Magnetic field dependence of the electron spin revival amplitude in periodically pulsed quantum dots

Periodic laser pulsing of singly charged semiconductor quantum dots in an external magnetic field leads to a synchronization of the spin dynamics with the optical excitation. The pumped electron spins partially rephase prior to each laser pulse, causing a revival of electron spin polarization with its maximum at the incidence time of a laser pulse. The amplitude of this revival is amplified by the frequency focusing of the surrounding nuclear spins. Two complementary theoretical approaches for simulating up to 20 million laser pulses are developed and employed that are able to bridge between 11 orders of magnitude in time: a fully quantum mechanical description limited to small nuclear bath sizes and a technique based on the classical equations of motion applicable for a large number of nuclear spins. We present experimental data of the nonmonotonic revival amplitude as function of the magnetic field applied perpendicular to the optical axis. The dependence of the revival amplitude on the external field with a profound minimum at $4\mathrm{T}$ is reproduced by both of our theoretical approaches and is ascribed to the nuclear Zeeman effect. Since the nuclear Larmor precession determines the electronic resonance condition, it also defines the number of electron spin revolutions between pump pulses, the orientation of the electron spin at the incidence time of a pump pulse, and the resulting revival amplitude. The magnetic field of $4\mathrm{T}$, for example, corresponds to half a revolution of nuclear spins between two laser pulses.

Figure 1

Photoluminescence (PL) spectra of the two studied samples. The spectra are taken at a temperature of $4.7\mathrm{K}$ with a photon excitation energy of $1.631\mathrm{eV}$. The laser pulses used in the pump-probe experiment are shown in red. (a) Sample 1 and (b) sample 2.

Figure 2

Dynamics of the electron spin projection onto the optical axis in different magnetic fields. The Faraday ellipticity measured for sample 2 is plotted vs the pump-probe delay. The dynamics are obtained at a temperature of $4.7\mathrm{K}$. The curves are shifted vertically for clarity.

Figure 3

Dependence of the revival amplitude on the external magnetic field. The revival amplitudes (the symbols) were extracted from the pump-probe spectra for the two different samples at temperature $4.7\mathrm{K}$. The lines are guides to the eye.

Figure 4

Influence of the hyperfine coupling constants $A_k$. (a) The numerically calculated time evolution of the electron spin component $〈S^z\left(t\right)〉$. At $t=0$, the spin is directed in the negative $z$ direction. The Gaussian envelope function according to Eq. (10) is indicated by black dashed lines. Small deviations are due the finite number of nuclei $N=6\left(N_C=100\right)$. (b) The distribution $p\left(T_j^{*}\right)$ of the dephasing time $T_j^{*}$ within a single configuration for different numbers $N$ of nuclei. Here, we scale the coupling constants to $T^{*}=1\mathrm{ns}$ in 1000 sets with $N_C=100$ configurations each, in order to obtain an approximately continuous distribution $p\left(T_j^{*}\right)$.

Figure 5

(a) Time evolution of the electron spin component $〈S^z〉$. Various colors show the time evolution after different numbers $N_P$ of pump pulses. The time axis starts with the arrival of the $N_P\mathrm{th}$ pump pulse and ends before the arrival of the next pump pulse after the repetition time $T_R=13.2\mathrm{ns}$. (b) Time evolution directly before the next pump pulse. In addition to the electron spin component $〈S^z〉$ (solid lines), the electron spin component $〈S^y〉$ is depicted (dashed lines) to show the spin precession.

Figure 6

Evolution of the electron spin revival amplitude $S^{\perp }\left(N_P{T}_R\right)$ with the pulse number $N_P$. Various colors show the development for different external magnetic fields $B_{\mathrm{ext}}$.

Figure 7

Magnetic field dependency of the electron spin revival amplitude. The converged revival amplitude $S^{\perp }\left(N_P{T}_R\right)$ after $1.5\times {10}^6\le N_P\le 20\times {10}^6$ is depicted as the blue curve. The exact number of pump pulses depends on the magnetic field $B_{\mathrm{ext}}$. The dominating electron spin component $〈S^z\left(N_P{T}_R\right)〉$ in the full expression for $S^{\perp }\left(N_P{T}_R\right)$ is indicated by red crosses. Furthermore, the revival amplitudes calculated from the Overhauser field distributions according to Eq. (41) are added as green diamonds.

Figure 8

Overhauser field distribution $p\left(B_N^x\right)$ for $B_{\mathrm{ext}}=1.95\mathrm{T}$ [i.e., $n=1$ in Eq. (37)]. The Gaussian distribution given by Eq. (35) is indicated by the red dashed curve. The results for a quantum mechanical system with $N=6$ and $N_C=100$ are drawn as solid lines (blue). (a) The initial distribution $p_0\left(B_N^x\right)$ before the first pulse. We added $p_0\left(B_N^x\right)$ obtained for $N_C={10}^5$ in green for comparison as well. (b) $p\left(B_N^x\right)$ after 1.5 million pump pulses. Overhauser fields that correspond to an integer number of electron Larmor revolutions within $T_R$ are indicated by the gray dashed vertical lines.

Figure 9

Relative Overhauser field distribution $p_{\mathrm{rel}}\left(B_N^x\right)$ for distinct external magnetic fields $B_{\mathrm{ext}}$. Overhauser fields that correspond to an integer (half-integer) number of electron Larmor revolutions are indicated by gray dashed (green dotted) vertical lines, respectively. The number $N_P$ of pump pulses is in the range $1.5\times {10}^6\le N_P\le 20\times {10}^6$.

Figure 10

Evolution of the revival amplitude $S^{\perp }\left(N_P{T}_R\right)$ as function of the pulse number $N_P$ for various external magnetic fields $B_{\mathrm{ext}}$, averaged over 25 200 random initial configurations with $N_{\mathrm{eff}}=200$. The solid black lines show the fits (44), yielding the saturated value $S_{\lim }$ of the revival amplitude.

Figure 11

Dependence of the saturated revival amplitude $S_{\lim }$ on the external magnetic field. The saturated revival amplitudes are determined by fitting Eq. (44) to the data from the classical simulations. The lines are guides to the eye.

Figure 12

Distribution of the Overhauser field $x$ component obtained from the classical simulations. The calculations were performed for an ensemble of 25 200 random initial configurations with $N_{\mathrm{eff}}=200$. The vertical dashed lines indicate the integer resonance condition.

Figure 13

Evolution of the electron spin revival amplitude with the pulse number $N_P$ for Gaussian pump pulses in the quantum mechanical approach. Various colors show the development for different external magnetic fields $B_{\mathrm{ext}}$.

Figure 14

Magnetic field dependency of the electron spin revival amplitude calculated by the quantum mechanical approach with Gaussian pump pulses. The revival amplitude $S_G^{\perp }\left(N_P{T}_R\right)$ and the spin component $\left|〈S_G^z\left(N_P{T}_R\right)〉\right|$ are taken after a number of pump pulses $2.5\times {10}^6\le N_P\le 20\times {10}^6$ large enough such that they have converged. The exact value of $N_P$ depends on the magnetic field. For comparison, we added the revival amplitude $S_I^{\perp }\left(N_P{T}_R\right)$ with instantaneous pump pulses taken from Fig. 7.

Figure 15

Relative Overhauser field distribution $p_{\mathrm{rel}}\left(B_N^x\right)$ for various external magnetic fields $B_{\mathrm{ext}}$. The considered pulse sequence consists of Gaussian pump pulses (red solid lines). Overhauser fields that correspond to an integer (half-integer) number of electron spin revolutions during $T_R$ are indicated by gray dashed (green dotted) vertical lines respectively. The number $N_P$ of pump pulses is in the range $2.5\times {10}^6\le N_P\le 20\times {10}^6$. For comparison, we added the results for instantaneous pump pulses taken from Fig. 9 as blue dotted lines.

Figure 16

Evolution of the electron spin revival amplitude for small numbers $N_P$ of pump pulses, i.e., without nuclear frequency focusing. Various colors show the amplitudes for different external magnetic fields $B_{\mathrm{ext}}$. The black curve is calculated analytically from Eq. (B2). The analytic revival amplitude $\left|{〈S^z\left(N_P{T}_R\right)〉}_{\infty }\right|=\left|1/2-1/\sqrt{3}\right|\approx 0.077$ in the limit $N_P\to \infty$ is indicated by a gray dashed horizontal line.