Quadratic Hamiltonian Systems Research Articles

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.

Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems

In this paper we are devoted to considering the existence of homoclinic solutions for some second order non-autonomous Hamiltonian systems with the asymptotically quadratic potential at infinity. The proof is based on a variant version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.

TIME-DEPENDENT WIGNER DISTRIBUTION FUNCTION EMPLOYED IN SQUEEZED SCHRÖDINGER CAT STATES: |Ψ(t)〉 = N-1/2(|β〉+eiϕ|-β〉)

The Wigner distribution function (WDF) for the time-dependent quadratic Hamiltonian system is investigated in the squeezed Schrödinger cat states with the use of Lewis–Riesenfeld theory of invariants. The nonclassical aspects of the system produced by superposition of two distinct squeezed states are analyzed with emphasis on their application into special systems beyond simple harmonic oscillator. An application of our development to the measurement of quantum state by reconstructing the WDF via Autler–Townes spectroscopy is addressed. In addition, we considered particular models such as Cadirola–Kanai oscillator, frequency stable damped harmonic oscillator, and harmonic oscillator with time-variable frequency as practical applications with the object of promoting the understanding of nonclassical effects associated with the WDF.

Polynomial Hamiltonian systems with a nilpotent critical point

The study of Hamiltonian systems is important for space physics and astrophysics. In this paper, we study local behavior of an isolated nilpotent critical point for polynomial Hamiltonian systems. We prove that there are exact three cases: a center, a cusp or a saddle. Then for quadratic and cubic Hamiltonian systems we obtain necessary and sufficient conditions for a nilpotent critical point to be a center, a cusp or a saddle. We also give phase portraits for these systems under some conditions of symmetry.

Quantum phase problem for harmonic and time-dependent oscillator systems

We address generalized measurements of linear multimode operators and discuss some aspects relevant to constructing angle operators for arbitrary quadratic Hamiltonian systems via Weyl-ordered expansions in terms of position and momentum operators.

Dynamics of SU(1, 1) coherent states for the time-dependent quadratic Hamiltonian system

The dynamics of SU(1, 1) coherent states introduced by Perelomov are investigated for the time-dependent quadratic Hamiltonian system. SU(1, 1) generators we employed are closely related to the invariant operator theory while those of the previous work of Gerry et al. [C.C. Gerry, P.K. Ma, E.R. Vrscay, Phys. Rev. A 39 (1989) 668] are associated to the simple harmonic oscillator. This is the main difference between the two approaches. The merit of the method used in this paper is that it admits wide sphere of analytical description for quantum features of time-dependent quadratic Hamiltonian system. Our development is applied to the Caldirola–Kanai oscillator and compared the corresponding results with those of the Gerry et al. after correcting some miscalculations of theirs. We showed that the results of our theory are in good agreement with the results of the corrected work of Gerry et al. even if the form of the SU(1, 1) generators we employed are somewhat different from those of their work. The nontrivial zero-point energy plays a dominant role in the very low energy limit ( ξ ˜ 0 → 0 ) for the Caldirola–Kanai oscillator, leading the system to exhibit pure quantum effects as expected. On the other hand, it turn out for sufficiently high energy limit ( ξ ˜ 0 → 1 ) that the characteristic feature of dissipating quantum energy become very much the same as that of the classical energy.

Exact formulated approach to transition probabilities of linearly driven parametric oscillators

The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.

Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems

In this paper, we find new conditions, which are different from those used in previous related studies, to ensure the existence of infinitely many homoclinic orbits for the second order Hamiltonian systems of the form − q ̈ = V q ( t , q ) . Here, we assume that V ( t , q ) depends periodically on t , and assume, on q , that V ( t , q ) is asymptotically quadratic at q = 0 and is, as | q | → ∞ , either asymptotically quadratic or superquadratic, as well as the new conditions.

Time-dependent Wigner distribution function employed in coherent Schrödinger cat states: |Ψ(t)⟩=N−1/2(|α⟩+eiϕ|-α⟩)

The Wigner distribution function for the time-dependent quadratic Hamiltonian system in the coherent Schrödinger cat state is investigated. The type of state we consider is a superposition of two coherent states, which are by an angle of π out of phase with each other. The exact Wigner distribution function of the system is evaluated under a particular choice of phase, δc, q. Our development is employed for two special cases, namely, the Caldirola–Kanai oscillator and the frequency stable damped harmonic oscillator. On the basis of the diverse values of the Wigner distribution function that were plotted, we analyze the nonclassical behavior of the systems.

Limit Cycles Near Homoclinic and Heteroclinic Loops

In the study of near-Hamiltonian systems, the first order Melnikov function plays an important role. It can be used to study Hopf, homoclinic and heteroclinic bifurcations, and the so-called weak Hilbert’s 16th problem as well. The form of expansion of the first order Melnikov function at the Hamiltonian value h 0 such that the curve defined by the equation H(x, y) = h 0 contains a homoclinic loop has been known together with the first three coefficients of the expansion. In this paper, our main purpose is to give an explicit formula to compute the first four coefficients appeared in the expansion of the first order Melnikov function at the Hamiltonian value h 0 such that the curve defined by the equation H(x, y) = h 0 contains a homoclinic or heteroclinic loop, where the formula for the fourth coefficient is new, and to give a way to find limit cycles near the loops by using these coefficients. As an application, we consider polynomial perturbations of degree 4 of quadratic Hamiltonian systems with a heteroclinic loop, and find 3 limit cycles near the loop.

SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

The SU(1,1) coherent states, so-called Barut-Girardello coherent state and Perelomov coherent state, for the generalized two-mode time-dependent quadratic Hamiltonian system are investigated through SU(1,1) Lie algebraic formulation. Two-mode Schrodinger cat states defined as an eigenstate of \(\hat{K}_{-}^{2}\) are also studied. We applied our development to two-mode Caldirola-Kanai oscillator which is a typical example of the time-dependent quadratic Hamiltonian system. The time evolution of the quadrature distribution for the probability density in the coherent states are analyzed for the two-mode Caldirola-Kanai oscillator by plotting relevant figures.

Algebraic structure of the Feynman propagator and a new correspondence for canonical transformations

We investigate the algebraic structure of the Feynman propagator with a general time-dependent quadratic Hamiltonian system. Using the Lie-algebraic technique, we obtain a normal-ordered form of the time-evolution operator, and then the propagator is easily derived by a simple “integration within ordered product” technique. It is found that this propagator contains a classical generating function, which demonstrates a new correspondence between classical and quantum mechanics.

Comment on “Time-dependent general quantum quadratic Hamiltonian system”

It is shown that the propagator and wave function recently found by are not complete and the results are not correct for constant coefficients of a time-dependent general quantum quadratic Hamiltonian for the case of . We presented the correct evolution operator, propagator, and wave function of the general time-dependent quantum quadratic Hamiltonian system.

Reply to “Comment on ‘Time-dependent general quantum quadratic Hamiltonian system” ’

In reply to the Comment by Zheng [Phys. Rev. A 68, 052108 (2003)], it is reaffirmed that the propagator and wave function derived originally are correct.

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed.

SU(1,1) LIE ALGEBRA APPLIED TO THE TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM PERTURBED BY A SINGULARITY

We realized SU (1,1) Lie algebra in terms of the appropriate SU (1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU (1,1) Lie algebra. Various expectation values in two kinds of the generalized SU (1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola–Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU (1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU (1,1) coherent states for the CKOPS are very similar to the classical trajectory.

On the cyclicity of a 2-polycycle for quadratic systems

Abstract It has been already known that the maximum number of limit cycles near a homoclinic loop of a quadratic Hamiltonian system under quadratic perturbations is two. However, the problem of finding the maximum number of limit cycles in the 2-polycycle case is still open. This paper addresses the problem in some detail and solves it partially.

Unitary Transformation of the Time-dependent Hamilton System for the Linear, the V-shape and the Triangular Well Potentials into the Quadratic Hamiltonian System

Unitary Transformation of the Time-dependent Hamilton System for the Linear, the V-shape and the Triangular Well Potentials into the Quadratic Hamiltonian System

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q H and Q R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q H ∩ Q R . One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.

Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems

A time-dependent closed-form formulation of the linear unitary transformation for harmonic-oscillator annihilation and creation operators is presented in the Schrödinger picture using the Lie algebraic approach. The time evolution of the quantum mechanical system described by a general time-dependent quadratic Hamiltonian is investigated by combining this formulation with the time evolution equation of the system. The analytic expressions of the evolution operator and propagator are found. The motion of a charged particle with variable mass in the time-dependent electric field is considered as an illustrative example of the formalism. The exact time evolution wave function starting from a Gaussian wave packet and the operator expectation values with respect to the complicated evolution wave function are obtained readily.