This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries(KdV)system in Rn.We first prove,by the Leray-Schauder principle and the vanishing viscosity method,that any initial da...This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries(KdV)system in Rn.We first prove,by the Leray-Schauder principle and the vanishing viscosity method,that any initial data N-dimensional vector value function u0(x)in Sobolev space H^(s)(R^(n))(s≥1)leads to a global weak solution.Second,we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in R^(2)and R^(3).展开更多
In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 bou...In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.展开更多
We investigate the rotating wave approximation applied in the high-spin quantum system driven by a linearly polarized alternating magnetic field in the presence of quadrupole interactions.The conventional way to apply...We investigate the rotating wave approximation applied in the high-spin quantum system driven by a linearly polarized alternating magnetic field in the presence of quadrupole interactions.The conventional way to apply the rotating wave approximation in a driven high-spin system is to assume the dynamics being restricted in the reduced Hilbert space.However,when the driving strength is relatively strong or the driving is off resonant,the leakage from the target resonance subspace cannot be neglected for a multi-level quantum system.We propose the correct formalism to apply the rotating wave approximation in the full Hilbert space by taking this leakage into account.By estimating the operator fidelity of the time propagator,our formalism applied in the full Hilbert space unambiguously manifests great advantages over the conventional method applied in the reduced Hilbert space.展开更多
We develop a quantum optical description of radiation from a two-level system(TLS)in strong laser fields,which provides a clear insight into the final states of the TLS and the harmonics field.It is shown that there a...We develop a quantum optical description of radiation from a two-level system(TLS)in strong laser fields,which provides a clear insight into the final states of the TLS and the harmonics field.It is shown that there are two emission channels:the Rayleigh-like channel and the Raman-like channel,which correspond to the TLS ending up in the ground state and excited state after the emission,respectively.The numerical result shows that the harmonics are mainly produced by the Rayleigh-like channel.In addition,according to the coherence of emission among the emitters,the radiation is divided into coherent parts that result from the semi-classical dipole oscillation and incoherent parts that result from the quantum fluctuations of the dipole moment.In the weak field limits,the Rayleigh-like channel corresponds to the coherent parts,and the Raman-like channel corresponds to the incoherent parts.However,in strong laser fields,both channels contribute to coherent and incoherent radiation,and how much they contribute depends on the final excitation.By manipulating the laser field,we can make the Rayleigh-like channel produce either coherent or incoherent radiation.展开更多
The creation and relaxation of double K-hole states 1s^(0)2s^(2)2p^(6)np(n≥3)of Ne^(1+)in the interaction with ultraintense ultrafast x-ray pulses are theoretically investigated.The x-ray photon energies are selected...The creation and relaxation of double K-hole states 1s^(0)2s^(2)2p^(6)np(n≥3)of Ne^(1+)in the interaction with ultraintense ultrafast x-ray pulses are theoretically investigated.The x-ray photon energies are selected so that x-rays first photoionize1s^(22)s^(22)p^(6) of a neon atom to create a single K-hole state of 1s2s^(22)p^(6) of Ne^(1+),which is further excited resonantly to double K-hole states of ls^(0)2s^(2)2p^(6)np(n≥3).A time-dependent rate equation is used to investigate the creation and relaxation processes of 1s^(0)2s^(2)2p^(6)np,where the primary microscopic atomic processes including photoexcitation,spontaneous radiation,photoionization and Auger decay are considered.The calculated Auger electron energy spectra are compared with recent experimental results,which shows good agreement.The relative intensity of Auger electrons is very sensitive to the photon energy and bandwidth of x-ray pulses,which could be used as a diagnostic tool for x-ray free electron laser and atom experiments.展开更多
Accurate and fast prediction of thermal radiation properties of materials is crucial for their potential applications.However,some models assume that the media are made up of pure water droplets,which do not account f...Accurate and fast prediction of thermal radiation properties of materials is crucial for their potential applications.However,some models assume that the media are made up of pure water droplets,which do not account for the increasing deviations caused by volcanic eruptions,pollution,and human activities that exacerbate dust production.The distinct radiation properties of water and dust particles make it challenging to determine the thermal radiation properties of water droplets containing dust particles.To address this issue,we investigate the influence of dust particles on light transmission and energy distribution in water droplets using the multiple sphere T-matrix method.By considering different droplet and dust diameters,volume fractions,and position distributions,we analyze how extinction regulation is achieved in dust-containing water droplets.Our results reveal the significant role of dust particles in the thermal radiation effect and provide insights into the electromagnetic properties of colloidal suspensions.Moreover,the dust-induced reestablishment of energy balance raises concerns about environmental management and climate change.This research highlights the importance of accounting for dust particles in atmospheric models and their potential impact on radiative balance.展开更多
In this paper, we prove the existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations in two-dimension with finite energy.The main techniques is the Faedo-Galerkin app...In this paper, we prove the existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations in two-dimension with finite energy.The main techniques is the Faedo-Galerkin approximation and weak compactness theory.展开更多
Isolated attosecond pulses with a duration of 88 as are generated in the spectral range of 29–72 eV using double optical gating technique.The gate width is set to be shorter than half the optical cycle to avoid carri...Isolated attosecond pulses with a duration of 88 as are generated in the spectral range of 29–72 eV using double optical gating technique.The gate width is set to be shorter than half the optical cycle to avoid carrier envelop phase stabilization of the 4.2 fs driving laser pulses centered at 800 nm.The attosecond pulse duration is measured with the technique of frequency resolved optical gating for complete reconstruction of attosecond bursts.展开更多
In the absorption chamber of a high-energy laser energy meter, water is directly used as an absorbing medium and the interaction of the high-power laser and the water flow can produce a variety of physical phenomena s...In the absorption chamber of a high-energy laser energy meter, water is directly used as an absorbing medium and the interaction of the high-power laser and the water flow can produce a variety of physical phenomena such as phase transitions. The unit difference method is adopted to deduce the phase transition model for water flow irradiated by a high-energy laser. In addition, the model is simulated and verified through experiments. Among them, the experimental verification uses the photographic method, shooting the distribution and the form of the air mass of water flow in different operating conditions, which are compared with the simulation results. The research shows that it is achievable to reduce the intensity of the phase transition by increasing the water flow, reducing the power intensity of the beam, shortening the distance the beam covers, reducing the initial water temperature or adopting a shorter wavelength laser. The study's results will provide the reference for the design of a water-direct-absorption-type high-energy laser energy meter as well as an analysis of the interaction processes of other similar high-power lasers and water flow.展开更多
The elaborate energy and momentum spectra of ionized electrons from atoms in laser fields suggest that the ionization dynamics described by tunneling theory should be modified. Although great efforts have been carried...The elaborate energy and momentum spectra of ionized electrons from atoms in laser fields suggest that the ionization dynamics described by tunneling theory should be modified. Although great efforts have been carried out within semiclassical models, there are few discussions describing the multiphoton absorption process within a quantum framework. Comparing the results obtained with the time-dependent Schr?dinger equation(TDSE)and the Keldysh–Faisal–Reiss(KFR) theory, we study the nonperturbative effects of ionization dynamics beyond the KFR theory. The difference in momentum spectra between multiphoton and tunneling regimes is understood in a unified picture with virtual multiphoton absorption processes. For the multiphoton regime, the momentum spectra can be obtained by coherent interference of each periodic contribution. However, the interference of multiphoton absorption peaks will result in a complex structure of virtual multiphoton bands in the tunneling regime. It is shown that the virtual spectra will be almost continuous in the tunneling regime instead of the discrete levels found in the multiphoton regime. Finally, with a model combining the TDSE and the KFR theory,we try to understand the different effects of virtual multiphoton processes on ionization dynamics.展开更多
The author considers the global existence and global nonexistence of theinitial-boundary value problem for some degenerate hyperbolic equation of the formutt ? div(|?u|p?2?u) = |u|mu, (x, t) ∈ [0, +∞) × ?with p...The author considers the global existence and global nonexistence of theinitial-boundary value problem for some degenerate hyperbolic equation of the formutt ? div(|?u|p?2?u) = |u|mu, (x, t) ∈ [0, +∞) × ?with p > 2 and m > 0. He deals with the global solutions by D.H.Sattinger’s potential wellideas. At the same time, when the initial energy is positive, but appropriately bounded,the global nonexistence of solutions is verified by using the analysis method.展开更多
In this paper,we consider spatial-temporal correlation functions of the turbulent velocities.With numerical simulations on the Gledzer-Ohkitani-Yamada (GOY) shell model,we show that the correlation function decays exp...In this paper,we consider spatial-temporal correlation functions of the turbulent velocities.With numerical simulations on the Gledzer-Ohkitani-Yamada (GOY) shell model,we show that the correlation function decays exponentially.The advecting velocity field is regarded as a colored noise field,which is spatially and temporally correlative.For comparison,we are also given the scaling exponents of passive scalars obtained by the Gaussian random velocity field,the multi-dimensional normal velocity field and the She-Leveque velocity field,introduced by She,et al.We observe that extended self-similarity scaling exponents H(p)/H(2) of passive scalar obtained by the colored noise field are more anomalous than those obtained by the other three velocity fields.展开更多
We study how the decoherence of macroscopic objects originates intrinsically from the relativistic effect. With the degree of freedom of the center of mass(CM) characterizing the collective quantum state of a macrosco...We study how the decoherence of macroscopic objects originates intrinsically from the relativistic effect. With the degree of freedom of the center of mass(CM) characterizing the collective quantum state of a macroscopic object(MO),it is found that an MO consisting of N particles can decohere with a time scale of no more than p (N^(1/2))^-1. Here, the special relativity can induce the coupling of the collective motion mode and the relative motion modes in an order of 1/c^2, which intrinsically results in the above minimum decoherence.展开更多
A three-wave interaction (3WI) code is developed to study the stimulated Raman scattering (SRS) in both absolute and convective regimes. In the simulations, the time and spatial evolutions of a plasma wave are describ...A three-wave interaction (3WI) code is developed to study the stimulated Raman scattering (SRS) in both absolute and convective regimes. In the simulations, the time and spatial evolutions of a plasma wave are described by temporal growth rate and spatial factor, respectively. The spatial factors in different phases and different instability regimes are investigated. It is found that the spatial factor is caused by the finite velocity of the pump wave in the first phase and by damping in the last phase. With inclusion of the spatial factor, the temporal growth rate decreases and the threshold for SRS for a finite frequency mismatch increases. Meanwhile, the effects of wave frequency mismatch on the temporal growth rate are also discussed.展开更多
Minimizing the effect of noise is essential for quantum computers.The conventional method to protect qubits against noise is through quantum error correction.However,for current quantum hardware in the so-called noisy...Minimizing the effect of noise is essential for quantum computers.The conventional method to protect qubits against noise is through quantum error correction.However,for current quantum hardware in the so-called noisy intermediate-scale quantum(NISQ)era,noise presents in these systems and is too high for error correction to be beneficial.Quantum error mitigation is a set of alternative methods for minimizing errors,including error extrapolation,probabilistic error cancella-tion,measurement error mitigation,subspace expansion,symmetry verification,virtual distillation,etc.The requirement for these methods is usually less demanding than error correction.Quantum error mitigation is a promising way of reduc-ing errors on NISQ quantum computers.This paper gives a comprehensive introduction to quantum error mitigation.The state-of-art error mitigation methods are covered and formulated in a general form,which provides a basis for comparing,combining and optimizing different methods in future work.展开更多
文摘This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries(KdV)system in Rn.We first prove,by the Leray-Schauder principle and the vanishing viscosity method,that any initial data N-dimensional vector value function u0(x)in Sobolev space H^(s)(R^(n))(s≥1)leads to a global weak solution.Second,we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in R^(2)and R^(3).
基金supported by the National NaturalScience Foundation of China(11971069 and 12126307)。
文摘In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.
基金the National Key Research and Development Program of China(Grant Nos.2017YFA0304202 and 2017YFA0205700)the National Natural Science Foundation of China(Grant Nos.11875231 and 11935012)the Fundamental Research Funds for the Central Universities(Grant No.2018FZA3005).
文摘We investigate the rotating wave approximation applied in the high-spin quantum system driven by a linearly polarized alternating magnetic field in the presence of quadrupole interactions.The conventional way to apply the rotating wave approximation in a driven high-spin system is to assume the dynamics being restricted in the reduced Hilbert space.However,when the driving strength is relatively strong or the driving is off resonant,the leakage from the target resonance subspace cannot be neglected for a multi-level quantum system.We propose the correct formalism to apply the rotating wave approximation in the full Hilbert space by taking this leakage into account.By estimating the operator fidelity of the time propagator,our formalism applied in the full Hilbert space unambiguously manifests great advantages over the conventional method applied in the reduced Hilbert space.
基金the National Key R&D Program of China(Grant No.2019YFA0307703)the National Natural Science Foundation of China(Grant Nos.12234020,12274384,and 11774415)the Major Research Plan of the National Natural Science Foundation of China(Grant No.91850201)。
文摘We develop a quantum optical description of radiation from a two-level system(TLS)in strong laser fields,which provides a clear insight into the final states of the TLS and the harmonics field.It is shown that there are two emission channels:the Rayleigh-like channel and the Raman-like channel,which correspond to the TLS ending up in the ground state and excited state after the emission,respectively.The numerical result shows that the harmonics are mainly produced by the Rayleigh-like channel.In addition,according to the coherence of emission among the emitters,the radiation is divided into coherent parts that result from the semi-classical dipole oscillation and incoherent parts that result from the quantum fluctuations of the dipole moment.In the weak field limits,the Rayleigh-like channel corresponds to the coherent parts,and the Raman-like channel corresponds to the incoherent parts.However,in strong laser fields,both channels contribute to coherent and incoherent radiation,and how much they contribute depends on the final excitation.By manipulating the laser field,we can make the Rayleigh-like channel produce either coherent or incoherent radiation.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12074430 and 11974423)。
文摘The creation and relaxation of double K-hole states 1s^(0)2s^(2)2p^(6)np(n≥3)of Ne^(1+)in the interaction with ultraintense ultrafast x-ray pulses are theoretically investigated.The x-ray photon energies are selected so that x-rays first photoionize1s^(22)s^(22)p^(6) of a neon atom to create a single K-hole state of 1s2s^(22)p^(6) of Ne^(1+),which is further excited resonantly to double K-hole states of ls^(0)2s^(2)2p^(6)np(n≥3).A time-dependent rate equation is used to investigate the creation and relaxation processes of 1s^(0)2s^(2)2p^(6)np,where the primary microscopic atomic processes including photoexcitation,spontaneous radiation,photoionization and Auger decay are considered.The calculated Auger electron energy spectra are compared with recent experimental results,which shows good agreement.The relative intensity of Auger electrons is very sensitive to the photon energy and bandwidth of x-ray pulses,which could be used as a diagnostic tool for x-ray free electron laser and atom experiments.
基金the National Natural Science Foundation of China(Grant No.12035004)the Science and Technology Commission of Shanghai Municipality(Grant No.20JC1414700)。
文摘Accurate and fast prediction of thermal radiation properties of materials is crucial for their potential applications.However,some models assume that the media are made up of pure water droplets,which do not account for the increasing deviations caused by volcanic eruptions,pollution,and human activities that exacerbate dust production.The distinct radiation properties of water and dust particles make it challenging to determine the thermal radiation properties of water droplets containing dust particles.To address this issue,we investigate the influence of dust particles on light transmission and energy distribution in water droplets using the multiple sphere T-matrix method.By considering different droplet and dust diameters,volume fractions,and position distributions,we analyze how extinction regulation is achieved in dust-containing water droplets.Our results reveal the significant role of dust particles in the thermal radiation effect and provide insights into the electromagnetic properties of colloidal suspensions.Moreover,the dust-induced reestablishment of energy balance raises concerns about environmental management and climate change.This research highlights the importance of accounting for dust particles in atmospheric models and their potential impact on radiative balance.
文摘In this paper, we prove the existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations in two-dimension with finite energy.The main techniques is the Faedo-Galerkin approximation and weak compactness theory.
基金National Key Research and Development Program of China under Grant No.2019YFA0307703the Major Research Plan of the National Natural Science Foundation of China under Grant No.91850201the National Natural Science Foundation of China under Grant No.11974426.
文摘Isolated attosecond pulses with a duration of 88 as are generated in the spectral range of 29–72 eV using double optical gating technique.The gate width is set to be shorter than half the optical cycle to avoid carrier envelop phase stabilization of the 4.2 fs driving laser pulses centered at 800 nm.The attosecond pulse duration is measured with the technique of frequency resolved optical gating for complete reconstruction of attosecond bursts.
文摘In the absorption chamber of a high-energy laser energy meter, water is directly used as an absorbing medium and the interaction of the high-power laser and the water flow can produce a variety of physical phenomena such as phase transitions. The unit difference method is adopted to deduce the phase transition model for water flow irradiated by a high-energy laser. In addition, the model is simulated and verified through experiments. Among them, the experimental verification uses the photographic method, shooting the distribution and the form of the air mass of water flow in different operating conditions, which are compared with the simulation results. The research shows that it is achievable to reduce the intensity of the phase transition by increasing the water flow, reducing the power intensity of the beam, shortening the distance the beam covers, reducing the initial water temperature or adopting a shorter wavelength laser. The study's results will provide the reference for the design of a water-direct-absorption-type high-energy laser energy meter as well as an analysis of the interaction processes of other similar high-power lasers and water flow.
基金the National Natural Science Foundation of China under Grant Nos 11725417 and 11575027the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics under Grant No U1730449,and the Science Challenge Project under Grant No TZ2018005
文摘The elaborate energy and momentum spectra of ionized electrons from atoms in laser fields suggest that the ionization dynamics described by tunneling theory should be modified. Although great efforts have been carried out within semiclassical models, there are few discussions describing the multiphoton absorption process within a quantum framework. Comparing the results obtained with the time-dependent Schr?dinger equation(TDSE)and the Keldysh–Faisal–Reiss(KFR) theory, we study the nonperturbative effects of ionization dynamics beyond the KFR theory. The difference in momentum spectra between multiphoton and tunneling regimes is understood in a unified picture with virtual multiphoton absorption processes. For the multiphoton regime, the momentum spectra can be obtained by coherent interference of each periodic contribution. However, the interference of multiphoton absorption peaks will result in a complex structure of virtual multiphoton bands in the tunneling regime. It is shown that the virtual spectra will be almost continuous in the tunneling regime instead of the discrete levels found in the multiphoton regime. Finally, with a model combining the TDSE and the KFR theory,we try to understand the different effects of virtual multiphoton processes on ionization dynamics.
文摘The author considers the global existence and global nonexistence of theinitial-boundary value problem for some degenerate hyperbolic equation of the formutt ? div(|?u|p?2?u) = |u|mu, (x, t) ∈ [0, +∞) × ?with p > 2 and m > 0. He deals with the global solutions by D.H.Sattinger’s potential wellideas. At the same time, when the initial energy is positive, but appropriately bounded,the global nonexistence of solutions is verified by using the analysis method.
基金Project supported by the Major Program of the National Natural Science Foundation (Grant No 10335010)the National Natural Science Foundation-the Science Foundation of China Academy of Engineering Physics NSAF(Grant No 10576005)
文摘In this paper,we consider spatial-temporal correlation functions of the turbulent velocities.With numerical simulations on the Gledzer-Ohkitani-Yamada (GOY) shell model,we show that the correlation function decays exponentially.The advecting velocity field is regarded as a colored noise field,which is spatially and temporally correlative.For comparison,we are also given the scaling exponents of passive scalars obtained by the Gaussian random velocity field,the multi-dimensional normal velocity field and the She-Leveque velocity field,introduced by She,et al.We observe that extended self-similarity scaling exponents H(p)/H(2) of passive scalar obtained by the colored noise field are more anomalous than those obtained by the other three velocity fields.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11421063 and 11534002)the National Key Basic Research Program of China(Grant No.2014CB921403)+1 种基金the National Key Research and Development Program of China(Grant No.2016YFA0301201)and the NSAF(Grant No.U1530401)
文摘We study how the decoherence of macroscopic objects originates intrinsically from the relativistic effect. With the degree of freedom of the center of mass(CM) characterizing the collective quantum state of a macroscopic object(MO),it is found that an MO consisting of N particles can decohere with a time scale of no more than p (N^(1/2))^-1. Here, the special relativity can induce the coupling of the collective motion mode and the relative motion modes in an order of 1/c^2, which intrinsically results in the above minimum decoherence.
基金supported by National Natural Science Foundation-the Science Foundation of China Academy of Engineering Physics (NSAF) under Grant No. 10576076the Major Projects of National Natural Science Foundation of China under Grant No. 10335010
文摘在这篇论文， f <SUB>0</SUB>-δ 参数表面上的动力学行为为 Gledzer Ohkitani Yamada 模型被调查。我们显示 intermittency 混乱转变的类型是僵绳节点分叉。我们在 f <SUB>0</SUB>-δ 参数表面上阴谋阶段图，它被划分成周期、伪周期、断断续续的混乱区域。借助于改变泰勒米克罗塞尔·雷纳兹数字，我们计算速度结构功能的扩大自我类似。
基金supported by Sci. & Tech.Funds of CAEP(Nos.2010A0102004 and 2010B0102018)National Natural Science Foundation of China(Nos.11075025,10975023,10935003,10835003)
文摘A three-wave interaction (3WI) code is developed to study the stimulated Raman scattering (SRS) in both absolute and convective regimes. In the simulations, the time and spatial evolutions of a plasma wave are described by temporal growth rate and spatial factor, respectively. The spatial factors in different phases and different instability regimes are investigated. It is found that the spatial factor is caused by the finite velocity of the pump wave in the first phase and by damping in the last phase. With inclusion of the spatial factor, the temporal growth rate decreases and the threshold for SRS for a finite frequency mismatch increases. Meanwhile, the effects of wave frequency mismatch on the temporal growth rate are also discussed.
基金The project supported by National Natural Science Foundation of China, the Science Foundation of China Academy of Engineering Physics under Grant No. 10576076, the Major Projects of National Natural Science Foundation of China under Grant No. 10335010, and the Science Foundation of China Academy of Engineering Physics under Grant No. 20040430.We would like to thank Guo-Yong Yuan and Li-Bin Fu, for their useful discussions.
基金This work is supported by the National Natural Science Foundation of China(Grant Nos.11875050 and 12088101)NSAF(Grant No.U1930403).
文摘Minimizing the effect of noise is essential for quantum computers.The conventional method to protect qubits against noise is through quantum error correction.However,for current quantum hardware in the so-called noisy intermediate-scale quantum(NISQ)era,noise presents in these systems and is too high for error correction to be beneficial.Quantum error mitigation is a set of alternative methods for minimizing errors,including error extrapolation,probabilistic error cancella-tion,measurement error mitigation,subspace expansion,symmetry verification,virtual distillation,etc.The requirement for these methods is usually less demanding than error correction.Quantum error mitigation is a promising way of reduc-ing errors on NISQ quantum computers.This paper gives a comprehensive introduction to quantum error mitigation.The state-of-art error mitigation methods are covered and formulated in a general form,which provides a basis for comparing,combining and optimizing different methods in future work.
基金Project supported by the Major Program of the National Natural Science Foundation for （Grant No 10335010） and the National Natural Science Foundation-the Science Foundation of China Academy of Engineering Physics （NSAF） （Grant No 10576005）. We are grateful to Professor Li Jing-Hui and Dr Yuan Guo-Yong for valuable discussion.
