Testing the validity of the Ehrenfest theorem beyond simple static systems: Caldirola–Kanai oscillator driven by a time-dependent force

The relationship between quantum mechanics and classical mechanics is investigated by taking a Gaussian-type wave packet as a solution of the Schr o¨dinger equation for the Caldirola–Kanai oscillator driven by a sinusoidal force. For this time-dependent system, quantum properties are studied by using the invariant theory of Lewis and Riesenfeld. In particular,we analyze time behaviors of quantum expectation values of position and momentum variables and compare them to those of the counterpart classical ones. Based on this, we check whether the Ehrenfest theorem which was originally developed in static quantum systems can be extended to such time-varying systems without problems.     

On the quantum motion of a generalized time-dependent forced harmonic oscillator

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator in terms of solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between coordinate and momentum. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola-Kanai oscillator.     

Using a New Disentangling Technique in the Solution of the Master Equation of a Driven Damped Harmonic Oscillator by Thermo Entangled States Representation

In this study we aim to solve the amplitude damping model master equation for a driven damped harmonic oscillator under the action of a classical force with arbitrary time dependence. We use thermo entangled state representation for the density operator, but to avoid a complicated disentangling process in such solutions, we introduce a simple and concise method to extract the density operator from its thermo entangled state representation. Whereas time evolution is a classical process, this method can be effectively used.

     

Wigner Distribution Function for the Time-Dependent Quadratic-Hamiltonian Quantum System using the Lewis–Riesenfeld Invariant Operator

We investigate the Wigner distribution function of the general time-dependent quadratic-Hamiltonian quantum system with the Lewis–Riesenfeld invariant operator method. The Wigner distribution function of the system in Fock state, coherent state, squeezed state, and thermal state are derived. We apply our study to the one-dimensional motion of a Brownian particle and to the driven oscillator with strongly pulsating mass.PACS: 03.65.-w, 03.65.Ca     

囚禁单离子的量子阻尼运动

用包含偶极和四极虚势能项的非厄米哈密顿算符来描述Paul阱中囚禁阻尼单离子在静电场下的量子运动.通过导出和分析系统的精确解,得到在PT对称和不对称情形下的不同实能谱与稳定量子态,以及PT不对称情形的虚能谱和衰减量子态,同时给出相应于不同态的参数区域和存活概率.结果发现该非厄米系统外场参数能惟一确定量子稳定态并导致波函数形态变化,据此提出非相干操控相应量子跃迁的方法.让量子态衰减导致的离子位置期待值的衰减与经典阻尼谐振子的衰减一致,得到虚势能参数与经典阻尼参数的对应关系.所得结果将进一步丰富具有广泛应用背景的囚禁离子动力学.     

Energy evolution in the time-dependent harmonic oscillator with arbitrary external forcing

The classical Hamiltonian system of time-dependent harmonic oscillator driven by an arbi- trary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique based on WKB approach. Energy evolution for the ensemble of uniformly distributed w.r.t, the canonical angle initial conditions on the initial invariant torus is studied. Exact expressions for the energy moments of arbitrary order taken at arbitrary time moment are analytically derived. Corresponding character- istic function is analytically constructed in the form of infinite series and numerically evaluated for certain values of the system parameters. Energy distribution function is numerically obtained in some particular cases. In the limit of small initial ensemble's energy the relevant formula for the energy distribution function is analytically derived. Keywords: classical dynamics, Hamiltonian systems, linear oscillator, energy distribution func- tion, WKB approach.     

Squeezed states of light as self-similar structures

A new approach to investigating a broad class of dynamic states for a quantum oscillator is suggested. It is based on an invariant transformation of the equation to a new time determined by the quantum dispersion of the corresponding state. The squeezed states of a quantum system generated by the ground-state wave function are constructed. In coordinate representation, these states are described by a self-similar wave function localized near a classical trajectory. The statistics of the squeezed state of light is analyzed in the single-mode approximation. The parametric excitation of squeezed states for a quantum harmonic oscillator is considered.     

三种方法求解外场驱动含时谐振子系统的异同

讨论了外场驱动含量谐振子系统的三种解法的异同，发现利用相干态平均方法与量子不变量理论所得结果一致，而利用海森伯运动方程所求结果与上述两种结果相关一个相位因子。     

Eisenhart lifts and symmetries of time-dependent systems

Certain dissipative systems, such as Caldirola and Kannai’s damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with n$n$ degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in n+2$n+2$ dimensions, equipped with its covariantly constant null Killing vector field. Reparametrisation of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola–Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schrödinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space.     

The driven-oscillator evolution in the tomographic-probability representation

We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system??s phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.     

Recent results on time-dependent Hamiltonian oscillators

Time-dependent Hamilton systems are important in modeling the nondissipative interaction of the system with its environment. We review some recent results and present some new ones. In time-dependent, parametrically driven, one-dimensional linear oscillator, the complete analysis can be performed (in the sense explained below), also using the linear WKB method. In parametrically driven nonlinear oscillators extensive numerical studies have been performed, and the nonlinear WKB-like method can be applied for homogeneous power law potentials (which e.g. includes the quartic oscillator). The energy in time-dependent Hamilton systems is not conserved, and we are interested in its evolution in time, in particular the evolution of the microcanonical ensemble of initial conditions. In the ideal adiabatic limit (infinitely slow parametric driving) the energy changes according to the conservation of the adiabatic invariant, but has a Dirac delta distribution. However, in the general case the initial Dirac delta distribution of the energy spreads and we follow its evolution, especially in the two limiting cases, the slow variation close to the adiabatic regime, and the fastest possible change – a parametric kick, i.e. discontinuous jump (of a parameter), where some exact analytic results are obtained (the so-called PR property, and ABR property). For the linear oscillator the distribution of the energy is always, rigorously, the arcsine distribution, whose variance can in general be calculated by the linear WKB method, while in nonlinear systems there is no such universality. We calculate the Gibbs entropy for the ensembles of noninteracting nonlinear oscillator, which gives the right equipartition and thermostatic laws even for one degree of freedom.     

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.     

Propagator for a Time-Dependent Damped Harmonic Oscillator with a Force Quadratic in Velocity

The propagator for a time-dependent damped harmonic oscillator with a force quadratic in velocity isobtained by making a specific coordinate transformation and by using the method of time-dependent invariant.     

Quantum mechanics of a general driven time-dependent oscillator

Summary In this paper we investigate the time evolution of a general driven time-dependent oscillator using the evolution operator method developed by Chenget al. We obtain an exact form of the time evolution operator which, in turn, enables us to find the exact wave functions and coherent states at any timet. Our analyses indicate that the time-dependent coherent state is equivalent to the well-known squeezed state, while the time-dependent number state is equivalent to the displaced and squeezed number state. Besides, we also calculate the time-dependent transition probabilities among the coherent states and number states of a simple harmonic oscillator associated with the initial HamiltonianH(0).     

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.     

Time Evolution and Squeezing of Degenerate and Non-degenerate Coupled Parametric Down-Conversion with Driving Term

By properly selecting the time-dependent unitary transformation for the linear combination of the number operators, we construct a time-dependent invariant and derive the corresponding auxiliary equations for the degenerate and non-degenerate coupled parametric down-conversion system with driving term. By means of this invariant and the Lewis-Riesenfeld quantum invariant theory, we obtain closed formulae of the quantum state and the evolution operator of the system. We show that the time evolution of the quantum system directly leads to production of various generalized one- and two-mode combination squeezed states, and the squeezed effect is independent of the driving term of the Hamiltonian. In somespecial cases, the current solution can reduce to the results of the previous works.     

Coherent states of the inverted Caldirola-Kanai oscillator with time-dependent singularities

The coherent states for a system of time-dependent singular potentials coupled to inverted CK (Caldirola-Kanai) oscillator are investigated by employing invariant operator method and Lie algebraic approach. We considered Coulomb potential and inverse quadratic potential as singularities of the system. The spectrum of quantum states is discrete for λ < 0 while continuous for λ ? 0. The probability densities for both Fock state and coherent state are converged to the center as time goes by according to the dissipation of energy. We confirmed that the probability density in the coherent state oscillates back and forth like a classical wave packet.     

Study of Certain Aspects of Anharmonic,Time-Dependent and Damped Harmonic Oscillator Systems

The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.