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Ilya Vinogradov

Publications and source records attributed to Ilya Vinogradov.

15 recordsLinked to original sources
arXiv2026

Sub-Doppler laser cooling and optical transport of cesium with static magnetic fields

Laser cooling of alkali atoms typically requires time-varying magnetic fields, introducing unwanted coupling between atom preparation and coherent operations. Here we demonstrate sub-Doppler laser cooling and optical transport of alkali atoms in a fully static magnetic-field configuration. Using a blue-detuned Type-II magneto-optical trap (MOT) operating on the closed $F=3 \rightarrow F'=2$ transition of the D2 line in cesium, we achieve temperatures of 17(1) $μ$K without changing the magnetic-field gradient between cooling stages. This enables direct loading into a shallow optical lattice and transport over 17 cm within the same static-field environment. In contrast to conventional alkali cooling schemes with dynamic fields, our approach establishes a continuous cooling and transport protocol compatible with static-field platforms. These results validate Type-II cooling as a practical technique for alkali atoms and provide a new route toward continuous-operation architectures in sensing and quantum computing.

physics.atom-ph↗
arXiv2026

Demonstration of a Logical Architecture Uniting Motion and In-Place Entanglement

We demonstrate a logical neutral atom architecture that integrates atom motion with in-place entanglement to achieve lower overheads than entangling-zone approaches. Using a 114-qubit device, we perform three proof-of-principle logical-qubit experiments. First, we implement a pre-compiled, non-scalable variant of Shor's algorithm, observing improved logical-over-physical performance, including with loss correction and leakage detection, achieving up to a 2x reduction in TVD. Second, we construct constant-depth logical CX ladders; on current hardware these execute with serial entangling operations, yet still yield 2-4x lower error for 8 and 12 logical qubits. Third, we prepare the [[16,4,4]] code and perform single-round decoding with post-processed error correction, achieving 8x improvement on logical vs physical. These results demonstrate how combining motion with in-place entanglement offers lower overhead than entangling-zone approaches.

quant-ph↗
arXiv2016

Effective equidistribution of horocycle lifts

We give a rate of equidistribution of lifts of horocycles from the space $\mathrm{SL}(2,\mathbb Z)\backslash \mathrm{SL}(2, \mathbb R)$ to the space $\mathrm{ASL}(2,\mathbb Z)\backslash \mathrm{ASL}(2,\mathbb R)$, making effective a theorem of Elkies and McMullen. This result constitutes an effective version of Ratner's measure classification theorem for measures supported on general horocycle lifts. The method used relies on Weil's resolution of the Riemann hypothesis for function fields in one variable and generalizes the approach of Strömbergsson to the case of linear lifts and that of Browning and the author to rational quadratic lifts.

math.DS↗
arXiv2016

Central limit theorems for simultaneous Diophantine approximations

We study the distribution modulo $1$ of the values taken on the integers of $r$ linear forms in $d$ variables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii $n^{-\frac{r}{d}}$. By the Khintchine-Groshev theorem on Diophantine approximations, $\frac{r}{d}$ is the critical exponent for the infinite number of hits.

math.DS↗
arXiv2016

Spherical averages in the space of marked lattices

A marked lattice is a $d$-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on ${\mathbb Z}^d$. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit.

math.DS↗
arXiv2016

Effective Ratner theorem for ASL(2,R) and gaps in \sqrt{n} modulo 1

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Strömbergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of \sqrt{n} modulo 1.

math.DS↗
arXiv2015

Directions in hyperbolic lattices

It is well known that the orbit of a lattice in hyperbolic $n$-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and Kontorovich for general lattices in dimension $n=2$.

math.DS↗
arXiv2013

The distribution of directions in an affine lattice: two-point correlations and mixed moments

We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most $T$. Strömbergsson and the second author proved in [Annals of Math.~173 (2010), 1949--2033] that the distribution of gaps between the lattice directions has a limit as $T$ tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs distinctly from the gap distribution observed in a Poisson process, which is exponential. The present study shows that the limiting two-point correlation function of the projected lattice points exists and is Poissonian. This answers a recent question by Boca, Popa and Zaharescu [arXiv:1302.5067]. The existence of the limit is subject to a certain Diophantine condition. We also establish the convergence of more general mixed moments.

math.NT↗
arXiv2013

Ergodic Properties of $k$-Free Integers in Number Fields

Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences, equipped with an $\mathcal O_K$-invariant probability measure associated to $\mathcal F_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Sinai arXiv:1112.4691 [math.DS] where $K=\mathbf Q$ and $k=2$.

math.DS↗
arXiv2013

The two-point correlation function of the fractional parts of \sqrt{n} is Poisson

Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown that the gaps between the fractional parts of \sqrt n for n=1,\ldots,N, have a limit distribution as N tends to infinity. The limit distribution is non-standard and differs distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the two-point correlation function of the above sequence converges to a limit, which in fact coincides with the answer for independent random variables. We also establish the convergence of moments for the probability of finding r points in a randomly shifted interval of size 1/N. The key ingredient in the proofs is a non-divergence estimate for translates of certain non-linear horocycles.

math.NT↗
arXiv2012

Effective bisector estimate with application to Apollonian circle packings

Let Γ<\PSL(2,\C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent δ be greater than 1. We use representation theory of \PSL(2,\C) to prove an effective bisector counting theorem for Γ, which allows counting the number of points of Γ in general expanding regions in \PSL(2,\C) and provides an explicit error term. We apply this theorem to give power savings in the Apollonian circle packing problem and related counting problems.

math.NT↗
arXiv2011

A Generalization of a Result of Hardy and Littlewood

In this note we study the growth of \sum_{m=1}^M\frac1{\|mα\|} as a function of M for different classes of α\in[0,1). Hardy and Littlewood showed that for numbers of bounded type, the sum is \simeq M\log M. We give a very simple proof for it. Further we show the following for generic α. For a non-decreasing function ϕtending to infinity, \limsup_{M\to\infty}\frac1{ϕ(\log M)}\bigg[\frac1{M\log M}\sum_{m=1}^M\frac1{\|mα\|}\bigg] is zero or infinity according as \sum\frac1{kϕ(k)} converges or diverges.

math.NT↗
arXiv2010

Limiting distribution of visits of sereval rotations to shrinking intervals

We show that given $n$ normalized intervals on the unit circle, the numbers of visits of $d$ random rotations to these intervals have a joint limiting distribution as lengths of trajectories tend to infinity. If $d$ then tends to infinity, then the numbers of points in different intervals become asymptotically independent unless an arithmetic obstruction arises. This is a generalization of earlier results of J. Marklof.

math.DS↗
arXiv2008

Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation $$Λ_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)Λ_{q}Λ_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $Λ_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $Λ_p$ with $Λ_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.

math.DS↗