Increased coherence time in narrowed bath states in quantum dots

We study the influence of narrowed distributions of the nuclear Overhauser field on the decoherence of a central electron spin in quantum dots. We describe the spin dynamics in quantum dots by the central spin model. We use analytic solutions for uniform couplings and the time dependent density-matrix renormalization group (tDMRG) for nonuniform couplings. With these tools we calculate the dynamics of the central spin for large baths of nuclear spins with or without external magnetic field applied to the central spin. The focus of our study is the influence of initial mixtures with narrowed distributions of the Overhauser field and of applied magnetic fields on the decoherence of the central spin.

Figure 1

The relative partition function $Z/Z_0$ for $N=49$ and $\gamma =50$ depending on the spread parameter $x$ in (4). While the spread of the couplings changes considerably according to the exponential distribution in (4) the relative partition function drops by about $5\%$ only.

Figure 2

Variance ${\sigma }^2\left(\gamma \right)$ for three sets of parameters. The red solid curve is calculated for a nonuniform system with $x=1$ and $N=49$. It is approximated well by the thermodynamic limit (37) (black dashed curve) in the whole interval shown. A curve for $x=1$ and $N=50$ is not shown because it could not be distinguished from the $x=1$ and $N=49$ case. The blue curve is calculated for a uniform system ($x=0$) and $N=49$. For large values of $\gamma$ the curve saturates to the value $A_Q^2/196$ according to (41). The green curve is calculated for a uniform system with $x=0$ and $N=50$. The curve decays as $exp(-\gamma /N)$ for large values of $\gamma$ according to (41).

Figure 3

Relative deviation $\mathrm{\Delta }{\sigma }^2$ defined in (42) depending on $N$ for three different values of the narrowing factor $\gamma$ and uniform couplings ($x=0$). The black line depicts the power law $\propto 1/N$; note the doubly logarithmic scale.

Figure 4

Crossing variance ${\sigma }_{\text{c}}^2$ defined in (44) depending on the bath size $N$ for uniform couplings. This variance represents the lowest variance for which the uniform system does not show significant finite size effects.

Figure 5

Relative deviation $\mathrm{\Delta }{\sigma }^2$ in (42) depending on $N$ for three different values of the weight factor $\gamma$ and nonuniform couplings with the spread parameter $x=1$. The black line is a power law $\propto 1/N$.

Figure 6

Variance ${\sigma }^2$ for the narrowing factor $\gamma =200$ and four bath sizes $N$ depending on the spread parameter $x$. Both curves quickly reach the thermodynamic limit ${\sigma }_{\infty }^2(\gamma =200)=A_Q^2/404$ according to (37).

Figure 7

Coherence time $T_2$ for various external magnetic fields $h$ depending on the initial variance ${\sigma }^2$. The data points are calculated for $N=49$ bath spins and nonuniform couplings with $x=1$. The colored lines (dashed, dotted, and solid) are calculated for $N=999$ bath spins and uniform couplings. The black dash-dotted line is the curve for the effective high-field model in (47).

Figure 8

Time-dependent variance ${\sigma }^2\left(t\right)$ of the Overhauser field for nonuniform couplings with coupling spread $x=1$ in (4), narrowing factor $\gamma =100$, and external magnetic field $h=5A_Q$. In addition, we plot the value of the first maximum ${\sigma }_m^2$ of the time-dependent variance as a black dashed line.

Figure 9

First maximum of the dynamic variance ${\sigma }^2\left(t\right)$ at time $t_m$, see Fig. 8, depending on the initial variance ${\sigma }^2$ for external magnetic field $h=5A_Q$. The green crosses are calculated for $N=49$ bath spins and nonuniform couplings with $x=1$. The dashed line is the corresponding quadratic fit. The blue crosses are calculated for $N=999$ bath spins and uniform couplings. The dotted line is the corresponding linear fit. Both fits yield finite values of ${\sigma }_m^2$ for ${\sigma }^2\to 0$.

Figure 10

Absolute value of the correlation function $C\left(t\right)$ for $A_{Q}t=10$ and its dependence on the variance ${\sigma }^2$ for three magnetic fields and nonuniform couplings with $x=1$ in (4). The colored lines (dashed, dotted, and solid) are fits of the form (53) to the data points.

Figure 11

Fit parameter $C_0\left(t\right)$ in (53) depending on time $t$ for three different magnetic fields $h$ and nonuniform couplings with $x=1$ in (4). The parameter $C_0\left(t\right)$ is equal to the absolute value of the correlation function $C\left(t\right)$ in the limit ${\sigma }^2\to 0$. Hence the curves represent the correlation functions for optimally suppressed fluctuations in the spin bath and yield the maximum coherence time $T_2$.

Figure 12

Coherence time $T_2$ as function of the external magnetic field $h$ for various narrowing factors $\gamma$. The data points are calculated for $N=49$ bath spins and nonuniform couplings with $x=1$ in (4). The colored lines (dashed, dotted, and solid) are calculated for $N=999$ bath spins and uniform couplings. The dotted black lines are the corresponding coherence times $T_{2,\infty }$ in (48) for the three values of $\gamma$ in the high-field limit $h\to \infty$.

Figure 13

Relative error $\mathrm{\Delta }Z\left(w_2\right)$ for two values of the weight $w_1$.

Figure 14

Relative error $\mathrm{\Delta }Z\left(\gamma \right)$ as function of $\gamma$.

Figure 15

Relative error $\mathrm{\Delta }Z\left(N\right)$ as function of $N$.