arXiv subjects

Anand Prakash

Publications and source records attributed to Anand Prakash.

6 recordsLinked to original sources

arXiv2026

Chromatic numbers of circulants with indispensable generators

The Cayley graph $\text{Cay}(G,S)$ is the graph whose vertex set is the group $G$, where two vertices $x$ and $y$ are adjacent if and only if $xy^{-1}$ or $yx^{-1}$ lies in some fixed subset $S$ of $G$. We call the elements of $S$ generators. A circulant graph is a Cayley graph where $G$ is finite and cyclic. Chromatic numbers of circulant graphs have been studied by many authors. A general formula due to Heuberger for the chromatic number of a circulant graph is known when $S$ has two elements, but no such formula is known when $S$ has three or more elements. We say that an element $x$ of $S$ is indispensable if $S\setminus\{x\}$ does not generate $G$. We say that $S$ is minimal if every element of $S$ is indispensable. By a result of Garcia-Marco and Knauer from 2024, if $G$ is nilpotent and $S$ is minimal, then $\text{Cay}(G,S)$ is $3$-colorable. In this article, we prove three main results. First, we give an upper bound for the chromatic number of a circulant graph with three generators, one of which is indispensable. Second, we present an alternate proof of the theorem of Garcia-Marco and Knauer for the case of abelian groups. Third, we apply these methods to provide a considerably more systematic (and potentially generalizable) proof of Heuberger's theorem for the chromatic number of circulant graphs with two generators. Throughout this paper, our primary tool is the theory of Heuberger matrices, for which we provide a brief primer.

math.CO↗

arXiv2026

Endcap-Type Paul Trap for Precision Spectroscopy and Studies of Controlled Interactions

We present the design and fabrication of an endcap-type Paul trap. The trap is designed for studies with Ca$^{+}$ and Yb$^{+}$. The design, fabrication process, and characterization are presented in detail with a focus on trapping a single compensated ion at the rf node. A custom-built imaging system of $NA = 0.14$ and magnification $\approx 22 \times$ performs close to diffraction-limit and resolves multi-ion clusters. Controlled ion loading and characterization of the trap are performed using $^{40}$Ca$^{+}$. The experimentally determined quadrupole coefficient of the trap is $\approx 0.3$, which is very close to the design value. The relative frequency shift along the spectroscopy beam due to excess micromotion (EMM) is at the level of $3.5\times 10^{-18}$ for $^{40}$Ca$^{+}$. Applications of this trap encompass single-ion-based optical frequency standards, tests of fundamental physics, the study of mesoscopic Coulomb clusters, and the controlled interaction of a single ion with co-trapped atoms.

physics.atom-ph↗

arXiv2026

Symmetry-controlled thermal activation in pyramidal Coulomb clusters: Testing Kramers-Langer theory

Laser-cooled ions confined in electromagnetic traps provide a unique, tunable mesoscopic system where the interplay of the trapping potential, nonlinear Coulomb interactions, and laser-ion scattering generates rich, collective dynamics. In this work, we engineer thermally activated switching between two oppositely oriented, square-pyramidal configurations of five laser-cooled ions in a Paul trap. For identical ions ($^{40}\mathrm{Ca}^{+}$), the inversions proceed via a \textit{Berry pseudo-rotation} mechanism with a low activation barrier, enabled by the permutation symmetry, in contrast to the \textit{umbrella inversion} observed in ammonia. The experimentally measured inversion rates, spanning two orders of magnitude, are accurately captured by the multidimensional Kramers-Langer theory, enabling thermometry of the Doppler-cooled ion cluster at $1.8 \pm 0.1$ mK. By substituting the apex ion with a heavier isotope ($^{44}\mathrm{Ca}^{+}$), we break the permutation symmetry and observe a suppression of thermally activated inversions. Numerical analysis reveals that this symmetry breaking closes the low-barrier channel, forcing the system to invert through a high-barrier \textit{turnstile rotation}. Thus, we demonstrate a structural analogue of molecular kinetic isotope effects, establishing trapped ions as a versatile platform to explore symmetry-controlled collective dynamics.

physics.atom-ph↗

arXiv2025

Nuclear spin quenching of the $^2S_{1/2}\rightarrow {^2}F_{7/2} $ electric octupole transition in $^{173}$Yb$^+$

We report the coherent excitation of the highly forbidden $^2S_{1/2} \rightarrow {^2}F_{7/2}$ clock transition in the odd isotope $^{173}\mathrm{Yb}^+$ with nuclear spin $I = 5/2$, and reveal the hyperfine-state-dependent, nuclear spin induced quenching of this transition. The inferred lifetime of the $F_e = 4$ hyperfine state is one order of magnitude shorter than the unperturbed ${^2}F_{7/2}$ clock state of $^{171}\mathrm{Yb}^+$. This reduced lifetime lowers the required optical power for coherent excitation of the clock transition, thereby reducing the AC Stark shift caused by the clock laser. Using a 3-ion Coulomb crystal, we experimentally demonstrate an approximately 20-fold suppression of the AC Stark shift, a critical improvement for the scalability of future multi-ion $\mathrm{Yb}^+$ clocks. Furthermore, we report the $|^2S_{1/2},F_g=3\rangle~\rightarrow~|^2F_{7/2},F_e=6\rangle$ unquenched reference transition frequency as $642.11917656354(43)$ THz, along with the measured hyperfine splitting and calculated quadratic Zeeman sensitivities of the ${^2}F_{7/2}$ clock state. Our results pave the way toward multi-ion optical clocks and quantum computers based on $^{173}\mathrm{Yb}^+$.

physics.atom-ph↗

arXiv2025

On Matsushita $\pi_1^2$ discrete fundamental groups

The Matsushita fundamental groups of a graph $X$, denoted $\pi_1^r(X)$, are certain discrete versions of the fundamental group for topological spaces. For $r=2$, these groups have a nice combinatorial description, due to Sankar. In this paper we prove two results about $\pi_1^2$. First, we prove a Seifert-van Kampen-type theorem. Similar results have previously been obtained by Barcelo, et al. (and strengthened by Kapulkin and Mavinkurve) for a different notion of discrete fundamental group. Second, we prove that an arbitrary group $G$ can be realized as $\pi_1^2(X)$ for some graph $X$. Our construction works equally well for the aforementioned alternate discrete fundamental group $A_1(X)$, and our second result thus also provides an entirely different method of proof for a theorem of Kapulkin and Mavinkurve.

math.CO↗

arXiv2025

Observing the dynamics of octupolar structural transitions in trapped-ion clusters

Interacting many-particle systems can self-organize into a rich variety of crystalline structures. While symmetry provides a powerful framework for predicting whether transitions between crystal states are continuous or discontinuous, collective lattice dynamics offer complementary insights into the microscopic mechanisms that drive these transitions. Trapped laser-cooled ions present a pristine and highly controllable few-body system for studying this interplay of symmetry and dynamics. Here, we use real-time fluorescence imaging while deforming the trap potential to observe a variety of structural transitions in three-dimensional (3D), unit-cell-like ion clusters. We identify a set of transitions signaled by parity-odd octupole order parameters, and probe their distinct dynamical signatures. Our observations reveal the softening of a collective Higgs-like mode indicating spontaneous symmetry-breaking, hysteresis resulting from a catastrophe where a metastable state vanishes abruptly, and stochastic switching between metastable states of differing symmetries. We also uncover a remarkable coincidence of symmetry-breaking and discontinuous transitions, analogous to a thermodynamic triple point. Our results establish 3D trapped-ion clusters as a versatile platform to engineer complex potential energy landscapes, opening new avenues for studies of reaction kinetics, geometric frustration, and related phenomena in mesoscopic platforms.

physics.atom-ph↗