In this letter, a distributed protocol for sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays is proposed, and a brief procedure of convergence analysis for s...In this letter, a distributed protocol for sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays is proposed, and a brief procedure of convergence analysis for such algorithm over undirected connected graphs is provided. Furthermore, a simple yet generic criterion is also presented to guarantee synchronized oscillatory motions in coupled harmonic oscillators. Subsequently, the simulation results are worked out to demonstrate the efficiency and feasibility of the theoretical results.展开更多
The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated,including stability, synchronization, and stochastic resonance(SR). First, the synchronization condi...The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated,including stability, synchronization, and stochastic resonance(SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain(OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.展开更多
We study the Connes distance of quantum states of two-dimensional(2D)harmonic oscillators in phase space.Using the Hilbert–Schmidt operatorial formulation,we construct a boson Fock space and a quantum Hilbert space,a...We study the Connes distance of quantum states of two-dimensional(2D)harmonic oscillators in phase space.Using the Hilbert–Schmidt operatorial formulation,we construct a boson Fock space and a quantum Hilbert space,and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional(4D)quantum phase space.Based on the ball condition,we obtain some constraint relations about the optimal elements.We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators.We prove that these two-dimensional distances satisfy the Pythagoras theorem.These results are significant for the study of geometric structures of noncommutative spaces,and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.展开更多
We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propaga...We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propagator related to the system. New examples of time-dependent frequencies are presented.展开更多
In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0...In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0≠λ∈R,l>1 is an integer and the corresponding involution G is(−θ,x,−y)→(θ,x,y).The existence of response solutions of the above reversible systems has already been proved in[22]if[f2(ωt,0,0)]satisfies some non-zero average conditions(See the condition(H)in[22]),here[·]denotes the average of a continuous function on T^(d).However,discussing the existence of response solutions for the above systems encounters difficulties when[f_(2)(ωt,0,0)]=0,due to a degenerate implicit function must be solved.This article will be doing work in this direction.The purpose of this paper is to consider the case where[f2(ωt,0,0)]=0.More precisely,with 2p<l,if f_(2)satisfies[f_(2)(ωt,0,0)]=[∂f_(2)(ωt,0,0)/∂x]=[∂^(2)f_(2)(ωt,0,0)/∂x2]=···=[∂p−1f2(ωt,0,0)∂xp−1]=0,eitherλ−1[∂pf2(ωt,0,0)∂xp]<0 as l−p is even orλ−1[∂pf2(ωt,0,0)∂xp]=0 as l−p is odd,we obtain the following results:(1)Forλ>˜0(seeλ˜in(2.2))and sufficiently small,response solutions exist for eachωsatisfying a weak non-resonant condition;(2)Forλ<˜0 and∗sufficiently small,there exists a Cantor set E∈(0,∗)with almost full Lebesgue measure such that response solutions exist for each∈E ifωsatisfies a Diophantine condition.In the remaining case whereλ−1[∂pf2(ωt,0,0)∂xp]>0 and l−p is even,we prove the system admits no response solutions in most regions.展开更多
The quantum harmonic oscillator(QHO),one of the most important and ubiquitous model systems in quantum mechanics,features equally spaced energy levels or eigenstates.Here we present a new class of nearly ideal QHOs fo...The quantum harmonic oscillator(QHO),one of the most important and ubiquitous model systems in quantum mechanics,features equally spaced energy levels or eigenstates.Here we present a new class of nearly ideal QHOs formed by hydrogenic substitutional dopants in an AlGaAs/GaAs heterostructure.On the basis of model calculations,we demonstrate that,when aδ-doping Si donor substitutes the Ga/Al lattice site close to AlGaAs/GaAs heterointerface,a hydrogenic Si QHO,characterized by a restoring Coulomb force producing square law harmonic potential,is formed.This gives rise to QHO states with energy spacing of~8–9 meV.We experimentally confirm this proposal by utilizing gate tuning and measuring QHO states using an aluminum single-electron transistor(SET).A sharp and fast oscillation with period of~7–8 mV appears in addition to the regular Coulomb blockade(CB)oscillation with much larger period,for positive gate biases above 0.5 V.The observation of fast oscillation and its behavior is quantitatively consistent with our theoretical result,manifesting the harmonic motion of electrons from the QHO.Our results might establish a general principle to design,construct and manipulate QHOs in semiconductor heterostructures,opening future possibilities for their quantum applications.展开更多
The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limit...The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limits using different expansion procedures, for each limit. In the weak-relativistic limit, a Normal Form expansion is developed, which yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures. In the extreme-relativistic limit, an expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) →1 yields an excellent approximation for the solution throughout the whole period of oscillations. The variation of the coefficients of the Fourier expansion of the solution from the weak- to the extreme-relativistic limits is displayed.展开更多
Attempts to unify Gravity Theory and Quantum Field Theory (QFT) under Loop Quantum Gravity Theory (LQG), are diverse;a dividing line between classical and quantum is sought with Schrödinger cat-state experiments....Attempts to unify Gravity Theory and Quantum Field Theory (QFT) under Loop Quantum Gravity Theory (LQG), are diverse;a dividing line between classical and quantum is sought with Schrödinger cat-state experiments. A Primordial Field Theory-based alternative is presented, and a gravity-based harmonic oscillator developed. With quantum theory applied at micro-scales and gravity theory at meso- and macro-scales, this scale-gap contributes to the conceptual problems associated with Loop Quantum Gravity. Primordial field theory, spanning all scales, is used to conceptually stretch key ideas across this gap. An LQG interpretation of the wave function associated with the oscillator is explained.展开更多
Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transform...Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.展开更多
The quantization of circuits has received to be rather attractive in domains of solid state—molecular—and biophysics, since the quanta referred to as Q-bits play a significant role in the design of the quantum compu...The quantization of circuits has received to be rather attractive in domains of solid state—molecular—and biophysics, since the quanta referred to as Q-bits play a significant role in the design of the quantum computer and entangled structures. Quantized circuits cannot be applied without modifications, since the energy differences are not equidistant and the polarization of the excited states has to be accounted for having particular importance for the creation of virtual states. Applications of the presented theory are scanning methods in radiotherapy without multi-leaf collimators, which may be realized in tomo-scanning radiotherapy and in the keV domain, which provides a new design of CT. The problem of lateral scatter in the target and energy storage by heat production is significantly reduced by a multilayer system with focusing the impinging electrons at the walls and by a magnetic field. The verification of the Heisenberg-Euler scatter of crossing beams of 9 MV is a central problem of photon physics and can be solved by the new bremsstrahlung technique. A comparison with GEANT 4 Monte-Carlo data indicates that the presented method also works in the GeV domain, and a multi-target can improve the bremsstrahlung yield. GEANT 4 provides the spatial distribution, whereas the virtual oscillator states only show the created energy spectrum. In every case, the exploitation yield can be drastically improved by the superiority of the focused multitarget system compared to a single standard target, and the door to new technologies is opened.展开更多
Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The non...Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.展开更多
First, a Lagrangian is presented and authenticated for a Relativistic Harmonic Oscillator in 1 + 1 dimensions. It yields a two-component set of equations of motion. The time-component is the missing piece in all previ...First, a Lagrangian is presented and authenticated for a Relativistic Harmonic Oscillator in 1 + 1 dimensions. It yields a two-component set of equations of motion. The time-component is the missing piece in all previous discussions of this system! The second result is that this Oscillator Langrangian generalizes to Langrangians for a class of particles in 1 + 1 dimensions subject to an arbitrary potential <em>V</em> which is space dependent only.展开更多
The stochastic resonance phenomenon in a harmonic oscillator with fluctuating intrinsic frequency by asymmetric dichotomous noise is investigated in this paper. By using the random average method and Shapiro- Loginov ...The stochastic resonance phenomenon in a harmonic oscillator with fluctuating intrinsic frequency by asymmetric dichotomous noise is investigated in this paper. By using the random average method and Shapiro- Loginov formula, the exact solution of the average output amplitude gain (OAG) is obtained. Numerical results show that OAG depends non-monotonically on the noise characteristics: intensity, correlation time and asymmetry. The maximum OAG can be achieved by tuning the noise asymmetry and or the noise correlation time.展开更多
We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D(D ≥ 1) dimensions. The coherent states, defined as the eigenstat...We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D(D ≥ 1) dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D-dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.展开更多
Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachme...Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachment cross section of negative ion of hydrogen H-in the perturbed one dimensional linear harmonic potential via static electric field. To this end,the quantum formula for total photodetachment cross section of the H-ion is derived by calculating the dipole matrix element in spherical coordinates. In order to obtain the detached electron wave function, we have solved the time-independent Schr¨odinger wave equation for the perturbed Hamiltonian of the harmonic oscillator in momentum representation. To acquire the corresponding normalized final state detached electron wave function in momentum space, we have employed an approach analogous to the WKB(Wenzel–Kramers–Brillouin) approximation. The resulting analytical formula of total photodetachment cross section depicts interesting oscillator structure that varies considerably with incident-photon energy,oscillator potential frequency, and electric field strength as elucidated by the numerical results. The current problem having close analogy with the Stark effect in charged harmonic oscillator may have potential implications in atomic and molecular physics and quantum optics.展开更多
In this paper we find that a set of energy eigenstates of a two-dimensional anisotropic harmonic potential in a uniform magnetic field is classified as the atomic coherent states |τ) in terms of the spin values of ...In this paper we find that a set of energy eigenstates of a two-dimensional anisotropic harmonic potential in a uniform magnetic field is classified as the atomic coherent states |τ) in terms of the spin values of j in the Schwinger bosonic realization. The correctness of the above conclusions can be verified by virtue of the entangled state 〈η| representation of the state |τ).展开更多
The photodetachment cross section of H- in a linear harmonic oscillator potential is investigated. This system pro- vides a rare example that can be studied analytically by both quantum and semiclassical methods with ...The photodetachment cross section of H- in a linear harmonic oscillator potential is investigated. This system pro- vides a rare example that can be studied analytically by both quantum and semiclassical methods with some approxi- mations. The formulas of the cross section for different laser polarization directions are explicitly derived by both the traditional quantum approach and closed-orbit theory. In the traditional quantum approach, we calculate the cross sections in coordinate representation and momentum representation, and get the same formulas. We compare the quantum formulas with closed-orbit theory formulas, and find that when the detachment electron energy is larger than hco, where co is the frequency of the oscillator potential, the quantum results are shown to be in good agreement with the semiclassical results.展开更多
In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and...In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.展开更多
Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the ...Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.展开更多
The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="...The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em></span>), with the commutation relation <img src="Edit_28f5b839-7de4-41e5-9ed8-69dc1bf72c2c.bmp" alt="" />, and using a Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s like equation on these variable, and associating a linear operator to a constant of motion <em>K</em> (<em>x, v, t</em>) of the classical system, The comparison with the quantization in the space (<em>x, p</em>) is done with the usual Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s equation for the Hamiltonian <em>H</em><span style="white-space:normal;">(</span><em style="white-space:normal;">x, p, t</em><span style="white-space:normal;">)</span>, and with the commutation relation <img src="Edit_cca7e318-5b35-4c55-8f09-6089970ce9a2.bmp" alt="" />. It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization has fewer oscillations than the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, the average energy of the system is higher in (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, and the Boltzmann-Shannon entropy on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization is higher than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization.展开更多
基金partially supported by the National Science Foundation of China(11272791,61364003,and 61203006)the Innovation Program of Shanghai Municipal Education Commission(10ZZ61 and 14ZZ151)the Science and Technology Foundation of Guizhou Province(20122316)
文摘In this letter, a distributed protocol for sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays is proposed, and a brief procedure of convergence analysis for such algorithm over undirected connected graphs is provided. Furthermore, a simple yet generic criterion is also presented to guarantee synchronized oscillatory motions in coupled harmonic oscillators. Subsequently, the simulation results are worked out to demonstrate the efficiency and feasibility of the theoretical results.
基金supported by the National Natural Science Foundation of China for the Youth (Grant Nos. 11501385 and 11801385)。
文摘The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated,including stability, synchronization, and stochastic resonance(SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain(OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.
基金Project supported by the Key Research and Development Project of Guangdong Province,China(Grant No.2020B0303300001)the National Natural Science Foundation of China(Grant No.11911530750)+2 种基金the Guangdong Basic and Applied Basic Research Foundation,China(Grant No.2019A1515011703)the Fundamental Research Funds for the Central Universities,China(Grant No.2019MS109)the Natural Science Foundation of Anhui Province,China(Grant No.1908085MA16).
文摘We study the Connes distance of quantum states of two-dimensional(2D)harmonic oscillators in phase space.Using the Hilbert–Schmidt operatorial formulation,we construct a boson Fock space and a quantum Hilbert space,and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional(4D)quantum phase space.Based on the ball condition,we obtain some constraint relations about the optimal elements.We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators.We prove that these two-dimensional distances satisfy the Pythagoras theorem.These results are significant for the study of geometric structures of noncommutative spaces,and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.
文摘We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propagator related to the system. New examples of time-dependent frequencies are presented.
基金partially supported by the National Natural Science Foundation of China(Grant Nos.11971261,11571201)partially supported by the National Natural Science Foundation of China(Grant Nos.12001315,12071255)Shandong Provincial Natural Science Foundation,China(Grant No.ZR2020MA015)。
文摘In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0≠λ∈R,l>1 is an integer and the corresponding involution G is(−θ,x,−y)→(θ,x,y).The existence of response solutions of the above reversible systems has already been proved in[22]if[f2(ωt,0,0)]satisfies some non-zero average conditions(See the condition(H)in[22]),here[·]denotes the average of a continuous function on T^(d).However,discussing the existence of response solutions for the above systems encounters difficulties when[f_(2)(ωt,0,0)]=0,due to a degenerate implicit function must be solved.This article will be doing work in this direction.The purpose of this paper is to consider the case where[f2(ωt,0,0)]=0.More precisely,with 2p<l,if f_(2)satisfies[f_(2)(ωt,0,0)]=[∂f_(2)(ωt,0,0)/∂x]=[∂^(2)f_(2)(ωt,0,0)/∂x2]=···=[∂p−1f2(ωt,0,0)∂xp−1]=0,eitherλ−1[∂pf2(ωt,0,0)∂xp]<0 as l−p is even orλ−1[∂pf2(ωt,0,0)∂xp]=0 as l−p is odd,we obtain the following results:(1)Forλ>˜0(seeλ˜in(2.2))and sufficiently small,response solutions exist for eachωsatisfying a weak non-resonant condition;(2)Forλ<˜0 and∗sufficiently small,there exists a Cantor set E∈(0,∗)with almost full Lebesgue measure such that response solutions exist for each∈E ifωsatisfies a Diophantine condition.In the remaining case whereλ−1[∂pf2(ωt,0,0)∂xp]>0 and l−p is even,we prove the system admits no response solutions in most regions.
基金Profs.Y.Zhang,J.Chen,J.Zhao,and C.Lin are greatly appreciated.M.Feng thanks financial support from the National Key R&D Program of China(Grant No.2018YFA0305802)the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XD30000000)+1 种基金the National Natural Science Foundation of China(Grant Nos.11574364 and 11774267)L.Mao thanks financial support from the National Key R&D Program of China by the Ministry of Science and Technology of China(Grant No.2015C8932400).
文摘The quantum harmonic oscillator(QHO),one of the most important and ubiquitous model systems in quantum mechanics,features equally spaced energy levels or eigenstates.Here we present a new class of nearly ideal QHOs formed by hydrogenic substitutional dopants in an AlGaAs/GaAs heterostructure.On the basis of model calculations,we demonstrate that,when aδ-doping Si donor substitutes the Ga/Al lattice site close to AlGaAs/GaAs heterointerface,a hydrogenic Si QHO,characterized by a restoring Coulomb force producing square law harmonic potential,is formed.This gives rise to QHO states with energy spacing of~8–9 meV.We experimentally confirm this proposal by utilizing gate tuning and measuring QHO states using an aluminum single-electron transistor(SET).A sharp and fast oscillation with period of~7–8 mV appears in addition to the regular Coulomb blockade(CB)oscillation with much larger period,for positive gate biases above 0.5 V.The observation of fast oscillation and its behavior is quantitatively consistent with our theoretical result,manifesting the harmonic motion of electrons from the QHO.Our results might establish a general principle to design,construct and manipulate QHOs in semiconductor heterostructures,opening future possibilities for their quantum applications.
文摘The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limits using different expansion procedures, for each limit. In the weak-relativistic limit, a Normal Form expansion is developed, which yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures. In the extreme-relativistic limit, an expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) →1 yields an excellent approximation for the solution throughout the whole period of oscillations. The variation of the coefficients of the Fourier expansion of the solution from the weak- to the extreme-relativistic limits is displayed.
文摘Attempts to unify Gravity Theory and Quantum Field Theory (QFT) under Loop Quantum Gravity Theory (LQG), are diverse;a dividing line between classical and quantum is sought with Schrödinger cat-state experiments. A Primordial Field Theory-based alternative is presented, and a gravity-based harmonic oscillator developed. With quantum theory applied at micro-scales and gravity theory at meso- and macro-scales, this scale-gap contributes to the conceptual problems associated with Loop Quantum Gravity. Primordial field theory, spanning all scales, is used to conceptually stretch key ideas across this gap. An LQG interpretation of the wave function associated with the oscillator is explained.
文摘Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.
文摘The quantization of circuits has received to be rather attractive in domains of solid state—molecular—and biophysics, since the quanta referred to as Q-bits play a significant role in the design of the quantum computer and entangled structures. Quantized circuits cannot be applied without modifications, since the energy differences are not equidistant and the polarization of the excited states has to be accounted for having particular importance for the creation of virtual states. Applications of the presented theory are scanning methods in radiotherapy without multi-leaf collimators, which may be realized in tomo-scanning radiotherapy and in the keV domain, which provides a new design of CT. The problem of lateral scatter in the target and energy storage by heat production is significantly reduced by a multilayer system with focusing the impinging electrons at the walls and by a magnetic field. The verification of the Heisenberg-Euler scatter of crossing beams of 9 MV is a central problem of photon physics and can be solved by the new bremsstrahlung technique. A comparison with GEANT 4 Monte-Carlo data indicates that the presented method also works in the GeV domain, and a multi-target can improve the bremsstrahlung yield. GEANT 4 provides the spatial distribution, whereas the virtual oscillator states only show the created energy spectrum. In every case, the exploitation yield can be drastically improved by the superiority of the focused multitarget system compared to a single standard target, and the door to new technologies is opened.
基金supported by the National Natural Science Foundation of China (Grants 11402151 and 11572182)
文摘Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.
文摘First, a Lagrangian is presented and authenticated for a Relativistic Harmonic Oscillator in 1 + 1 dimensions. It yields a two-component set of equations of motion. The time-component is the missing piece in all previous discussions of this system! The second result is that this Oscillator Langrangian generalizes to Langrangians for a class of particles in 1 + 1 dimensions subject to an arbitrary potential <em>V</em> which is space dependent only.
文摘The stochastic resonance phenomenon in a harmonic oscillator with fluctuating intrinsic frequency by asymmetric dichotomous noise is investigated in this paper. By using the random average method and Shapiro- Loginov formula, the exact solution of the average output amplitude gain (OAG) is obtained. Numerical results show that OAG depends non-monotonically on the noise characteristics: intensity, correlation time and asymmetry. The maximum OAG can be achieved by tuning the noise asymmetry and or the noise correlation time.
基金Project supported by the National Natural Science Foundation of China (Grant No 60261004) and Yunnan Province Science Foundation (Grant No 2002E0008M).
文摘We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D(D ≥ 1) dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D-dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.
文摘Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachment cross section of negative ion of hydrogen H-in the perturbed one dimensional linear harmonic potential via static electric field. To this end,the quantum formula for total photodetachment cross section of the H-ion is derived by calculating the dipole matrix element in spherical coordinates. In order to obtain the detached electron wave function, we have solved the time-independent Schr¨odinger wave equation for the perturbed Hamiltonian of the harmonic oscillator in momentum representation. To acquire the corresponding normalized final state detached electron wave function in momentum space, we have employed an approach analogous to the WKB(Wenzel–Kramers–Brillouin) approximation. The resulting analytical formula of total photodetachment cross section depicts interesting oscillator structure that varies considerably with incident-photon energy,oscillator potential frequency, and electric field strength as elucidated by the numerical results. The current problem having close analogy with the Stark effect in charged harmonic oscillator may have potential implications in atomic and molecular physics and quantum optics.
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. Y2008A23)the Shandong Provincal Higher Educational Science and Technology Program of China (Grant Nos. J09LA07 and J10LA15)
文摘In this paper we find that a set of energy eigenstates of a two-dimensional anisotropic harmonic potential in a uniform magnetic field is classified as the atomic coherent states |τ) in terms of the spin values of j in the Schwinger bosonic realization. The correctness of the above conclusions can be verified by virtue of the entangled state 〈η| representation of the state |τ).
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11421063 and 11474079)the Natural Science Foundation of Shanxi Province,China(Grant No.2009011004)the Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi Province,China
文摘The photodetachment cross section of H- in a linear harmonic oscillator potential is investigated. This system pro- vides a rare example that can be studied analytically by both quantum and semiclassical methods with some approxi- mations. The formulas of the cross section for different laser polarization directions are explicitly derived by both the traditional quantum approach and closed-orbit theory. In the traditional quantum approach, we calculate the cross sections in coordinate representation and momentum representation, and get the same formulas. We compare the quantum formulas with closed-orbit theory formulas, and find that when the detachment electron energy is larger than hco, where co is the frequency of the oscillator potential, the quantum results are shown to be in good agreement with the semiclassical results.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10475001 and 10675001)the Program for New Century Excellent Talents in University of China (Grant No NCET-05-0558)+1 种基金the Program for Excellent Talents in Anhui Province Universitythe Education Committee Foundation of Anhui Province (Grant No 2006KJ259B)
文摘In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.
文摘Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.
文摘The quantization of the forced harmonic oscillator is studied with the quantum variable (<em>x</em>, <span style="white-space:nowrap;"><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em></span>), with the commutation relation <img src="Edit_28f5b839-7de4-41e5-9ed8-69dc1bf72c2c.bmp" alt="" />, and using a Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s like equation on these variable, and associating a linear operator to a constant of motion <em>K</em> (<em>x, v, t</em>) of the classical system, The comparison with the quantization in the space (<em>x, p</em>) is done with the usual Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span></span>dinger’s equation for the Hamiltonian <em>H</em><span style="white-space:normal;">(</span><em style="white-space:normal;">x, p, t</em><span style="white-space:normal;">)</span>, and with the commutation relation <img src="Edit_cca7e318-5b35-4c55-8f09-6089970ce9a2.bmp" alt="" />. It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization has fewer oscillations than the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, the average energy of the system is higher in (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization, and the Boltzmann-Shannon entropy on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>p</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization is higher than on the (<em style="white-space:normal;">x</em><span style="white-space:normal;">, </span><em style="white-space:normal;"><sub>v</sub><sup style="margin-left:-8px;">∧</sup></em>) quantization.