Persisting correlations of a central spin coupled to large spin baths

The decohering environment of a quantum bit is often described by the coupling to a large bath of spins. The quantum bit itself can be seen as a spin $S=\frac{1}{2}$ which is commonly called the central spin. The resulting central spin model describes an important mechanism of decoherence. We provide mathematically rigorous bounds for a persisting magnetization of the central spin in this model with and without magnetic field. In particular, we show that there is a well-defined limit of infinite number of bath spins. Only if the fraction of very weakly coupled bath spins tends to 100% does no magnetization persist.

Figure 1

Scheme of the central spin model (CSM) with couplings between the central spin ${\mathbf{S}}_0$ and the surrounding bath spins ${\mathbf{S}}_k$.

Figure 2

Example of the exponential coupling distribution defined in (5) for $N=32$ bath spins and $x=3$.

Figure 3

Example for the autocorrelation function $A\left(t\right)$ defined in (7) with a well-defined limit $A_{\infty }={lim}_{t\to \infty }A\left(t\right)$.

Figure 4

Lower bounds for $S_{\text{low}}$ calculated via the spin-spin (SS) and via the field-field (BB) autocorrelation for $x\in [1,2,3,4]$ and bath sizes up to $N_{\text{max}}=4096$. The bounds are compared to results from the stochastic evaluation of the Bethe ansatz (BA) equations for up to 48 bath spins.

Figure 5

Data points: extrapolated bounds $S_{\infty ,\text{BB}}\left(x\right)$ for an infinite spin bath. The solid line is a fit by $S_{\text{log}}\left(x\right)$ from Eq. (22) in the interval $x\in [6,64]$. We also included the extrapolated values $S_{\infty ,\text{BA}}\left(x\right)$ from the Bethe ansatz data and $S_{\infty ,\text{SS}}\left(x\right)$ from the spin-spin bounds for $x\in [1,2,3,4]$.

Figure 6

Comparison of lower bounds using single conserved quantities with an even number of summed spins so that $S_{\mathrm{low}}$ remains finite for $N\to \infty$. The coupling distribution is given by (5). To show how tight the bounds are, Chebyshev expansion data are included [30, 46]. Upper panel: $x=1$; lower panel: $x=4$.

Figure 7

Comparison of lower bounds resulting from single conserved quantities with an odd number of summed spins so that $S_{\mathrm{low}}\to 0$ holds. The coupling distribution is given by (5) with $x=4$. We point out that the lower bounds for $I^z$ and $I_Q^z$ do not depend on the couplings at all. The significance of the lower bound from $I^z{H}_0^2$ is rather low.

Figure 8

Improved lower bounds $S_{\mathrm{low}}$ using $I^z{H}_0^3$ (yellow) or $I^z{I}^2{H}_0$ (green) in addition to three constants (blue) already considered in Ref. [39]. The lower bound stemming from the combination of all these conserved quantities is depicted in red. All curves are obtained for the coupling distribution (5).

Figure 9

Extrapolated improved lower bounds $S_{\mathrm{low}}$. The lower bound obtained by using $I^z,I_Q^z$, and $I^z{H}_0$ is labeled $S_{\mathrm{low}}^{\left(3\right)}$. The bound $S_{\mathrm{low}}^{\left(4\right)}$ denotes the result from adding $I^z{H}_0^3$ and $S_{\mathrm{low}}^{\left(6\right)}$ shows the bound obtained on inclusion of $I^z{H}_0^2$ and $I^z{I}^2{H}_0$. The couplings are distributed according to (5). Results from stochastic Bethe ansatz are included as well as the approximate lower bounds from the field-field correlations (BB). The diamonds denote the bounds obtained from the Gaussian approximation applied to constants of motion containing high powers of $H_0$. This approximation becomes exact upon $N\to \infty$ (see main text).

Figure 10

Convergence of $S_{\mathrm{low}}^{\left(m\right)}$ as a function of $m$ for the constants of motion $\{I^z{H}_0,\cdots ,I^z{H}_0^{2m-1}\}$ for exponential couplings given by (5) and a bath size of $N=20$. Larger values of $N$ do not change the curves significantly.

Figure 11

Lower bounds and their dependence on the magnetic field $h$ for two finite bath sizes $N=9$ and 19 compared to the infinite bath limit for the exponential couplings in (5).

Figure 12

Lower bounds and their dependence on the magnetic field $h$ for two finite bath sizes $N=9$ and 199 computed for the constants of motion of type $H_l^z\left(h\right)$ for the exponential couplings in (5). Clearly, the inclusion of all constants of motion beyond $H_0^z\left(h\right)$ does not provide significant improvement, in particular for systems which are not very small. The bounds deteriorate for larger fields, indicating that the considered constants of motion matter only for small fields.

Figure 13

Comparison of the rigorous bounds from two and $2N+2$ conserved quantities for bath size $N=199$ and the exponential couplings in (5).

Figure 14

Comparison of different bounds generated by combining $H_0^z\left(h\right)$ and $H_0\left(h\right)$ and a third conserved quantity as denoted in the legend for bath size $N=999$ and the exponential couplings in (5).

Figure 15

Comparison of the bounds generated by combining all six conserved quantities (see legend of Fig. 14) to numerical data from Chebyshev polynomial expansion for bath size $N=19$ and the exponential couplings in (5).