral width.

Figure 4

Signal decay time for a fixed pulse duration of $2\mathrm{ps}$ in dependence on the external magnetic field $B_{\mathrm{ext}}$. The expected $1/B_{\mathrm{ext}}$ dependence is depicted by the dash-dotted gray line. The blue solid line shows a fit to the data proportional to $1/B_{\mathrm{ext}}^{\alpha }$.

Figure 5

Dephasing due to the nuclear spin fluctuations. We set the parameters to $T_N^{*}=3\mathrm{ns}$, $g^{\left(i\right)}=0.555$, and $J_{i,j}=0$. The $S_z$ component of the electron spin as function of the time $t$ after a resonant $\pi$ pulse is depicted as a blue curve for three different magnetic fields in (a), (b), and (c). Differences between homogeneous and exponentially distributed coupling constants cannot be resolved on this timescale. The dashed lines mark the analytical envelope functions according to Eq. (28).

Figure 6

Distributions of the trion excitation energy ${\epsilon }_{\mathrm{T}}^{\left(i\right)}$ and the electron $g$ factor $g_{\mathrm{e}}^{\left(i\right)}$. The trion excitation energy is Gaussian distributed ${\epsilon }_{\mathrm{T}}^{\left(i\right)}\sim \mathcal{N}({\epsilon }_{\mathrm{T},0},{\sigma }_{{\epsilon }_{\mathrm{T}}})$ (solid blue line) based on the experimental photoluminescence spectrum (cf. Fig. 1). The linear relation between the average $g$ factor ${\overline{g}}_{\mathrm{e}}^{\left(i\right)}$ and the trion excitation energy ${\epsilon }_{\mathrm{T}}^{\left(i\right)}$ as stated in Eq. (32) is shown as a black dashed line. The red dots depict 5000 pairs $({\epsilon }_{\mathrm{T}}^{\left(i\right)},g_{\mathrm{e}}^{\left(i\right)})$ with $g_{\mathrm{e}}^{\left(i\right)}\sim \mathcal{N}({\overline{g}}_{\mathrm{e}}^{\left(i\right)},{\sigma }_{g,0})$ generated randomly.

Figure 7

Dephasing due to the electron $g$-factor dispersion. The spin dynamic with the parameters $A_k^{\left(i\right)}=0$, $J_{i,j}=0$, $\mathrm{\Delta }E=5\mathrm{meV}$ is depicted for three different external magnetic fields in (a), (b), and (c). The dashed lines mark the Gaussian envelope functions with standard deviation $T_g^{*}$ according to Eq. (34).

Figure 8

Dependency of the electron $g$ factor induced dephasing time $T_g^{*}$ on the external magnetic field $B_{\mathrm{ext}}$. The data points (blue triangles) are extracted from numerical calculations with the parameters $A_k^{\left(i\right)}=0$, $J_{i,j}=0$, $\mathrm{\Delta }E=5\mathrm{meV}$. The red line depicts the dependency $\propto B_{\mathrm{ext}}^{-1}$ according to Eq. (34).

Figure 9

Dependency of the electron $g$ factor induced dephasing time $T_g^{*}$ on the laser bandwidth $\mathrm{\Delta }E$. The data points (blue triangles) are extracted from numerical calculations with the parameters $A_k^{\left(i\right)}=0$, $J_{i,j}=0$, and $B_{\mathrm{ext}}=1\mathrm{T}$. The analytical solution from Eq. (39) is depicted by the red line.

Figure 10

Dephasing due to the electronic spin-spin interaction. The blue line depicts the calculated spin dynamics with the parameters $A_k^{\left(i\right)}=0$, $g=0.555$, $\overline{J}=0.4{\mathrm{ns}}^{-1}$, and $B_{\mathrm{ext}}=1\mathrm{T}$. The envelope function (green line) is given by Eq. (42). The initial dephasing after the pulse can be approximated by a Gaussian envelope function (red line) with standard deviation $T_J^{*}$.

Figure 11

Testing the analytical results of the toy model with a numerical calculation. Two QDs are excited by an ideal $\pi$ pulse at $t=0$. The dephasing time with a strong interaction ($\overline{J}=200{\mathrm{ns}}^{-1}$, blue curve) is $\sqrt{2}$ times longer than the dephasing time without interaction ($\overline{J}=0$, green curve). The parameters $T_{\mathrm{N}}^{*}=3\mathrm{ns}$, $g^{\left(i\right)}=0.555$, and $B_{\mathrm{ext}}=1\mathrm{T}$ are used.

Figure 12

Dephasing time as a function of the spectral width of the laser for three different magnetic fields $0.25\mathrm{T}$ (a), $1\mathrm{T}$ (b), and $6\mathrm{T}$ (c). All three dephasing effects are included in the numerical calculation. There is a good agreement for $\overline{J}=0.4{\mathrm{ns}}^{-1}$ with the experimental data (cf. Fig. 3).

Figure 13

Comparison between the numerical results for $\overline{J}=0.4{\mathrm{ns}}^{-1}$ after a single laser pulse (solid lines) and the experimental measurements from Sec. 3 (x markers) with adjusted power. (a) Dephasing time $T^{*}$ as function of the spectral laser width $\mathrm{\Delta }E$. Results for various external magnetic fields are depicted in different colors. In addition to the numerical results after a single pulse, we added the calculations after 100 pulses as dashed lines. (b) Dependence of $T^{*}$ on the external magnetic field $B_{\mathrm{ext}}$.

Figure 14

Dephasing time as a function of the magnetic field $B_{\mathrm{ext}}$. All three dephasing effects are included in the numerical calculation with $\overline{J}=0.4{\mathrm{ns}}^{-1}$ and $\mathrm{\Delta }E=1.5\mathrm{meV}$. The data points of the numerical calculation (blue triangles) are compared to a dependency $\propto 1/B_{\mathrm{ext}}$ (green curve) and dependency $\propto 1/B_{\mathrm{ext}}^{\alpha }$ with $\alpha =0.7$ (red curve).