Arbitrary Time-dependent Frequency Research Articles

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

Dynamics of entanglement in three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants

The dynamics of mixedness and entanglement is examined by solving the time-dependent Schrodinger equation for three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants parameters. We assume that part of oscillators is inaccessible and remaining oscillators accessible. We compute the dynamics of entanglement between inaccessible and accessible oscillators. In order to show the dynamics pictorially, we introduce three quenched models. In the quenched models, both mixedness and entanglement exhibit oscillatory behavior in time with multi-frequencies. It is shown that the mixedness for the case of one inaccessible oscillator is larger than that for the case of two inaccessible oscillators in the most time interval. Contrary to the mixedness, entanglement for the case of one inaccessible oscillator is smaller than that for the case of two inaccessible oscillators in the most time interval.

Generation of Photons from Vacuum in a Cavity with Time-Dependent Eigenfrequency and Dumping Coefficients

This is a brief review of recent achievements in the theory of non-stationary Casimir effect in non-ideal cavities and of existing proposals to observe this effect in a laboratory. In this connection, a model of quantum damped oscillator with arbitrary time-dependent frequency and damping coefficient is developed and the influence of different parameters on the photon generation rate is analyzed.

The Heisenberg-Langevin model of a quantum damped harmonic oscillator with time-dependent frequency and damping coefficients

We develop a consistent model of a quantum damped harmonic oscillator with arbitrary time-dependent frequency and damping coefficients within the framework of the Heisenberg-Langevin equations with two noncommuting delta-correlated noise operators justifying the choice of the “minimal noise” set of damping coefficients and correlation functions.

The nonstationary Casimir effect in a cavity with periodical time-dependent conductivity of a semiconductor mirror

We consider the problem of photon creation from vacuum or thermal states inside a cavity with periodical time-dependent conductivity of a thin semiconductor boundary layer, simulating periodical displacements of the wall. Our approach is based on the consistent model of a quantum-damped harmonic oscillator with arbitrary time-dependent frequency and damping coefficients in the framework of the Heisenberg–Langevin equations with two noncommuting delta-correlated noise operators. We calculate the rate of photon generation under the resonance conditions, taking into account the internal dissipation inside the semiconductor. This rate depends mainly on two parameters: the total intensity of the laser pulse and the recombination time of photo-excited carriers in the semiconductor slab (for fixed mobility of carriers and geometry). Optimal values of these parameters and dimensions of the cavity are found for the TE and TM modes. The influence of temperature and detuning from strict resonance is analysed.

Time-dependent quantum damped oscillator with ‘minimal noise’: application to the nonstationary Casimir effect in nonideal cavities

We consider a consistent model of a quantum damped harmonic oscillator with arbitrary time-dependent frequency and damping coefficients in the frameworks of the Heisenberg–Langevin equations with two noncommuting delta-correlated noise operators. For the 'minimal noise' set of correlation functions, which have the same time dependence as the damping coefficient, we obtain the exact solution, which is the generalization of Husimi's solution for the undamped nonstationary oscillator. The model is applied to the problem of the photon creation from vacuum or thermal states due to the nonstationary Casimir effect inside the cavity with periodical time-dependent conductivity of the thin semiconductor boundary layer, simulating the periodical displacements of the wall. The general formula for the rate of photon generation under the resonance conditions in the presence of dissipation is obtained.

Quantum Harmonic Oscillator and Nonstationary Casimir Effect

We consider the relations between the theory of quantum nonstationary damped oscillator and nonstationary Casimir effect in view of the problem of photon creation from vacuum inside the cavity with periodical time-dependent conductivity of a thin semiconductor boundary layer, which simulates periodical displacements of the cavity boundaries. We develop a consistent model of quantum damped harmonic oscillator with arbitrary time-dependent frequency and damping coefficients within the framework of Heisenberg-Langevin equations with two noncommuting delta-correlated noise operators. For the minimum noise set of correlation functions, whose time dependence follows that of the damping coefficient, we obtain the exact solution, which is a generalization of the Husimi solution for undamped nonstationary oscillator. It yields the general formula for the photon-generation rate under the resonance condition in the presence of dissipation. We obtain a simple approximate formula for a time-dependent shift of the complex resonance frequency. It depends only on the total energy of a short laser pulse (which creates an effective time-dependent electron-hole “plasma mirror” on the semiconductor-slab surface) and the recombination time. We show that damping due to a finite conductivity of the material significantly diminishes the photon-generation rate in the selected field mode of the cavity. Nonetheless, we have found optimum values of the parameters (laser pulse power, recombination time, and cavity dimensions), for which the effect of photon generation from vacuum could be observed in the experimental set-up proposed in the University of Padua. We also provide with a list of publications from 2001 to 2005 devoted to the study on quantum-field interactions with moving boundaries (mirrors).

Exact results on the dynamics of a multicomponent Bose-Einstein condensate

We study the time evolution of a two-dimensional multicomponent Bose-Einstein condensate in an external harmonic trap with arbitrary time-dependent frequency. We show analytically that the time evolution of the total mean-square radius of the wave packet is determined in terms of the same solvable equation as in the case of a single-component condensate. The dynamics of the total mean-square radius is also the same for a rotating as well as a nonrotating multicomponent condensate. We determine the criteria for the collapse of the condensate at a finite time. Generalizing our previous work on a single-component condensate, we show explosion-implosion duality in the multicomponent condensate.

Functional determinants from Wronski Green functions

A general technique is developed for calculating functional determinants of second-order differential operators with Dirichlet, periodic, and antiperiodic boundary conditions, without the knowledge of spectral properties. As an example, we give explicit formulas for a harmonic oscillator with an arbitrary time-dependent frequency, where our result is a generalization of the Gel’fand–Yaglom famous formula for Dirichlet boundary conditions. Our technique is based on the Wronski’s construction of Green functions, which does not require spectral knowledge. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2×2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of the classical solution. Special properties of operators with a zero mode are exhibited.

Quantum states of an oscillator with periodic time-dependent frequency under quasiresonant condition: Unperturbed evolution, perturbative effects, and anharmonic effects

In a recent paper [L. Ferrari, Phys. Rev. A 57, 2347 (1998)], one of the authors has shown that the mean energy of a quantum oscillator with periodic time-dependent frequency diverges exponentially in time, under certain conditions. In the present paper, we study the explicit form of the evolving state, and compare the results obtained with the general expressions developed by other authors for an arbitrary time-dependent frequency. Then we approach the problem of the anharmonic effects. A first-order calculation is performed in the case of a short-range perturbation. The transition rates between different states, evolving with the unperturbed Hamiltonian, are shown to vanish at long times when the unperturbed oscillator's energy diverges exponentially. A nonperturbative approach must be adopted, in the presence of anharmonic potentials of the form $V(q)\ensuremath{\propto}{q}^{j}, j>2$. In the case of weak anharmonicity $(j\ensuremath{-}2\ensuremath{\ll}1)$, a mean-field procedure can be used to show that the mean energy does actually saturate at long times, with the possible exception of periodic peaks, having nonsaturating height, that we call ``special quantum effects.''

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution. In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's. Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.

Quantum system under periodic perturbation: Effect of environment

In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with arbitrary time dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. Specific application is made to the case of periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: Effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists

Exact green function of a damped oscillator

Abstract The explicit Green functions for a damped oscillator with an arbitrary time-dependent frequency and a perturbative force are obtained in coordinate and coherent states representation. This illustrates the application of a general method for the construction of Green functions based on the employment of the integrals of the motion of quantum systems.