Correlated many-body noise and emergent $1/f$ behavior

Fluctuating electric fields emanating from surfaces are a major possible source of decoherence in a number of quantum applications, including trapped ions and near-surface nitrogen-vacancy diamond qubits. Here, we show that at low temperatures, due to superradiant decay, phonon-induced excitation exchange between adsorbed atoms can counterintuitively mitigate the electric field noise. We derive an exact mapping between the noise spectrum of $N$ interacting fluctuators with $M$ vibrational levels to $\left(\genfrac{}{}{0pt}{}{N+M-1}{N}\right)$-1 noninteracting two-level dipoles. The anharmonic interaction of the fluctuators with the surface is semiempirical and physically motivated. This anharmonicity affects the noise spectral power intensity at higher temperatures. We describe conditions for which the ubiquitous $1/f$ noise emerges naturally from the coupled dynamics of identical fluctuators and whose behavior depends critically on correlation among the fluctuators.

Figure 1

(a) The coefficients $C_k$ in Eq. (5) are shown as a function of eigenvalues ${\lambda }_k$ for $N=3$. The up (blue) and down (red) pointing triangles correspond to numerically exact calculations at $T/{\omega }_0=0.2,1$, respectively. The widespread distributions of weights $C_k$ over ten orders of magnitude suggest an approximation scheme utilizing the largest $C_k$. (b) The noise power spectrum illustrates the point: With only the highest weighted ($C_k,{\lambda }_k$) pairs, here up to four pairs, results converge to the exact values for $T/{\omega }_0=0.2$. The convergence will only be more rapid for higher temperatures. Here, we chose $U_0=0.6$ eV, $m=100$ amu, ${\beta }_0=1.86$ Å${}^{-1}$, and $z_0=3.1$ Åas an example of a tightly bound adatom species. We note that qualitatively similar results are obtained for weakly bound adatoms, where the noise spectrum is shifted to lower frequencies.

Figure 2

The noise spectral function for adatoms of mass $m=100$ amu interacting with the surface potential at depth $U_0=250$ meV at $T/{\omega }_0=0.1,0.4$, and 1 with ${\beta }_0$ and $z_0$ as before. (a) The correlated noise spectrum is shown for different patch sizes $N$. At $T/{\omega }_0<1$, correlated superradiant decay results in a lowering of the white noise with an increasing number of correlated adatoms $N$. Thus, in this regime one adatom is a larger source of noise than two or more correlated adatoms. With $T/{\omega }_0\sim 1$, superradiant decay does not prevent an increasing noise magnitude as a function of increasing $N$. (b) The magnitude of the white noise spectra per adatom at different $T/{\omega }_0$ is shown to decrease with increasing $N$ where in the low-temperature limit the $N^{-2}$ scaling appears as predicted by Eq. (6). The solid lines are guides for the eye.

Figure 3

The emergence of $1/f$ noise with a $\mathcal{D}\left(N\right)=\frac{1}{N}$ patch distribution. The black bottommost curve results from a sum over $N_{\max }=10$ correlated adatoms in $S_{\text{tot}}\left(\omega \right)$ while the blue curve is the $N_{\max }\to \infty$ limit in Eq. (7). The latter has a universal functional form in $\omega /{\mathrm{\Gamma }}_0$. With increasing numbers of correlated adatoms in each patch, the noise spectrum turns from $1/f^2$ to $1/f$.