arXiv subjects
Publications and source records attributed to Chenhao Du.
Let $\tau_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$ \sum_{1\leq n_1,n_2, \dots, n_{\ell}\leq X^{\frac{1}{r}} \atop 1\leq n_{\ell+1}\le X^{\frac{1}{s}}}\tau_k(n_1^r+n_2^r+\dots +n_{\ell}^r+n_{\ell+1}^s), $$ where $k\geq 4$, $r\geq 2$, $s\geq 2$ and $\ell\geq 2$ are integers. Previously only special cases are studied.
Stabilizing important physical quantities to atom-based standards lies at the heart of modern atomic, molecular and optical physics, and is widely applied to the field of precision metrology. Of particular importance is the atom-based microwave field amplitude stabilizer, the so-called atomic candle. Previous atomic candles are realized with atoms in their ground state, and hence suffer from the lack of frequency band tunability and small stabilization bandwidth, severely limiting their development and potential applications. To tackle these limitations, we employ microwave-dressed Rydberg atoms to realize a novel atomic candle that features multi-band frequency tunability and large stabilization bandwidth. We demonstrate amplitude stabilization of microwave field from C-band to Ka-band, which could be extended to quasi-DC and terahertz fields by exploring abundant Rydberg levels. Our atomic candle achieves stabilization bandwidth of 100 Hz, outperforming previous ones by more than two orders of magnitude. Our simulation indicates the stabilization bandwidth can be further increased up to 100 kHz. Our work paves a route to develop novel electric field control and applications with a noise-resilient, miniaturized, sensitive and broadband atomic candle.