The ability to tailor laser light on a chip using integrated photonics has allowed for extensive control over fundamental light-matter interactions in manifold quantum systems including atoms, trapped ions, quantum dots, and defect centers. Free elec
trons, enabling high-resolution microscopy for decades, are increasingly becoming the subject of laser-based quantum manipulation. Using free-space optical excitation and intense laser pulses, this has led to the observation of free-electron quantum walks, attosecond electron pulses, and imaging of electromagnetic fields. Enhancing the interaction with electron beams through chip-based photonics promises unique applications in nanoscale quantum control and sensing, but has yet to enter electron microscopy. Here, we merge integrated photonics with electron microscopy, demonstrating coherent phase modulation of an electron beam using a silicon nitride microresonator driven by a continuous-wave laser. The high-Q factor (~$10^6$) cavity enhancement and a waveguide designed for phase matching lead to efficient electron-light scattering at unprecedentedly low, few-microwatt optical powers. Specifically, we fully deplete the initial electron state at a cavity-coupled power of 6 $mu$W and create >500 photon sidebands for only 38 mW in the bus waveguide. Moreover, we demonstrate $mu$eV electron energy gain spectroscopy (EEGS). Providing simultaneous optical and electronic spectroscopy of the resonant cavity, the fiber-coupled photonic structures feature single-mode electron-light interaction with full control over the input and output channels. This approach establishes a versatile framework for exploring free-electron quantum optics, with future developments in strong coupling, local quantum probing, and electron-photon entanglement. Our results highlight the potential of integrated photonics to efficiently interface free electrons and light.
We study invariants and quotient categories of fixed subrings of Artin-Schelter regular algebras under Hopf algebra actions.
We provide a construction of minimal injective resolutions of simple comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau condition of algebras, we introduce the Calabi-Yau condition to coalgebras. Then we give some descriptions
of Calabi-Yau coalgebras with lower global dimensions. An appendix is included for listing some properties of cohom functors.
A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual coalgebra. If $A
$ is Artin-Schelter regular, then the local cohomology of $A$ is isomorphic to a shift of twisted bimodule ${}_1C_{sigma^*}$ with $sigma$ a coalgebra automorphism. This yields that the balanced dualinzing complex of $A$ is a shift of the twisted bimodule ${}_{sigma^*}A_1$. If $sigma$ is an inner automorphism, then $A$ is Calabi-Yau.
The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal en
veloping algebra of a finite dimensional Lie algebra $g$ with a finite subgroup $G$ of automorphisms of $g$ is Calabi-Yau if and only if the universal enveloping algebra itself is Calabi-Yau and $G$ is a subgroup of the special linear group $SL(g)$. The Noetherian cocommutative Calabi-Yau Hopf algebras of dimension not larger than 3 are described. The Calabi-Yau property of Sridharan enveloping algebras of finite dimensional Lie algebras is also discussed. We obtain some equivalent conditions for a Sridharan enveloping algebra to be Calabi-Yau, and then partly answer a question proposed by Berger. We list all the nonisomorphic 3-dimensional Calabi-Yau Sridharan enveloping algebras.
The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul
DG algebra $A$ is AS-regular, the Ext-algebra $E$ of $A$ is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left $E$-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG $A$-modules consisting of all compact DG modules modulo the full triangulated subcategory consisting of all the right DG modules with finite dimensional cohomology. The classical BGG correspondence can derived from the DG version.
