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Journal of Theoretical and Computational Chemistry

QUANTUM PARAMETRIC RESONANCE: ANALYTICAL STUDIES

Jiushu Shao

Published Aug 1, 2008• 1 citation
Publisher

Abstract

The quantum version of parametric resonance of the harmonic oscillator is studied in terms of Feynman's propagator method and a discrete map. The eigenfunctions and eigenvalues of the Floquet operator are derived explicitly. The solutions follow the general laws: In the regions of parametric resonance the spectrum of the Floquet operator becomes continuous, while it is pointlike in the stable regimes. The Gaussian wave packet undergoes spreading exponentially and linearly in the two regimes, respectively.

DOI
10.1142/s0219633608004015

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The quantum version of parametric resonance of the harmonic oscillator is studied in terms of Feynman's propagator method and a discrete map. The eigenfunctions and eigenvalues of the Floquet operator are derived explicitly. The solutions follow the general laws: In the regions of parametric resonance the spectrum of the Floquet operator becomes continuous, while it is pointlike in the stable regimes. The Gaussian wave packet undergoes spreading exponentially and linearly in the two regimes, respectively.
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically unstable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically instable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.
The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined via the solution of a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, tunnelling currents, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied via linearization of the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. Furthermore, these two solutions also provide the time-dependent Green function for the systems propagation for any given initial condition, not only for Gaussians. It is also possible to obtain the corresponding Wigner function and generalized creation and annihilation operators in this way. In searching for problems with analytical solutions, the crucial point is to identify systems that have either a wave packet solution with time-dependent width (in closed form) that obeys a Riccati equation, or a classical parametric oscillator with frequency ω(t) that still enables an analytical solution of the Newtonian equation. Comparing the wave packets with the solution of the diffusion equation – also a Gaussian with time-dependent width – leads to a real Riccati equation. A first attempt at "complexifying" this equation turns out to be problematic. Nevertheless, it provides a clue to time-dependent frequencies inversely proportional to time that lead to Newtonian equations with analytical solutions. Following a detailed analysis of the corresponding quantum dynamics and energetics, possible extensions of the method are indicated.
AbstractWe demonstrate a formal equivalence between Gaussian Resonance R(x) and the squared quantum wave function |ψ(x)|², and derive a connection to the Schrödinger equation. The key step: replacing the real Gaussian parameter ε² with the imaginary quantity i transforms the Gaussian Resonance metric into the quantum probability ℏ amplitude. The time evolution of this amplitude under an Hamiltonian operator yields exactly the Schrödinger equation. Gaussian Resonance is therefore the real (dissipative) version of quantum mechanics; quantum mechanics is the complex (conservative) extension of Gaussian Resonance. Natural systems oscillate within a resonance neighborhood — not at a fixed point — and this oscillation is governed by the same principle at all scales, from fluid dynamics to quantum mechanics, with ε as the interpolating parameter between classical and quantum regimes.Central result: the substitution ε² i in the Gaussian Resonance → ℏ metric R(x) = exp(-f(x)²/2ε²) yields the quantum amplitude ψ(x) = exp(-f(x)²/2i ). The time evolution of ψ under a Hamiltonian H gives ℏ ℏ i ∂ψ/∂t = Hψ — the Schrödinger equation
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
Using the parametrically driven harmonic oscillator as a working example, we study two different Markovian approaches to the quantum dynamics of a periodically driven system with dissipation. In the simpler approach, the driving enters the master equation for the reduced density operator only in the Hamiltonian term. An improved master equation is achieved by treating the entire driven system within the Floquet formalism and coupling it to the reservoir as a whole. The different ensuing evolution equations are compared in various representations, particularly as Fokker-Planck equations for the Wigner function. On all levels of approximation, these evolution equations retain the periodicity of the driving, so that their solutions have Floquet form and represent eigenfunctions of a nonunitary propagator over a single period of the driving. We discuss asymptotic states in the long-time limit as well as the conservative and the high-temperature limits. Numerical results obtained within the different Markov approximations are compared with the exact path-integral solution. The application of the improved Floquet-Markov scheme becomes increasingly important when considering stronger driving and lower temperatures.