Spin noise in the anisotropic central spin model

Spin-noise measurements can serve as a direct probe for the microscopic decoherence mechanism of an electronic spin in semiconductor quantum dots (QDs). We have calculated the spin-noise spectrum in the anisotropic central spin model using a Chebyshev expansion technique which exactly accounts for the dynamics up to an arbitrary long but fixed time in a finite-size system. In the isotropic case, describing QD charge with a single electron, the short-time dynamics is in good agreement with quasistatic approximations for the thermodynamic limit. The spin-noise spectrum, however, shows strong deviations at low frequencies with a power-law behavior of ${\omega }^{-3/4}$ corresponding to a $t^{-1/4}$ decay at intermediate and long times. In the Ising limit, applicable to QDs with heavy-hole spins, the spin-noise spectrum exhibits a threshold behavior of ${(\omega -{\omega }_L)}^{-1/2}$ above the Larmor frequency ${\omega }_L=g{\mu }_{B}B$. In the generic anisotropic central spin model we have found a crossover from a Gaussian type of spin-noise spectrum to a more Ising-type spectrum with increasing anisotropy in a finite magnetic field. In order to make contact with experiments, we present ensemble averaged spin-noise spectra for QD ensembles charged with single electrons or holes. The Gaussian-type noise spectrum evolves to a more Lorentzian shape spectrum with increasing spread of characteristic time scales and $g$ factors of the individual QDs.

Figure 1

Probability density $P\left(A\right)$ vs $A_s/A_{\text{max}}$ for three different radii $R/L_0=2,3,4$. The dashed lines of the same color depict the histogram of the distribution generated by ${10}^8$ random picks for $A_k$.

Figure 2

Benchmark for the CET calculating the time evolution of the spin correlation function $S\left(t\right)$ for uniform coupling constants $A_k=\frac{A_s}{N}$. Two different orders $N_C$ of the expansion are compared to ED for (a) $N=2$ and (b) $N=6$ bath spins. After the time scale indicated by the arrows the CET error exceeds $O\left({10}^{-3}\right)$. In (a) and (b) the occurring traces have been calculated exactly and no error from the statistical evaluation of traces enters. (c) Treats this error for $N=10$ by comparing results for a varying number $N_s$ of random states to a result taking the full trace into account with $N_C=66$.

Figure 3

The spin correlation function $S\left(t\right)$ for randomly generated coupling constants $A_k$ calculated via CET. Each shown curve has been averaged over $n=50$ different random realizations of couplings. (a) Short-time evolution of $S\left(t\right)$ for increasing $r_0$ with $N=18$. (b) The analytical result $M\left(t\right)$ in comparison to calculated data for increasing bath size $N$ based on a fixed ratio $r_0=R/L_0=1.5$. The inset shows the area marked by the box.

Figure 4

(a) The behavior of $S\left(t\right)$ on long-time scales for different values of $r_0$ and $N=18$. Each shown curve has been averaged over $n=50$ different random configurations $\left\{A_k\right\}$ of couplings. The colored arrows indicate the long-time values for $S_{\infty }$ predicted [8] from the fluctuation ratio $u=\langle A^2\rangle /{\langle A\rangle }^2$. (b) The nondecaying fraction $S_{\infty }$, obtained by averaging the data over the calculated data points in the interval $\frac{t}{T^{*}}\in [450:500]$ vs the cutoff $r_0$ for two different system sizes $N=14,20$. The straight lines are linear fits. The analytical predictions using the fluctuation ratio $u$ are added for comparison.

Figure 5

The spin-noise spectrum $S\left(\omega \right)$ in the absence of an external field for $r_0=1.5$. (a) Illustrates the convergence properties of the CET for varying $N_C$ and the effect of averaging results for several ($n$) configurations of the coupling constants $A_k$ based on $N=18$ bath spins and $r_0=1.5$. (b) Focuses on the finite bath size effects where we have averaged over $n=50$ different configurations $\left\{A_k\right\}$. $M\left(\omega \right)$ denotes the Fourier transformation of the semiclassical result where the $\delta$ peak in $M\left(\omega \right)$ at $\omega =0$ has been approximated by a Lorentzian.

Figure 6

$S\left(\omega \right)$ vs $\omega T^{*}$ on a log-log scale for different values of $r_0=1.5,1.75,2$. The dashed line is a fit to the low-frequency behavior above the resolution-broadened $\delta \left(\omega \right)$ peak. The smallest accessible frequency ${\omega }_{\min }$ and the expectation value of the smallest contributing coupling constant ${\overline{A}}_{\text{min}}$ are indicated by arrows. Parameters: $N_C=1000$, $n=100$, $N=18$, and $b=0$.

Figure 7

All shown simulations have been averaged over $n=50$ configurations. The spectral functions are all based on $N_C=100$. (a) The correlation function $S\left(t\right)$ for two different values of $b$. To distinguish the results for different $b$ an offset of $0.5$ has been added to the upper curve. The function $E\left(t\right)=\frac{1}{4}\text{exp}[-\frac{1}{2}{\left(\frac{t}{2T^{*}}\right)}^2]$ approximates the envelope of the signals. (b) $S\left(t\right)$ for small external field strengths. (c) The renormalized spin-noise function $S\left(\omega \right)$ for $b=2,4,\cdots ,10$. (d) The same data as shown in (c) shifted by ${\omega }^{*}=\sqrt{b^2+\frac{1}{2}}$, pointing out that the width of the occurring maxima is given by $T^{*}$ and that ${\omega }^{*}$ is the exact frequency of the occurring spin precession. Parameters: $N=20$ and $r_0=1.5$.

Figure 8

(a) The spin correlation function $S\left(t\right)$ for increasing applied longitudinal fields, $N=18$ and an average over $n=10$ different configurations. (b) Corresponding calculated spectral functions $S\left(\omega \right)$. The inset depicts the spectral weight of the $\delta \left(\omega \right)$ peak as function of $b$ corresponding to the nondecaying fraction of $S\left(t\right)$. Parameters: $N_C=100$, $n=50$, and $r_0=1.5$.

Figure 9

The spin-noise function $S\left(t\right)$ for the Ising limit $\lambda \to \infty$ of the anisotropic central spin model plotted vs the dimensionless time $\tau =t/4b{T}^{*}$ for three values of the external magnetic field $b=2,4,8$. An offset of $0.5$ has been added to distinguish the three individual curves. The envelope of $E_z\left(\tau \right)$ given by Eq. (57) has been added as a black line to each simulation. All curves have been calculated for $r_0=1.5$, $n=20$, and $N=18$.

Figure 10

The spin-noise spectra $S\left(\omega \right)$ for $(\omega -{\omega }_L)\ge 0$ in the Ising limit calculated by exact Fourier transformation of (58) for different external magnetic fields $b$. The inset shows the nondecaying fraction $S_{\infty }$ which defines the spectral weight of the $\delta$ peak at $\omega =0$. Parameters: $N=18$ and $r_0=1.5$.

Figure 11

(a) $S\left(t\right)$ vs $t/T_{\lambda }^{*}$ for different values of the anisotropy factor $\lambda =1,2,4,8$. (b) Spin-noise spectra vs $\omega T^{*}$ for the same parameters as in (a) and $N_C=100$. A $\delta$ peak at zero frequency resolved by the CET with the same number of moments has been added as reference. Parameters: $N=18$, $n=50$, and $r_0=1.5$.

Figure 12

Comparison between the analytical prediction (blue) of Eq. ((26)c) of Ref. [12] using a QSA and the CET approach for $b/\lambda >1$. All curves have been calculated for $b=10$ and $\lambda =2$ or $\lambda =10$. We have added an offset of $0.5$ to the $\lambda =2$ curves for better comparison. Parameters: $N=18$ and $r_0=1.5$.

Figure 13

$S\left(t\right)$ in a transversal magnetic field: (a) for a fixed value $\lambda =10$ and $b=2,6,10$, and (b) for a fixed value $b=4$ and four different values of $\lambda =2,20,50,100$. For all cases we have added the envelope function $0.25exp[-{(t/T_{\lambda }^{*})}^2/8]$ to reveal the difference to the short-time dynamics of the isotropic CSM. (c) Rescaled spin correlation function $S\left(t\right){(1+{\tau }^2)}^{1/4}$ vs $t/T_{\lambda }^{*}$ for the same parameters as in (b) with $\tau =t/4b{T}_1^{*}$. Parameters: $N=18$, $n=20$, and $r_0=1.5$.

Figure 14

Spin-noise spectra $S\left(\omega \right)$ for $b=4$ and $\lambda =2,10,20,50$, plotted vs $(\omega -{\omega }_L)T_{\lambda }^{*}$. Since the characteristic time-scale $T_{\lambda }^{*}$ increases with increasing $\lambda$, we adjusted the number of Chebychev momenta to $N_C(\lambda =2)=100$, $N_C(\lambda =10)=300$, $N_C(\lambda =20)=600$, and $N_C(\lambda =50)=1500$ to provide an adequate frequency resolution. The chosen parameters are equivalent to those in Fig. 13.

Figure 15

(a) Ensemble averaged spin-noise spectra vs $\omega$ for five different values of $B$ for a dot-confined electron spin, i.e., $\lambda =1$. Defining the average time scale ${\overline{T}}^{*}=\langle T^{*}\rangle =1$ ns as reference, and using $|g_e|=0.54$ as in Ref. [19], $S\left(\omega \right)$ has been obtained by averaging over $n=100$ different spectra generated from a constant distribution of $T^{*}\in [0.8\mathrm{ns},1.2\mathrm{ns}]$ of characteristic single QD time scales. For better visibility, an offset proportional to $b={\omega }_L{\overline{T}}^{*}$ has been added to $S\left(\omega \right)$. Parameters: $N_C=15$ and $N=18$. The black lines indicate Lorentzian fits to the individual curves. (b) Evolution of the ensemble noise spectrum with increasing $N_C$ at fixed $B=300$ G.

Figure 16

(a) Ensemble averaged spin-noise spectra in the asymmetric CSM for $\lambda =5$ (red) and $\lambda =50$ (blue) for different $b$. $S\left(\omega \right)$ has been obtain by averaging over $n=2000$ different single QD spectra generated from a constant distribution of $T_i^{*}\in [0.8\overline{T^{*}},1.2\overline{T^{*}}]$, and the individual $g$ factors are taken from a Gaussian distribution with $\Delta g/g=0.2$. (b) Influence of the $g$-factor distribution function on the shape of the spin-noise spectrum: Comparing the $\lambda =5$ data of (a) with an ensemble averaged $S\left(\omega \right)$ where the individual $g$ factors $g_i$ are taken from a constant distribution of $g_i/g\in [0.8,1.2]$ (green curves). For better visibility, an offset proportional to $b$ has been added to $S\left(\omega \right)$. Parameters: $N=18$, $N_C=50$, and $r_0=1.5$.