Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator

Abstract

A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamic behaviors of a driven and damped parametric oscillator to obtain the optimal solutions by means of invariant parameters of Ks to combine with Lewis-Riesenfeld invariant method. This approach can discriminate the external dynamics from the internal evolution of a wave packet by producing independent parametric equations that dramatically facilitate the parametric control on the quantum state evolution in a dissipative system. In order to show the advantages of this method, several time-dependent models proposed in the quantum control field are analyzed in details.

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

“…A very interesting class of minimum wave packets was exhaustively studied by Nieto [142][143][144][145][146][147]. On the other hand, the dynamics of many physical systems can be described by using the quantum time-dependent harmonic oscillator [148][149][150][151][152][153][154][155][156][157][158], where the construction of minimum wave packets is relevant [159][160][161][162][163][164][165][166]. In a more general situation, wave packets with time-dependent width may occur for systems with different initial conditions, time-dependent frequency, or in contact with a dissipative environment [167][168][169][170].…”

The time-dependent harmonic oscillator revisited

“…For instance, the boundedness and stability of the solutions, or the knowledge of the tiny evolution of the adiabatic invariants under slowly varying ω's after millions or even billions of cycles T = 2π/ω, are crucial for many phenomena and problems in electrodynamics (in vacuum and plasmas) 15 applied to geo-and astro-physics [24], accelerator physics [3,32,4], plasma confinement in nuclear fusion reactors [11]; in particular, rigorous mathematical results [16,15,23,14,34,26] 16 on the adiabatic invariance of I = H/ω have allowed to dramatically increase the predictive power for these and other phenomena. A number of important classical and quantum control problems (like the stability of atomic clocks [31], the behaviour of parametric amplifiers based on electronic [13] or superconducting devices [22], or of parametrically excited oscillations in microelectromechanical systems [33]) are ruled by linear oscillator equations reducible to (1) (as sketched in section 2), where the interplay between time-dependent driving (parametric and/or external) and/or damping plays a crucial role [35]. In the quantum framework (1) arises e.g.…”

“…If q 1 , q 2 are proportional then L = 0, θ =const, and the particle oscillates along a straight line passing through O; otherwise L = 0, and the particle goes around O. Finally, the invariant (65) is related also to the symmetries of the equation (1) (see [29,35] and references therein). Probably it is worth underlining that in general ρ = 2I/ω, θ = ψ and, as already noted, I = I, albeit (4a), (61) look similar, and both 2I/ω, ρ(t) -contrary to q(t) -keep their sign, so that they can play the role of modulating amplitudes (envelopes) for the solutions of (1) resp.…”

“…Here we present a rather general and effective method for approximating the solutions (1), as well as a number of useful bounds or qualitative properties of the latter, that we have not been able to find in the very broad literature (see e.g. the already cited references, the more mathematical ones [7,28,16,23,21,34,32,27,30,35,10,29,17,2], and the references therein) on the subject. Our key observation is that, passing (section 4.1) from the canonical coordinates q, p to the angle-action variables ψ, I = H/ω, via…”

Lie transformation on shortcut to adiabaticity in parametric driving quantum system

“…and we can introduce three independent unitary transformations based on SU(2) by [18] Û± (t) = e iθ±(t) Ĵ± , Û0 (t) = e 2θ0(t) Ĵ0 ,…”