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Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s

The question of whether there exists a finite mobility in the standard Holstein model with one vibrational mode on each site remains unclear. In this letter, we approach this problem by employing the hierarchical equation of motion (HEOM) method to simulate model systems where the vibrational modes are dissipative. It is found that, as the friction becomes smaller, the charge carrier mobility increases significantly and a friction free limit can not be obtained. The current autocorrelation functions are also calculated for the friction free Holstein model, and converged results can not be obtained with the increase of the number of sites. Based on these observations, we conclude that a finite mobility can not be defined for the standard Holstein model, which is consistent with the recent finding by Kolss {\it et al.} (Phys. Rev. Lett. 123, 126601).

Singlet fission (SF) is a very significant photophysical phenomenon and possesses potential applications. In this work, we try to give the rather detailed theoretical investigation of the SF process in the stacked polyacene dimer by combining the high-level quantum chemistry calculations, and the quantum dynamics simulations based on the tensor train decomposition method. Starting from the construction of the linear vibronic coupling model, we explore the pure electronic dynamics and the vibronic dynamics in the SF processes. The role of vibrational modes in nonadiabatic dynamics is addressed. The results show that the super-exchange mechanism mediated by the charge-transfer state is found in both pure electronic dynamics and the nonadiabatic dynamics. Particularly, the vibrational modes with the frequency resonance with the adiabatic energy gap play very import roles in the SF dynamics. This work not only provides a deep and detailed understanding of the SF process, but also verifies the efficiency of the tensor train decomposition method that can serve as the reference dynamics method to explore the dynamics behaviors of complex systems.

Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that $$\liminf_{n\rightarrow\infty}r_{1,k}(A,n)=\infty$$ providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu's result and obtained that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{\log n}>0.$$ In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{n}>0.$$ Our bound reflects the correct order of magnitude of the representation function $r_{1,k}(A,n)$ under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le n/k.$

For any given set $A$ of nonnegative integers and for any given two positive integers $k_1,k_2$, $R_{k_1,k_2}(A,n)$ is defined as the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we prove that if integer $k\geq2$ and set $A\subseteq\mathbb{N}$ such that $R_{1,k}(A,n)=R_{1,k}(\mathbb{N}\setminus A,n)$ holds for all integers $n\geq n_0$, then $R_{1,k}(A,n)\gg \log n$.

Allowing the generation of effective interactions between distant quantum emitters (QEs) via flying photons, quantum interconnect (QI) is essentially a light-matter interface and acts as a building block in quantum technologies. A surface plasmon polariton (SPP) supported by a metallic waveguide provides an ideal interface to explore strong light-matter couplings and to realize QI. However, the loss of SPP in metal makes the mediated entanglement of the QEs damp with the increase of the distance and time, which hinders its applications. We propose a scheme of non-Markovian QI formed by the SPP of a metallic nanowire. A mechanism to make the generated entanglement of the QEs persistent is discovered. We find that, as long as bound states are formed in the energy spectrum of total QE-SPP system, the damping of the SPP-mediated entanglement is overcome even in the presence of the metal absorption to the SPP. Our finding enriches our understanding of light-matter couplings in absorptive medium and paves the way for using the SPP in designing QI.