Nuclear spin dynamics and Zeno effect in quantum dots and defect centers

We analyze nuclear-spin dynamics in quantum dots and defect centers with a bound electron under electron-mediated coupling between nuclear spins due to the hyperfine interaction (“$J$ coupling” in NMR). Our analysis shows that the Overhauser field generated by the nuclei at the position of the electron has short-time dynamics quadratic in time for an initial nuclear-spin state without transverse coherence. The quadratic short-time behavior allows for an extension of the Overhauser-field lifetime through a sequence of projective measurements (quantum Zeno effect). We analyze the requirements on the repetition rate of measurements and the measurement accuracy to achieve such an effect. Further, we calculate the long-time behavior of the Overhauser field for effective electron Zeeman splittings larger than the hyperfine coupling strength and find, both in a Dyson-series expansion and a generalized master-equation approach, that for a nuclear-spin system with a sufficiently smooth polarization the electron-mediated interaction alone leads only to a partial decay of the Overhauser field by an amount of the order of the inverse number of nuclear spins interacting with the electron.

Figure 1

Effect of projective measurements at time intervals ${\tau }_m={\tau }_e/10$ on the time evolution of the Overhauser-field expectation value $⟨h_z\left(t\right)⟩$. Due to the Zeno effect, the decay with measurements is $1-ct/{\tau }_{\text{Zeno}}$ rather than $1-c{t}^2/{\tau }_e^2$ without measurements, where ${\tau }_{\text{Zeno}}={\tau }_e^2/{\tau }_m$. The formula $1-ct/{\tau }_{\text{Zeno}}$ for the decay with measurement is only strictly valid at times $t=m{\tau }_m$ with $m$ being a positive integer. After the measurement at $t=m{\tau }_m$ the decay is again quadratic with time dependence $⟨h_z\left(m{\tau }_m\right)⟩/⟨h_z\left(0\right)⟩-c{\left(t-m{\tau }_m\right)}^2/{\tau }_e^2$ (broken lines).

Figure 2

Numerical prefactor $c$ [given in Eq. (18)] of the $t^2$ term in the decay of the Overhauser-field mean value $⟨h_z\left(t\right)⟩$. While $c$ turns out to be independent (for $N\gtrsim 100$ in the case shown according to numerical summation) of the number $N$ of nuclear spins within a Bohr radius of the electron envelope wave function, it does depend on the type of structure and the initial polarization. We show the case of a two-dimensional quantum dot with a Gaussian electron envelope $\left(d/q=1\right)$. The dependence on the initial polarization is parametrized by $N/N_p$, where $N_p$ is the number of nuclear spins that are polarized substantially (see text). Inset: Dependence of $c$ on the ratio $d/q$ for $N/N_p=1$. We see that, e.g., for a donor impurity with a hydrogenlike wave function $\left(d/q=3\right)$ the prefactor $c$ is more than three orders of magnitude smaller compared to the two-dimensional lateral quantum dot with $d/q=1$.

Figure 3

In this figure we show the $N$ dependence of $1-{⟨h_z⟩}_{\text{stat}}/⟨h_z\left(0\right)⟩$ [see Eq. (37)], i.e., the part by which $⟨h_z⟩$ decays in units of $p{A}^2/N{\omega }^2$, in the regime $\omega ⪢A$. This plot is for a three-dimensional defect center with a hydrogenlike electron envelope $\left(d/q=3\right)$ and the initial polarization is parameterized by $N/N_p=0.5$ as described in Sec. 4. For this choice of polarization distribution the decay is $O\left(1/N\right)$. The inset shows the full time dynamics of $⟨h_z\left(t\right)⟩/⟨h_z\left(0\right)⟩$ as given in Eq. (35) for $d/q=1$, $N⪢1$, and $N/N_p=1$. We see that the decay occurs on a time scale of ${\tau }_c=N/A$.