Remarks on hard Lefschetz conjectures on Chow groups

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.Comment: to appear in Sciences in China, Ser. A Mathematic

Valley filtering and spatial maps of coupling between silicon donors and quantum dots

Exchange coupling is a key ingredient for spin-based quantum technologies since it can be used to entangle spin qubits and create logical spin qubits. However, the influence of the electronic valley degree of freedom in silicon on exchange interactions is presently the subject of important open questions. Here we investigate the influence of valleys on exchange in a coupled donor/quantum dot system, a basic building block of recently proposed schemes for robust quantum information processing. Using a scanning tunneling microscope tip to position the quantum dot with sub-nm precision, we find a near monotonic exchange characteristic where lattice-aperiodic modulations associated with valley degrees of freedom comprise less than 2~\% of exchange. From this we conclude that intravalley tunneling processes that preserve the donor's $\pm x$ and $\pm y$ valley index are filtered out of the interaction with the $\pm z$ valley quantum dot, and that the $\pm x$ and $\pm y$ intervalley processes where the electron valley index changes are weak. Complemented by tight-binding calculations of exchange versus donor depth, the demonstrated electrostatic tunability of donor/QD exchange can be used to compensate the remaining intravalley $\pm z$ oscillations to realise uniform interactions in an array of highly coherent donor spins.Comment: 6 pages, 4 figures, 6 pages Supplemental Materia

Certification of spin-based quantum simulators

Quantum simulators are engineered devices controllably designed to emulate complex and classically intractable quantum systems. A key challenge is to certify whether the simulator truly mimics the Hamiltonian of interest. This certification step requires the comparison of a simulator's output to a known answer, which is usually limited to small systems due to the exponential scaling of the Hilbert space. Here, in the context of Fermi-Hubbard spin-based analog simulators, we propose a modular many-body spin to charge conversion scheme that scales linearly with both the system size and the number of low-energy eigenstates to discriminate. Our protocol is based on the global charge state measurement of a 1D spin chain performed at different detuning potentials along the chain. In the context of semiconductor-based systems, we identify realistic conditions for detuning the chain adiabatically to avoid state mixing while preserving charge coherence. Large simulators with vanishing energy gaps, including 2D arrays, can be certified block-by-block with a number of measurements scaling only linearly with the system size

Optical properties of an ensemble of G-centers in silicon

We addressed the carrier dynamics in so-called G-centers in silicon (consisting of substitutional-interstitial carbon pairs interacting with interstitial silicons) obtained via ion implantation into a silicon-on-insulator wafer. For this point defect in silicon emitting in the telecommunication wavelength range, we unravel the recombination dynamics by time-resolved photoluminescence spectroscopy. More specifically, we performed detailed photoluminescence experiments as a function of excitation energy, incident power, irradiation fluence and temperature in order to study the impact of radiative and non-radiative recombination channels on the spectrum, yield and lifetime of G-centers. The sharp line emitting at 969 meV ($\sim$1280 nm) and the broad asymmetric sideband developing at lower energy share the same recombination dynamics as shown by time-resolved experiments performed selectively on each spectral component. This feature accounts for the common origin of the two emission bands which are unambiguously attributed to the zero-phonon line and to the corresponding phonon sideband. In the framework of the Huang-Rhys theory with non-perturbative calculations, we reach an estimation of 1.6$\pm$0.1 \angstrom for the spatial extension of the electronic wave function in the G-center. The radiative recombination time measured at low temperature lies in the 6 ns-range. The estimation of both radiative and non-radiative recombination rates as a function of temperature further demonstrate a constant radiative lifetime. Finally, although G-centers are shallow levels in silicon, we find a value of the Debye-Waller factor comparable to deep levels in wide-bandgap materials. Our results point out the potential of G-centers as a solid-state light source to be integrated into opto-electronic devices within a common silicon platform

Measurement of miniband parameters of a doped superlattice by photoluminescence in high magnetic fields

We have studied a 50/50\AA superlattice of GaAs/Al$_{0.21}$Ga$_{0.79}$As composition, modulation-doped with Si, to produce $n=1.4\times 10^{12}$ cm$^{-2}$ electrons per superlattice period. The modulation-doping was tailored to avoid the formation of Tamm states, and photoluminescence due to interband transitions from extended superlattice states was detected. By studying the effects of a quantizing magnetic field on the superlattice photoluminescence, the miniband energy width, the reduced effective mass of the electron-hole pair, and the band gap renormalization could be deduced.Comment: minor typing errors (minus sign in eq. (5)

Effective algebraic degeneracy

We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}

Holomorphic symplectic geometry: a problem list

A list of open problems on holomorphic symplectic, contact and Poisson manifolds