Exponential decay in a spin bath

We show that the coherence of an electron spin interacting with a bath of nuclear spins can exhibit a well-defined purely exponential decay for special (“narrowed”) bath initial conditions in the presence of a strong applied magnetic field. This is in contrast to the typical case, where spin-bath dynamics have been investigated in the non-Markovian limit, giving superexponential or power-law decay of correlation functions. We calculate the relevant decoherence time $T_2$ explicitly for free-induction decay and find a simple expression with dependence on bath polarization, magnetic field, shape of the electron wave function, dimensionality, total nuclear spin $I$, and isotopic concentration for experimentally relevant heteronuclear-spin systems.

Figure 1

(Color online) Exponential decay $x_t=\exp (-t∕T_2)$ (solid line) and maximum error bounds $x_t\pm {\mid ϵ\left(t\right)\mid }_{\max }$ (dashed lines), found by the numerical integration of Eq. (9) with parameters for a two-dimensional quantum dot [before Eq. (14)], $I=3∕2$ and $A∕b=1∕20$. For comparison, we show the decay curves for superexponential forms $\exp [-{(t∕T_2)}^2]$ and $\exp [-{(t∕T_2)}^4]$ (dot-dashed lines) and rapidly decaying bath correlation function $C\left(t\right)∕C\left(0\right)$ [dotted line, see Eqs. (10, 11)].

Figure 2

(Color online) Geometrical factor $f(d∕q)$ from Eq. (13), where $d=1,2,3$ is the dimension and $q$ characterizes the electron envelope function $\psi \left(r\right)=\psi \left(0\right)\exp [-{(r∕r_0)}^q∕2]$.

Figure 3

(Color online) Decay rates for an ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ quantum dot with In doping $x=0.05$. Here, we have assumed $N={10}^5$ and used values of ${\nu }_i$ and $A^i$ for GaAs from Ref. 41: $A^{{}^{75}\mathrm{As}}=86\mu \mathrm{eV}$, $A^{{}^{69}\mathrm{Ga}}=74\mu \mathrm{eV}$, $A^{{}^{71}\mathrm{Ga}}=96\mu \mathrm{eV}$, ${\nu }_{{}^{75}\mathrm{As}}=0.5$, ${\nu }_{{}^{69}\mathrm{Ga}}=0.3(1-x)$, and ${\nu }_{{}^{71}\mathrm{Ga}}=0.2(1-x)$. The hyperfine coupling for In in InAs was taken from Ref. 42: $A^{{}^{113}\mathrm{In}}\approx A^{{}^{115}\mathrm{In}}\approx A^{\mathrm{In}}=170\mu \mathrm{eV}$, ${\nu }_{\mathrm{In}}=x∕2$.