Einstein-Podolsky-Rosen entanglement in time-dependent periodic pumping non-degenerate optical parametric amplifier

The solution of the time-dependent periodic pumping non-degenerate optical parametric amplifier （NOPA） is derived when the pump depletion is considered both above and below thresholds. Based on this solution, the quantum fluctuation calculated shows that a high entanglement and a good squeezing degree of the parametric light beams are achieved near and above thresholds. We adopt two kinds of pump fields： （i） a continuously modulated pump with a sinusoidal envelope; （ii） a sequence of laser pulses with Gaussian envelopes. We analyse the time evolution of continuous variable entanglement by analytical and numerical calculations, and show that the periodic driven pumping also improves the degree of entanglement. The squeezing and Einstein-Podolsky-Rosen （EPR） entanglement by using the two pumping driven functions are investigated from below to above the threshold regions, the tendencies are nearly the same, and the Caussian driven function is superior to that of the sine driven function, when the maximum squeezing and the minimum variance of quantum fluctuation are considered. In the meantime, the generalization of frequency degenerate OPA to frequency non-degenerated OPA problem is investigated.

Einstein-Podolsky-Rosen entanglement in time-dependent broadband pumping frequency non-degenerate optical parametric amplifier

This paper investigates quantum fluctuations characteristic of time-dependent broadband pumping frequency non-degenerate optical parametric amplifier for below and above threshold regions. It finds that a high squeezing and entanglement can be achieved.

The propagation characteristic of the EPR entanglement for a composite non-degenerate parametric optical amplification system

This paper applies the minimum variance V1 criterion to monitor the evolution of signal and idler modes of a composite non-degenerate optical parametric amplification （NOPA） system. The analytics and numerical calculation show the influence of the transition time, the vacuum fluctuations, and the thermal noise level on the EPR entanglement of the composite NOPA system. It finds that the entanglement and the squeezing degrade as the minimum variance V1 increases.

Higher-order squeezing of quantum field and higher-order uncertainty relations in non-degenerate four-wave mixing

It is found that the field of the combined mode of the probe wave and the phase conjugate wave in the process of non-degenerate four-wave mixing exhibits higher-order squeezing to all even orders. The higher-order squeezed parameter and squeezed limit due to the modulation frequency are investigated. The smaller the modulation frequency is, the stronger the degree of higher-order squeezing becomes. Furthermore, the hlgher-order uncertainty relations in the process of non-degenerate four-wave mixing are presented for the first time. The product of higher-order noise moments is related to even order number N and the length L of the medium.

The solution of the time-dependent Fokker-Planck equation of non-degenerate optical parametric amplification and its application to the optimum realization of EPR paradox

In this paper, the solution of the time-dependent Fokker-Planck equation of non-degenerate optical parametric amplification is used to deduce the condition demonstrating the Einstein-Podolsky-Rosen （EPR） paradox. The analytics and numerical calculation show the influence of pump depletion on the error in the measurement of continuous variables. The optimum realization of EPR paradox can be achieved by adjusting the parameter of squeezing. This result is of practical importance when the realistic experimental conditions are taken into consideration .

Circular Scale of Time and Its Use in Calculating the Schrödinger Perturbation Energy of a Non-Degenerate Quantum State

The paper presents a circular scale of time—and its diagrams—which can be successfully applied in calculating the Schrödinger perturbation energy of a non-degenerate quantum state. This seems to be done in a more simple way than with the aid of any other of the perturbation approaches of a similar kind. As an example of the theory suitable to comparison is considered the Feynman diagrammatic method based on a straight-linear scale of time which represents a much more complicated formalism than the present one. All diagrams of the approach outlined in the paper can obtain as their counterparts the algebraic formulae which can be easily extended to an arbitrary Schrödinger perturbation order. The calculations and results descending from the perturbation orders N between N = 1 and N = 7 are reported in detail.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

Solitons in nonlinear systems and eigen-states in quantum wells

We study the relations between solitons of nonlinear Schro¨dinger equation and eigen-states of linear Schro¨dinger equation with some quantum wells. Many different non-degenerated solitons are re-derived from the eigen-states in the quantum wells. We show that the vector solitons for the coupled system with attractive interactions correspond to the identical eigen-states with the ones of the coupled systems with repulsive interactions. Although their energy eigenvalues seem to be different, they can be reduced to identical ones in the same quantum wells. The non-degenerated solitons for multi-component systems can be used to construct much abundant degenerated solitons in more components coupled cases.Meanwhile, we demonstrate that soliton solutions in nonlinear systems can also be used to solve the eigen-problems of quantum wells. As an example, we present the eigenvalue and eigen-state in a complicated quantum well for which the Hamiltonian belongs to the non-Hermitian Hamiltonian having parity–time symmetry. We further present the ground state and the first exited state in an asymmetric quantum double-well from asymmetric solitons. Based on these results, we expect that many nonlinear physical systems can be used to observe the quantum states evolution of quantum wells, such as a water wave tank, nonlinear fiber, Bose–Einstein condensate, and even plasma, although some of them are classical physical systems. These relations provide another way to understand the stability of solitons in nonlinear Schro¨dinger equation described systems, in contrast to the balance between dispersion and nonlinearity.

Gelfand-Naimark Theorem of Commutative Hopf C^＊ -Algebras

Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the ease of Hopf C^＊-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C^＊-algebra are proposed.

Theoretical investigation of light generated by optical parametric down-conversion and its application in the demonstration of the EPR paradox

Parametric down-conversion is a useful method to obtain non-classical state in quantum optics. （1） An analytical solution to the Fokker-Planck equation of non-degenerate optical parametric amplification （NOPA） for generation of squeezed light is presented. The maximum intra-cavity compression of squeezed light derived from the analytical solution is 1/16 （vacuum fluctuations 1/4）. To compare it with the previous 1/8 of degenerate optical parametric amplification （DOPA）, it seems that squeezing for NOPA is superior to DOPA.

Tripartite continuous-variable entanglement of NOPA system

We investigate the fundamental limits to the achievable tripartite continuous-variable （CV） entanglement criterion of a generalized Vl criterion. Our numerical simulation results show that the non-degenerate eigenvalues do effect the performances of the estimated minimum variances. From below the threshold to above the threshold, with the increase of the pump parameter, the tripartite CV entanglement gradually disappears. The different off-diagonal elements seriously distort the weights for entanglement. We can obtain a good tripartite CV entanglement by appropriately controlling the values of off-diagonal elements eij.

Circular Scale of Time as a Guide of the Schrödinger’s Perturbation Theory

The paper is a kind of a review which considers an investigation of the scale of time suggested by an application of the Schr&ouml;dinger perturbation method, especially when the perturbation of a non-degenerate quantum state is examined. In fact the method was applied in numerous cases—also by Schr&ouml;dinger himself—without any use of the notion of time. Simultaneously, because of the development of computers, their use in solving the perturbation problems gradually decreased. However, the point of importance in the paper became the time. We demonstrate that collisions of a quantum system with the perturbation potential can be arranged along a circular scale of time whose properties provide us precisely with the energy terms obtained by the Schr&ouml;dinger perturbation theory. This validity of results is checked till the perturbation order N = 7.

Circular Scale of Time and Energy of a Quantum State Calculated from the Schrodinger Perturbation Theory

The main facts about the scale of time considered as a plot of a sequence of events are submitted both to a review and a more detailed calculation. Classical progressive character of the time variable, present in the everyday life and in the modern science, too, is compared with a circular-like kind of advancement of time. This second kind of the time behaviour can be found suitable when a perturbation process of a quantum-mechanical system is examined. In fact the paper demonstrates that the complicated high-order Schrodinger perturbation energy of a non-degenerate quantum state becomes easy to approach of the basis of a circular scale. For example for the perturbation order N = 20 instead of 19! ≈ 1.216 × 1017 Feynman diagrams, the contribution of which should be derived and calculated, only less than 218 ≈ 2.621 × 105 terms belonging to N = 20 should be taken into account to the same purpose.

Circular Scale of Time and Construction of the Schrodinger Perturbation Series for Energy Made Simple

Physically the examined perturbation problem can be regarded as a set of collision events of a time-independent perturbation potential with a quantum system. As an effect of collisions there is an expected definite change of energy of an initially unperturbed state of the system to some stationary perturbed state. The collision process certainly occupies some intervals of time which, however, do not enter the formalism. A striking property is the result of a choice of the sequence of collisions according to the applied circular scale of time: the scale produces almost automatically the energy terms predicted by the Schr&ouml;dinger perturbation theory which usually is attained in virtue of complicated mathematical transformations. Beyond of the time scale and its rules—strictly connected with the perturbation order N introduced by Schr&ouml;dinger—a partition process of the number N-1 is applied. This process, combined with contractions of the time points on the scale, provides us precisely with the perturbation terms entering the Schr&ouml;dinger theory.

Planck’s Oscillators at Low Temperatures and Haken’s Perturbation Approach to the Quantum Oscillators Reconsidered

In the first step the extremal values of the vibrational specific heat and entropy represented by the Planck oscillators at the low temperatures could be calculated. The positions of the extrema are defined by the dimensionless ratios between the quanta of the vibrational energy and products of the actual temperature multiplied by the Boltzmann constant. It became evident that position of a local maximum obtained for the Planck’s average energy of a vibration mode and position of a local maximum of entropy are the same. In the next step the Haken’s time-dependent perturbation approach to the pair of quantum non-degenerate Schr?dinger eigenstates of energy is re-examined. An averaging process done on the time variable leads to a very simple formula for the coefficients entering the perturbation terms.

Transformations and non-degenerate maps

We shall prove the equivalences of a non-degenerate circle-preserving map and a M(o)bius transformation in ^Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn of a non-degenerate line-preserving map and an affine transformation in Rn. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.

Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Suppose that X ={Xt, t≥0;Pμ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X. Then Mt := e-λ0t〈φ0,Xt〉 is a positive martingale. Let M∞ be the limit of Mt. It is known(see Liu et al.(2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0,∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E,lim/t→∞γt〈φ0,Xt〉=W, a.s.-Pμ.We also give the almost sure limit of γt〈f, Xt〉for a class of general test functions f.

Minimization of eigenvalues and construction of non-degenerate potentials for the one-dimensional p-Laplacian

We first use the Schwarz rearrangement to solve a minimization problem on eigenvalues of the one-dimensional p-Laplacian with integrable potentials. Then we construct an optimal class of non-degenerate potentials for the one-dimensional p-Laplacian with the Dirichlet boundary condition. Such a class of nondegenerate potentials is a generalization of many known classes of non-degenerate potentials and will be useful in many problems of nonlinear differential equations.

A Class of Non-degenerate Solvable Lie Algebras and Their Derivations

The author constructs a class of indecomposable non-degenerate solvable Lie Algebras corresponding to a Cartan matrix A over the field of complex numbers and we determine all their derivations.

Infinite Horizon Backward Doubly Stochastic Differential Equations with Non-degenerate Terminal Functions and Their Stationary Property

In this paper we consider infinite horizon backward doubly stochastic differential equations （BDS- DEs for short） coupled with forward stochastic differential equations, whose terminal functions are non-degenerate. For such kind of BDSDEs, we study the existence and uniqueness of their solutions taking values in weighted Lp（dx）¤L2（dx）space （p _〉 2）, and obtain the stationary property for the solutions.