arXiv subjects

Shasha Zheng

Publications and source records attributed to Shasha Zheng.

11 recordsLinked to original sources

arXiv2026

Asymptotic enumeration of minimally transitive permutation groups

We prove that Pyber's upper bound $2^{O(n\log(n))}$ for the number of minimally transitive subgroups of $S_n$ is best possible along the powers of every fixed prime, even when the groups are counted up to permutational isomorphism. As a byproduct, our construction shows that, along the powers of every fixed prime, the maximum order of a minimally transitive permutation group of degree $n$ is $2^{\Theta(n)}$. For completeness, we also present Pyber's previously unpublished proof of his upper bound. We further deduce that the numbers of labelled vertex-transitive graphs and digraphs of order $n$ are both $2^{\Theta(n\log(n))}$, and discuss the implications of our results for approaches to the McKay--Praeger conjecture.

math.GR↗

arXiv2026

Almost all standard double covers of abelian Cayley graphs have smallest possible automorphism groups

The standard double cover of a graph $\Gamma$ is the direct product $\Gamma\times K_2$. A graph $\Gamma$ is said to be stable if all the automorphisms of $\Gamma\times K_2$ come from its factors. Although the study of stability has attracted significant attention, particularly regarding Cayley graphs of abelian groups, a complete classification remains elusive even for Cayley graphs of cyclic groups. In this paper, we study the asymptotic enumeration of both labeled and unlabeled Cayley graphs of abelian groups whose standard double cover has the smallest possible automorphism group. As a corollary, in both the labeled and unlabeled settings, we conclude that the proportion of stable Cayley graphs of an abelian group of order $r$ approaches $1$ as $r\rightarrow\infty$, proving that almost all Cayley graphs of finite abelian groups are stable.

math.CO↗

arXiv2023

Asymptotic enumeration of graphical regular representations

We estimate the number of graphical regular representations (GRRs) of a given group with large enough order. As a consequence, we show that almost all finite Cayley graphs have full automorphism groups 'as small as possible'. This confirms a conjecture of Babai-Godsil-Imrich-Lovasz on the proportion of GRRs, as well as a conjecture of Xu on the proportion of normal Cayley graphs, among Cayley graphs of a given finite group.

math.CO↗

arXiv2023

Tutorial: Nonlinear magnonics

Nonlinear magnonics studies the nonlinear interaction between magnons and other physical platforms (phonon, photon, qubit, spin texture) to generate novel magnon states for information processing. In this tutorial, we first introduce the nonlinear interactions of magnons in pure magnetic systems and hybrid magnon-phonon and magnon-photon systems. Then we show how these nonlinear interactions can generate exotic magnonic phenomena. In the classical regime, we will cover the parametric excitation of magnons, bistability and multistability, and the magnonic frequency comb. In the quantum regime, we will discuss the single magnon state, Schrödinger cat state and the entanglement and quantum steering among magnons, photons and phonons. The applications of the hybrid magnonics systems in quantum transducer and sensing will also be presented. Finally, we outlook the future development direction of nonlinear magnonics.

cond-mat.mes-hall↗

arXiv2022

Cubic graphical regular representations of some classical simple groups

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}Ω_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}Ω_{n}^{-}(q)$ ($n=8, 10, 12$), where $q = 2^f$ with $f \ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q \rightarrow \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^{i}-1$ for each $1 \le i < ef$ together with a random involution $y$ gives rise to a cubic GRR, where $e=n-2$ for $\mathrm{P}Ω_{n}^{+}(q)$ and $e=n$ for other groups. Moreover, for sufficiently large $q$, there are elements $x$ satisfying these conditions, and for each of them there exists an involution $y$ such that $\{x,x^{-1},y\}$ produces a cubic GRR. This result together with certain known results in the literature implies that except for $\mathrm{PSL}_2(q)$, $\mathrm{PSL}_3(q)$, $\mathrm{PSU}_3(q)$ and a finite number of other cases, every finite non-abelian simple group contains an element $x$ and an involution $y$ such that $\{x,x^{-1},y\}$ produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR.

math.GR↗

arXiv2022

Cubic Graphical Regular Representations of $\mathrm{PSU}_3(q)$

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that $\mathrm{PSU}_3(q)$ has a cubic GRR if and only if $q\geq4$. Moreover, a cubic GRR of $\mathrm{PSU}_3(q)$ is constructed for each of these $q$.

math.GR↗

arXiv2021

Unconventional magnon excitation by off-resonant microwaves

It is widely recognized that a physical system can only respond to a periodic driving significantly when the driving frequency matches the normal mode frequency of the system, which leads to resonance. Off-resonant phenomena are rarely considered because of the difficulty to realize strong coupling between physical systems and off-resonant waves. Here we examine the response of a magnetic system to squeezed light and surprisingly find that the magnons are maximally excited when the effective driving frequency is several orders of magnitude larger than the resonant frequency. The generated magnons are squeezed which brings the advantage of tunable squeezing through an external magnetic field. Furthermore, we demonstrate that such off-resonant quasi-particle excitation is universal in all the hybrid systems in which the coherent and parametric interaction of bosons exists and that it is purely a quantum effect, which is rooted in the quantum fluctuations of particles in the squeezed vacuum. Our findings may provide an unconventional route to study off-resonant phenomena and may further benefit the use of hybrid matter-light systems in continuous variable quantum information.

cond-mat.mes-hall↗

arXiv2020

Nonperturbative leakage elimination for a logical qubit encoded in a mechanical oscillator

Continuous-variable (CV) systems are attracting increasing attention in the realization of universal quantum computation. Several recent experiments have shown the feasibility of using CV systems to, e.g., encode a qubit into a trapped-ion mechanical oscillator and perform logic gates [Nature 566, 513-517 (2019)]. The essential next step is to protect the encoded qubit from quantum decoherence, e.g., the motional decoherence due to the interaction between a mechanical oscillator and its environment. Here we propose a scheme to suppress quantum decoherence of a single-mode harmonic oscillator used to encode qubits by introducing a nonperturbative leakage elimination operator (LEO) specifically designed for this purpose. Remarkably, our nonperturbative LEO can be used to analytically derive exact equations of motion without approximations. It also allows us to prove that the effectiveness of these LEOs only depends on the integral of the pulse sequence in the time domain, while details of the pulse shape does not make a significant difference when the time period is chosen appropriately. This control method can be applied to a system at an arbitrary temperature and arbitrary system-bath coupling strength which makes it extremely useful for general open quantum systems.

quant-ph↗

arXiv2019

Steady Bell state generation via magnon-photon coupling

We show that parity-time ($\mathcal{PT}$) symmetry can be spontaneously broken in the recently reported energy level attraction of magnons and cavity photons. In the $\mathcal{PT}$-broken phase, magnon and photon form a high-fidelity Bell state with maximum entanglement. This entanglement is steady and robust against the perturbation of environment, in contrast to the general wisdom that expects instability of the hybridized state when the symmetry is broken. This anomaly is further understood by the compete of non-Hermitian evolution and particle number conservation of the hybridized system. As a comparison, neither $\mathcal{PT}$-symmetry broken nor steady magnon-photon entanglement is observed inside the normal level repulsion case. Our results may open a novel window to utilize magnon-photon entanglement as a resource for quantum technologies.

cond-mat.mes-hall↗

arXiv2019

Enhancement of antiferromagnetic magnon-magnon entanglement by cavity cooling

Magnon-photon coupling has been experimentally realized inside a cavity and the emerging field known as cavity spintronics has attracted significant attention for its potential docking with quantum information science. However, one seldom knows whether this coupling implies an entanglement state among magnons and photons or not, which is crucial for its usage in quantum information. Here we study the entanglement properties among magnons and photons in an antiferromagnet-light system and find that the entanglement between magnon and photon is nearly zero while the magnon-magnon entanglement is very strong and it can be even further enhanced through the coupling with the cavity photons. The maximum enhancement occurs when the antiferromagnet reaches resonant with the light. The essential physics can be well understood within the picture of cavity induced cooling of magnon-magnon state near its joint vacuum with stronger entanglement. Our results are significant to extend the cavity spintronics to quantum manipulation and further provide an alternate to manipulate the deep strong correlations of continuous variable entanglement with a generic stable condition and easy tunability.

cond-mat.mes-hall↗

arXiv2018

Manipulation and enhancement of asymmetric steering via interference effects induced by closed-loop coupling

We present a phase control method for a general three-mode system with closed-loop in coupling that drives the system into an entangled steady state and produces directional steering between two completely symmetric modes via quantum interference effects. In the scheme, two modes are coupled with each other both by a direct binary interaction and by an indirect interaction through a third intermediate damping mode, creating interference effects determined by the relative phase between the two physical interaction paths. By calculating the populations and correlations of the two modes, we show that depending on the phase, two modes can be prepared into an entangled steady state with asymmetric and directional steering even if they possess completely symmetric decoherence properties. Meanwhile, entanglement and steering can be significantly enhanced due to constructive interference, and thus more robust to thermal noises. This provides an active method to manipulate the asymmetry of steering instead of adding asymmetric losses or noises on subsystems at the cost of reducing steerability. Moreover, we show that the interference effects can also enhance and control the correlations between other pair of modes in the loop with opposite phase dependent behavior, indicating monogamy constraints for distributing correlations among multipartite. The present model could be applied in cavity optomechanical systems or in antiferromagnets where all components can mutually interact.

quant-ph↗