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arXiv:1006.3362 · Journal of Physics A Mathematical and Theoretical

The degenerate parametric oscillator and Ince's equation

Ricardo Cordero–Soto, Sergeĭ K. Suslov

Published Dec 7, 2010• 36 citations
arXivHTMLPDF

Abstract

We construct Green's function for the quantum degenerate parametric oscillator in the coordinate representation in terms of standard solutions of Ince's equation in a framework of a general approach to variable quadratic Hamiltonians. Exact time-dependent wavefunctions and their connections with dynamical invariants and SU (1, 1) group are also discussed. An extension to the degenerate parametric oscillator with time-dependent amplitude and phase is also mentioned.

DOI
10.1088/1751-8113/44/1/015101

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We construct Green's function for the quantum degenerate parametric oscillator in the coordinate representation in terms of standard solutions of Ince's equation in a framework of a general approach to variable quadratic Hamiltonians. Exact time-dependent wavefunctions and their connections with dynamical invariants and SU (1, 1) group are also discussed. An extension to the degenerate parametric oscillator with time-dependent amplitude and phase is also mentioned.
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.
The solution of the quantum dynamical equation $\frac{i\ensuremath{\hbar}\mathrm{dT}}{\mathrm{dt}}=HT$, for the time-displacement operator TTT, is given, when the Hamiltonian HHH is a polynomial of the second degree in canonically conjugate variables, with arbitrary time-dependent coefficients. Heisenberg's equations of motion are then solved, and the general integral of Schr\"odinger's equation in coordinate space is expressed by the Green's function corresponding to TTT. An example is given.
We describe a general method to solve a time-dependent Hamiltonian system exhibiting dynamical symmetry under the Lie algebra SU(1,1). The instantaneous eigenstates can be obtained by applying a displacement operator which is constructed explicitly. The construction proves to be equivalent to the solution of the classical equations of motion. One can also obtain the exact solution of the quantum Hamiltonian using this displacement operator. We define and discuss the differences between the cyclic evolution in the space of the physical states and in the parameter space. This leads to a method to define and find the T-cyclic and periodic states and their phases for a system with dynamical symmetry SU(1,1) just by using its time evolution operator. For SU(1,1) one can show that T-cyclic states only exist for some values of T which can be determined through an inequality which is found explicitly and discussed. The example used to check these results has been the degenerate optical parametric oscillator with dynamical symmetry SU(1,1). The system has no periodic states but it does possess T-cyclic states. The characteristic interaction times are distributed in forbidden and allowed regions as in the band theory of particles in a periodic potential. We also define exact phases and adiabatic phases along the lines of the above discussion on T-cyclic and periodic states. For sufficiently low frequencies the initial eigenstates of the system acquire non-trivial Berry phases in each cycle of the evolution of the functional parameters on which H depends. Such a phase can be measured experimentally and from the discussion of its possible values under various circumstances emerges an interesting relationship with the golden mean.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
Abstract 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 by solving 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, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing 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. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.