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We observe a considerable hysteresis effect of low-field microwave absorption (LFMA) in copper-substituted lead apatite. By continuously rotating samples under external magnetic field, this effect is diminished which can not be renewed by a strong magnetic field but will be spontaneously recovered after two days, indicating its glassy features and excluding possibility of any ferromagnetism. The intensity of LFMA is found to sharply decrease at around 250K, suggesting a phase transition takes place. A lattice gauge model is then employed to assign these effects to the transition between superconducting Meissner phase and vortex glass, and the slow dynamics wherein is calculated as well., Comment: 13 pages, 3 figures

Two entangled qubits emerge as an essential resource for quantum control, which are normally quantum confined with atomic precision. It seems inhibitive that in the macroscopic scope collective qubit pairs manifest long coherence and quantum entanglement, especially at high temperature. Here, we report this exotic ensemble effect in solid-state sintering lead apatite samples with copper substitution, which have been repeatedly duplicated with superior stability and low cost. An extraordinarily low-field absorption signal of cw electron paramagnetic resonance (EPR) spectroscopy stems from low-mass conduction electrons implying the coherence of cuprate radicals can be long-termly protected. The pulsed EPR experiments exhibit triplet Rabi oscillation from paired cuprate diradicals with the coherence time exceeding 1 microsecond at 85K. We believe these appealing effects are sufficiently promising to be applied for scalable quantum control and computation., Comment: 16 pages, 4 figures

This work is devoted to averaging principle of a two-time-scale stochastic partial differential equation on a bounded interval $[0, l]$, where both the fast and slow components are directly perturbed by additive noises. Under some regular conditions on drift coefficients, it is proved that the rate of weak convergence for the slow variable to the averaged dynamics is of order $1-\varepsilon$ for arbitrarily small $\varepsilon>0$. The proof is based on an asymptotic expansion of solutions to Kolmogorov equations associated with the multiple-time-scale system.

This article deals with the weak errors for averaging principle for a stochastic wave equation in a bounded interval $[0,L]$, perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. Under suitable conditions, it is proved that the rate of weak convergence to the averaged effective dynamics is of order $1$ via an asymptotic expansion approach., Comment: arXiv admin note: text overlap with arXiv:1701.07983

The present article deals with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equation. Under suitable conditions, the weak error is expanded in powers of timescale parameter. It is proved that the rate of weak convergence to the averaged dynamics is of order $1$. This reveals the rate of weak convergence is essentially twice that of strong convergence.

In this article, we investigate averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, in which both the slow and fast components are perturbed by multiplicative noises. Particularly, we prove that the rate of strong convergence for the slow component to the averaged dynamics is of order $1/2$, which significantly improves the order $1/4$ established in our previous work.