In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms： the Gaussian term and the interference term with the...In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms： the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius ｜β｜ = ｜α｜ with a separate angle of 2π/N, and the second term is composed of 1/2N（N - 1） Gaussian-cosine surfaces evenly centred in a circular region of radius ｜β｜ 〈 ｜α｜. Here, a is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.展开更多
We find a new x-parameter squeezed coherent state （p, q）κ representation, which possesses well-behaved features, i.e., its Wigner function＇s marginal distribution in the ＂q-direction＂ and in the ＂p-direction＂ ...We find a new x-parameter squeezed coherent state （p, q）κ representation, which possesses well-behaved features, i.e., its Wigner function＇s marginal distribution in the ＂q-direction＂ and in the ＂p-direction＂ is the Gauss/an form exp（-κ（q＇ - q）2}, and exp{（p＇ - p）2/κ}, respectively. Based on this, the Husimi function of（p, q）κ is also obtained, which is a Gauss/an broaden version of the Wigner function. The （P, q）κ state provides a good representative space for studying various properties ot the Husimi operator.展开更多
Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner function for the Hermite polynomial state （HP...Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner function for the Hermite polynomial state （HPS）. The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.展开更多
This paper discusses some statistical properties of the superposition of two coherent states with a vacuum state, such as sub-Poissonian photon statistics and negativity of the Wigner function. Phase probability distr...This paper discusses some statistical properties of the superposition of two coherent states with a vacuum state, such as sub-Poissonian photon statistics and negativity of the Wigner function. Phase probability distribution and phase variance are calculated. Special cases of the constructed superposition states are presented. The results show that depending on the vacuum state coefficient γ and the coherent state coefficient a, it can generate a variety of nonclassical states.展开更多
By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. Th...By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. The technique of integration within an ordered product （IWOP） of operators is employed in our discussions.展开更多
This paper proves a new theorem on the relationship between optical field Wigner function＇s two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field ψ （x＇） propagates th...This paper proves a new theorem on the relationship between optical field Wigner function＇s two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field ψ （x＇） propagates through an optical [D （-B） （-C） A] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D, B. It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A, C.展开更多
By using the technique of integration within an ordered product of operators, the normal ordered density operator of the photon-subtracted squeezed thermal state （PSSTS） is derived. Then the corresponding Wigner fun...By using the technique of integration within an ordered product of operators, the normal ordered density operator of the photon-subtracted squeezed thermal state （PSSTS） is derived. Then the corresponding Wigner function is presented by using the coherent state representation of the Wigner operator. The nonclassical properties of the PSSTS are discussed based on the negativity of the Wigner function.展开更多
Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner functions for the photon-depleted even and odd...Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner functions for the photon-depleted even and odd coherent states （PDEOCSs）. Moreover, in terms of the Wigner functions with respect to the complex parameter a the nonclassical properties of the PDEOCSs are discussed. The results show that the nonclassicality for the state ｜β, m〉o （or ｜β,m〉e） is more pronounced when m is even （or odd）. According to the marginal distributions of the Wigner functions, the physical meaning of the Wigner functions is given. Further, the tomograms of the PDEOCSs are calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.展开更多
We show that the Wigner function (an ensemble average of the density operator ρ, Δ is the Wigner operator) can be expressed as a matrix element of ρ in the entangled pure states. In doing so, converting from quant...We show that the Wigner function (an ensemble average of the density operator ρ, Δ is the Wigner operator) can be expressed as a matrix element of ρ in the entangled pure states. In doing so, converting from quantum master equations to time-evolution equation of the Wigner functions seems direct and concise. The entangled states are defined in the enlarged Fock space with a fictitious freedom.展开更多
By using the technique of integration within an ordered product （IWOP） of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we...By using the technique of integration within an ordered product （IWOP） of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we derive the Wigner function of squeezed number state, which yields negative binomial distribution by virtue of the entangled state representation and the entangled Wigner operator.展开更多
Based on the Wigner function in local equilibrium, we derive hydrodynamical quantities for a system of polarized spin-1/2 particles: the particle number current density, the energy-momentum tensor, the spin tensor, an...Based on the Wigner function in local equilibrium, we derive hydrodynamical quantities for a system of polarized spin-1/2 particles: the particle number current density, the energy-momentum tensor, the spin tensor, and the dipole moment tensor. Compared with ideal hydrodynamics without spin, additional terms at the first and second orders in the Knudsen number Κ_(n) and the average spin polarization Χ_(s) have been derived. The Wigner function can be expressed in terms of matrix-valued distributions, whose equilibrium forms are characterized by thermodynamical parameters in quantum statistics. The equations of motion for these parameters are derived by conservation laws at the leading and next-to-leading order Κ_(n) and Χ_(s).展开更多
Objective Repetitive transcranial magnetic stimulation(rTMS)has demonstrated efficacy in enhancing neurocognitive performance in Alzheimer’s disease(AD),but the neurobiological mechanisms linking synaptic pathology,n...Objective Repetitive transcranial magnetic stimulation(rTMS)has demonstrated efficacy in enhancing neurocognitive performance in Alzheimer’s disease(AD),but the neurobiological mechanisms linking synaptic pathology,neural oscillatory dynamics,and brain network reorganization remain unclear.This investigation seeks to systematically evaluate the therapeutic potential of rTMS as a non-invasive neuromodulatory intervention through a multimodal framework integrating clinical assessments,molecular profiling,and neurophysiological monitoring.Methods In this prospective double-blind trial,12 AD patients underwent a 14-day protocol of 20 Hz rTMS,with comprehensive multimodal assessments performed pre-and postintervention.Cognitive functioning was quantified using the mini-mental state examination(MMSE)and Montreal cognitive assessment(MOCA),while daily living capacities and neuropsychiatric profiles were respectively evaluated through the activities of daily living(ADL)scale and combined neuropsychiatric inventory(NPI)-Hamilton depression rating scale(HAMD).Peripheral blood biomarkers,specifically Aβ1-40 and phosphorylated tau(p-tau181),were analyzed to investigate the effects of rTMS on molecular metabolism.Spectral power analysis was employed to investigate rTMS-induced modulations of neural rhythms in AD patients,while brain network analyses incorporating topological properties were conducted to examine stimulus-driven network reorganization.Furthermore,systematic assessment of correlations between cognitive scale scores,blood biomarkers,and network characteristics was performed to elucidate cross-modal therapeutic associations.Results Clinically,MMSE and MOCA scores improved significantly(P<0.05).Biomarker showed that Aβ1-40 level increased(P<0.05),contrasting with p-tau181 reduction.Moreover,the levels of Aβ1-40 were positively correlated with MMSE and MOCA scores.Post-intervention analyses revealed significant modulations in oscillatory power,characterized by pronounced reductions in delta(P<0.05)and theta bands(P<0.05),while concurrent enhancements were observed in alpha,beta,and gamma band activities(all P<0.05).Network analysis revealed frequency-specific reorganization:clustering coefficients were significantly decreased in delta,theta,and alpha bands(P<0.05),while global efficiency improvement was exclusively detected in the delta band(P<0.05).The alpha band demonstrated concurrent increases in average nodal degree(P<0.05)and characteristic path length reduction(P<0.05).Further research findings indicate that the changes in the clinical scale HAMD scores before and after rTMS stimulation are negatively correlated with the changes in the blood biomarkers Aβ1-40 and p-tau181.Additionally,the changes in the clinical scales MMSE and MoCA scores were negatively correlated with the changes in the node degree of the alpha frequency band and negatively correlated with the clustering coefficient of the delta frequency band.However,the changes in MMSE scores are positively correlated with the changes in global efficiency of both the delta and alpha frequency bands.Conclusion 20 Hz rTMS targeting dorsolateral prefrontal cortex(DLPFC)significantly improves cognitive function and enhances the metabolic clearance ofβ-amyloid and tau proteins in AD patients.This neurotherapeutic effect is mechanistically associated with rTMS-mediated frequency-selective neuromodulation,which enhances the connectivity of oscillatory networks through improved neuronal synchronization and optimized topological organization of functional brain networks.These findings not only support the efficacy of rTMS as an adjunctive therapy for AD but also underscore the importance of employing multiple assessment methods—including clinical scales,blood biomarkers,and EEG——in understanding and monitoring the progression of AD.This research provides a significant theoretical foundation and empirical evidence for further exploration of rTMS applications in AD treatment.展开更多
Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function(WF) for the squeezed negative binomial state(SNBS) and the analytical evolution law of density operator...Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function(WF) for the squeezed negative binomial state(SNBS) and the analytical evolution law of density operator in the amplitude decay channel.The results show that the analytical WF is related to the square of the module of single-variable Hermite polynomials, which leads to a new two-variable special function and its generating function, and the parameters s and γplay opposite roles in the WF distributions.Besides, after undergoing this channel, the initial pure SNBS evolves into a new mixed state related to two operator Hermite polynomials within normal ordering, and fully loses its nonclassicality and decays to vacuum at long decay time.展开更多
Both the negativity of Wigner function and the phase sensitivity of an SU(1,1) interferometer are investigated in this paper. In the case that the even coherent state and squeezed vacuum state are input into the inter...Both the negativity of Wigner function and the phase sensitivity of an SU(1,1) interferometer are investigated in this paper. In the case that the even coherent state and squeezed vacuum state are input into the interferometer, the Heisenberg limit can be approached with parity detection. At the same time, the negativity volume of Wigner function of detection mode comes entirely from the input state and varies periodically with the encoding phase. In addition, the negativity volume of Wigner function is positively correlated with the phase sensitivity of the SU(1,1) interferometer. The positive correlation may mean that the non-classicality indicated by negative Wigner function is a kind of resource that can verify some related research results of phase estimation.展开更多
In this paper we address the possibility of using the Wigner function to capture the quantum entanglement present in a multi-qubit system. For that purpose, we calculate both the degree of entanglement and the Wigner ...In this paper we address the possibility of using the Wigner function to capture the quantum entanglement present in a multi-qubit system. For that purpose, we calculate both the degree of entanglement and the Wigner function for mixed tripartite squeezed states of Greenberger–Horne–Zeilinger(GHZ) type then we compare their behaviors. We show that the role of Wigner function in detecting and quantifying bipartite quantum correlation [Int. J. Mod. Phys. B30(2016) 1650187] may be generalized to the multipartite case.展开更多
Two physical interpretations of chirp transform related to Fresnel diffraction and Wigner distribution function are given. The chirp transform can be regarded as a Fresnel diffraction observed on a spherical tangent t...Two physical interpretations of chirp transform related to Fresnel diffraction and Wigner distribution function are given. The chirp transform can be regarded as a Fresnel diffraction observed on a spherical tangent to the diffraction plane, or a rotation and stretching transformation of the Wigner distribution function space. A general fast algorithm for the numerical calculation of chirp transform is developed by employing two fast Fourier transform algorithms. The algorithm, by which a good evaluation can be achieved, unifies the calculations of Fresnel diffraction, arbitrary fractional- order Fourier transforms and other scalar diffraction systems. The algorithm is used to calculate the Fourier transform of a Gaussian function and the Fourier transform, the Fresnel transform, the Fractional-order Fourier transforms of a rectangle function to evaluate the performance of this algorithm. The calculated results are in good agreement with the analytical results, both in the amplitude and phase.展开更多
In this paper,we obtain a normality criterion for families of meromorphic functions concerning‘wandering’shared functions,which generalizes and improves Montel’s criterion and the related results due to Schwick,Xu-...In this paper,we obtain a normality criterion for families of meromorphic functions concerning‘wandering’shared functions,which generalizes and improves Montel’s criterion and the related results due to Schwick,Xu-Fang,Xu-Qiu,and Grahl-Nevo.Also,a normality relationship between two families is given.展开更多
This paper introduces the generalized excited pair coherent state （GEPCS）. Using the entangled state 〈η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, ...This paper introduces the generalized excited pair coherent state （GEPCS）. Using the entangled state 〈η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q, α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state ｜η1, η2, τ1, τ2｜. The entangled states ｜η〉 and ｜η1, η2, τ1, τ2〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.展开更多
We propose a scheme for the direct measurement of Wigner function in two-mode cavity QED. The atoms are sent to resonantly interact with two orthogonally polarized cavity modes in the presence of strong classical fiel...We propose a scheme for the direct measurement of Wigner function in two-mode cavity QED. The atoms are sent to resonantly interact with two orthogonally polarized cavity modes in the presence of strong classical field. The probability of measuring the atom in the ground state directly gives the useful information of the cavity field. This method can be used for quantum non-demolition measurement of the photon number.展开更多
Using thermal entangled state representation,we solve the master equation of a diffusive anharmonic oscillator（AHO） to obtain the exact time evolution formula for the density operator in the infinitive operator-sum ...Using thermal entangled state representation,we solve the master equation of a diffusive anharmonic oscillator（AHO） to obtain the exact time evolution formula for the density operator in the infinitive operator-sum representation.We present a new evolution formula of the Wigner function（WF） for any initial state of the diffusive AHO by converting the WF calculation into an overlap between two pure states in an enlarged Fock space.It is found that this formula is very convenient in investigating the WF＇s evolution of any known initial state.As applications,this formula is used to obtain the evolution of the WF for a coherent state and the evolution of the photon-number distribution of diffusive AHOs.展开更多
文摘In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms： the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius ｜β｜ = ｜α｜ with a separate angle of 2π/N, and the second term is composed of 1/2N（N - 1） Gaussian-cosine surfaces evenly centred in a circular region of radius ｜β｜ 〈 ｜α｜. Here, a is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.
基金＊The project supported by the Specialized Research Fund for the Doctorial Progress of.Higher Education of China under Grant No. 20040358019
文摘We find a new x-parameter squeezed coherent state （p, q）κ representation, which possesses well-behaved features, i.e., its Wigner function＇s marginal distribution in the ＂q-direction＂ and in the ＂p-direction＂ is the Gauss/an form exp（-κ（q＇ - q）2}, and exp{（p＇ - p）2/κ}, respectively. Based on this, the Husimi function of（p, q）κ is also obtained, which is a Gauss/an broaden version of the Wigner function. The （P, q）κ state provides a good representative space for studying various properties ot the Husimi operator.
基金Project supported by the National Natural Science Foundation of China （Grant No 10574060） and the Natural Science Foundation of Shandong Province of China （Grant No Y2004A09）.
文摘Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner function for the Hermite polynomial state （HPS）. The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10674038 and 10974039)the National Basic Research Program of China (Grant No. 2006CB302901)
文摘This paper discusses some statistical properties of the superposition of two coherent states with a vacuum state, such as sub-Poissonian photon statistics and negativity of the Wigner function. Phase probability distribution and phase variance are calculated. Special cases of the constructed superposition states are presented. The results show that depending on the vacuum state coefficient γ and the coherent state coefficient a, it can generate a variety of nonclassical states.
基金The project supported by the Natural Science Foundation of Heze University of Shandong Province of China under Grant Nos.XY07WL01 and XY05WL01the University Experimental Technology Foundation of Shandong Province of China under Grant No.S04W138
文摘By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. The technique of integration within an ordered product （IWOP） of operators is employed in our discussions.
基金supported by the National Natural Science Foundation of China (Grant Nos 10775097 and 10874174)
文摘This paper proves a new theorem on the relationship between optical field Wigner function＇s two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field ψ （x＇） propagates through an optical [D （-B） （-C） A] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D, B. It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A, C.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10775097 and 10874174)the Research Foundation of the Education Department of Jiangxi Province of China
文摘By using the technique of integration within an ordered product of operators, the normal ordered density operator of the photon-subtracted squeezed thermal state （PSSTS） is derived. Then the corresponding Wigner function is presented by using the coherent state representation of the Wigner operator. The nonclassical properties of the PSSTS are discussed based on the negativity of the Wigner function.
基金Project supported by the National Natural Science Foundation of China (Grant No 10574060)the Natural Science Foundation of Shandong Province of China (Grant No Y2004A09)
文摘Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner functions for the photon-depleted even and odd coherent states （PDEOCSs）. Moreover, in terms of the Wigner functions with respect to the complex parameter a the nonclassical properties of the PDEOCSs are discussed. The results show that the nonclassicality for the state ｜β, m〉o （or ｜β,m〉e） is more pronounced when m is even （or odd）. According to the marginal distributions of the Wigner functions, the physical meaning of the Wigner functions is given. Further, the tomograms of the PDEOCSs are calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.
文摘We show that the Wigner function (an ensemble average of the density operator ρ, Δ is the Wigner operator) can be expressed as a matrix element of ρ in the entangled pure states. In doing so, converting from quantum master equations to time-evolution equation of the Wigner functions seems direct and concise. The entangled states are defined in the enlarged Fock space with a fictitious freedom.
基金the Natural Science Foundation of Heze University of Shandong Province of China under Grant Nos.XY07WL01 and XY05WL01the University Experimental Technology Foundation of Shandong Province of China under Grant No.S04W138
文摘By using the technique of integration within an ordered product （IWOP） of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we derive the Wigner function of squeezed number state, which yields negative binomial distribution by virtue of the entangled state representation and the entangled Wigner operator.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 11890713, 11890710, 11947301, 11935007, 11221504,11861131009, 11890714, 11890710, and 12047528)the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No. XDB34030102)。
文摘Based on the Wigner function in local equilibrium, we derive hydrodynamical quantities for a system of polarized spin-1/2 particles: the particle number current density, the energy-momentum tensor, the spin tensor, and the dipole moment tensor. Compared with ideal hydrodynamics without spin, additional terms at the first and second orders in the Knudsen number Κ_(n) and the average spin polarization Χ_(s) have been derived. The Wigner function can be expressed in terms of matrix-valued distributions, whose equilibrium forms are characterized by thermodynamical parameters in quantum statistics. The equations of motion for these parameters are derived by conservation laws at the leading and next-to-leading order Κ_(n) and Χ_(s).
文摘Objective Repetitive transcranial magnetic stimulation(rTMS)has demonstrated efficacy in enhancing neurocognitive performance in Alzheimer’s disease(AD),but the neurobiological mechanisms linking synaptic pathology,neural oscillatory dynamics,and brain network reorganization remain unclear.This investigation seeks to systematically evaluate the therapeutic potential of rTMS as a non-invasive neuromodulatory intervention through a multimodal framework integrating clinical assessments,molecular profiling,and neurophysiological monitoring.Methods In this prospective double-blind trial,12 AD patients underwent a 14-day protocol of 20 Hz rTMS,with comprehensive multimodal assessments performed pre-and postintervention.Cognitive functioning was quantified using the mini-mental state examination(MMSE)and Montreal cognitive assessment(MOCA),while daily living capacities and neuropsychiatric profiles were respectively evaluated through the activities of daily living(ADL)scale and combined neuropsychiatric inventory(NPI)-Hamilton depression rating scale(HAMD).Peripheral blood biomarkers,specifically Aβ1-40 and phosphorylated tau(p-tau181),were analyzed to investigate the effects of rTMS on molecular metabolism.Spectral power analysis was employed to investigate rTMS-induced modulations of neural rhythms in AD patients,while brain network analyses incorporating topological properties were conducted to examine stimulus-driven network reorganization.Furthermore,systematic assessment of correlations between cognitive scale scores,blood biomarkers,and network characteristics was performed to elucidate cross-modal therapeutic associations.Results Clinically,MMSE and MOCA scores improved significantly(P<0.05).Biomarker showed that Aβ1-40 level increased(P<0.05),contrasting with p-tau181 reduction.Moreover,the levels of Aβ1-40 were positively correlated with MMSE and MOCA scores.Post-intervention analyses revealed significant modulations in oscillatory power,characterized by pronounced reductions in delta(P<0.05)and theta bands(P<0.05),while concurrent enhancements were observed in alpha,beta,and gamma band activities(all P<0.05).Network analysis revealed frequency-specific reorganization:clustering coefficients were significantly decreased in delta,theta,and alpha bands(P<0.05),while global efficiency improvement was exclusively detected in the delta band(P<0.05).The alpha band demonstrated concurrent increases in average nodal degree(P<0.05)and characteristic path length reduction(P<0.05).Further research findings indicate that the changes in the clinical scale HAMD scores before and after rTMS stimulation are negatively correlated with the changes in the blood biomarkers Aβ1-40 and p-tau181.Additionally,the changes in the clinical scales MMSE and MoCA scores were negatively correlated with the changes in the node degree of the alpha frequency band and negatively correlated with the clustering coefficient of the delta frequency band.However,the changes in MMSE scores are positively correlated with the changes in global efficiency of both the delta and alpha frequency bands.Conclusion 20 Hz rTMS targeting dorsolateral prefrontal cortex(DLPFC)significantly improves cognitive function and enhances the metabolic clearance ofβ-amyloid and tau proteins in AD patients.This neurotherapeutic effect is mechanistically associated with rTMS-mediated frequency-selective neuromodulation,which enhances the connectivity of oscillatory networks through improved neuronal synchronization and optimized topological organization of functional brain networks.These findings not only support the efficacy of rTMS as an adjunctive therapy for AD but also underscore the importance of employing multiple assessment methods—including clinical scales,blood biomarkers,and EEG——in understanding and monitoring the progression of AD.This research provides a significant theoretical foundation and empirical evidence for further exploration of rTMS applications in AD treatment.
基金Project supported by the National Natural Science Foundation of China(Grant No.11347026)the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2016AM03 and ZR2017MA011)
文摘Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function(WF) for the squeezed negative binomial state(SNBS) and the analytical evolution law of density operator in the amplitude decay channel.The results show that the analytical WF is related to the square of the module of single-variable Hermite polynomials, which leads to a new two-variable special function and its generating function, and the parameters s and γplay opposite roles in the WF distributions.Besides, after undergoing this channel, the initial pure SNBS evolves into a new mixed state related to two operator Hermite polynomials within normal ordering, and fully loses its nonclassicality and decays to vacuum at long decay time.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11574092,61775062,61378012,91121023,and 60978009)the National Basic Research Program of China(Grant No.2013CB921804)the Innovation Project of Graduate School of South China Normal University(Grant No.2017LKXM088)
文摘Both the negativity of Wigner function and the phase sensitivity of an SU(1,1) interferometer are investigated in this paper. In the case that the even coherent state and squeezed vacuum state are input into the interferometer, the Heisenberg limit can be approached with parity detection. At the same time, the negativity volume of Wigner function of detection mode comes entirely from the input state and varies periodically with the encoding phase. In addition, the negativity volume of Wigner function is positively correlated with the phase sensitivity of the SU(1,1) interferometer. The positive correlation may mean that the non-classicality indicated by negative Wigner function is a kind of resource that can verify some related research results of phase estimation.
文摘In this paper we address the possibility of using the Wigner function to capture the quantum entanglement present in a multi-qubit system. For that purpose, we calculate both the degree of entanglement and the Wigner function for mixed tripartite squeezed states of Greenberger–Horne–Zeilinger(GHZ) type then we compare their behaviors. We show that the role of Wigner function in detecting and quantifying bipartite quantum correlation [Int. J. Mod. Phys. B30(2016) 1650187] may be generalized to the multipartite case.
文摘Two physical interpretations of chirp transform related to Fresnel diffraction and Wigner distribution function are given. The chirp transform can be regarded as a Fresnel diffraction observed on a spherical tangent to the diffraction plane, or a rotation and stretching transformation of the Wigner distribution function space. A general fast algorithm for the numerical calculation of chirp transform is developed by employing two fast Fourier transform algorithms. The algorithm, by which a good evaluation can be achieved, unifies the calculations of Fresnel diffraction, arbitrary fractional- order Fourier transforms and other scalar diffraction systems. The algorithm is used to calculate the Fourier transform of a Gaussian function and the Fourier transform, the Fresnel transform, the Fractional-order Fourier transforms of a rectangle function to evaluate the performance of this algorithm. The calculated results are in good agreement with the analytical results, both in the amplitude and phase.
基金Supported by the National Natural Science Foundation of China(Grant No.11471163)。
文摘In this paper,we obtain a normality criterion for families of meromorphic functions concerning‘wandering’shared functions,which generalizes and improves Montel’s criterion and the related results due to Schwick,Xu-Fang,Xu-Qiu,and Grahl-Nevo.Also,a normality relationship between two families is given.
基金supported by the National Natural Science Foundation of China (Grant No 10574060)the Natural Science Foundation of Shandong Province of China (Grant No Y2004A09)
文摘This paper introduces the generalized excited pair coherent state （GEPCS）. Using the entangled state 〈η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q, α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state ｜η1, η2, τ1, τ2｜. The entangled states ｜η〉 and ｜η1, η2, τ1, τ2〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.
基金Project supported by the National Natural Science Foundation of China(Grant No.10974028)the Doctoral Foundation of the Ministry of Education of China(Grant No.20093514110009)+1 种基金the Natural Science Foundation of Fujian Province of China(Grant No.2009J06002)the Funds from the State Key Laboratory Breeding Base of Photocatalysis,Fuzhou University
文摘We propose a scheme for the direct measurement of Wigner function in two-mode cavity QED. The atoms are sent to resonantly interact with two orthogonally polarized cavity modes in the presence of strong classical field. The probability of measuring the atom in the ground state directly gives the useful information of the cavity field. This method can be used for quantum non-demolition measurement of the photon number.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11147009 and 11244005)the Natural Science Foundation of Shandong Province,China (Grant No. ZR2012AM004)
文摘Using thermal entangled state representation,we solve the master equation of a diffusive anharmonic oscillator（AHO） to obtain the exact time evolution formula for the density operator in the infinitive operator-sum representation.We present a new evolution formula of the Wigner function（WF） for any initial state of the diffusive AHO by converting the WF calculation into an overlap between two pure states in an enlarged Fock space.It is found that this formula is very convenient in investigating the WF＇s evolution of any known initial state.As applications,this formula is used to obtain the evolution of the WF for a coherent state and the evolution of the photon-number distribution of diffusive AHOs.
