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arXiv:2306.02249 · Scientific Reports

Quantum dynamics of a driven parametric oscillator in a Kerr medium

Elmira Bolandhemmat, Fardin Kheirandish

Published Jun 3, 2023• 5 citations
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Abstract

In this paper, we first analyze a parametric oscillator with both mass and frequency time-dependent. We show that the evolution operator can be obtained from the evolution operator of another parametric oscillator with a constant mass and time-dependent frequency followed by a time transformation [Formula: see text]. Then we proceed by investigating the quantum dynamics of a parametric oscillator with unit mass and time-dependent frequency in a Kerr medium under the influence of a time-dependent force along the motion of the oscillator. The quantum dynamics of the time-dependent oscillator is analyzed from both analytical and numerical points of view in two main regimes: (i) small Kerr parameter [Formula: see text], and (ii) small confinement parameter k. In the following, to investigate the characteristics and statistical properties of the generated states, we calculate the autocorrelation function, the Mandel Q parameter, and the Husimi Q-function.

DOI
10.1038/s41598-023-36075-7

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In this paper, we first analyze a parametric oscillator with both mass and frequency time-dependent. We show that the evolution operator can be obtained from the evolution operator of another parametric oscillator with a constant mass and time-dependent frequency followed by a time transformation [Formula: see text]. Then we proceed by investigating the quantum dynamics of a parametric oscillator with unit mass and time-dependent frequency in a Kerr medium under the influence of a time-dependent force along the motion of the oscillator. The quantum dynamics of the time-dependent oscillator is analyzed from both analytical and numerical points of view in two main regimes: (i) small Kerr parameter [Formula: see text], and (ii) small confinement parameter k. In the following, to investigate the characteristics and statistical properties of the generated states, we calculate the autocorrelation function, the Mandel Q parameter, and the Husimi Q-function.
We study the temporal evolution of a coherent state under the action of a parametric oscillator and a nonlinear Kerr-like medium. We make use of the interaction-picture representation and use an exact time-evolution operator for the time-independent part of the Hamiltonian. We approximate the interaction-picture Hamiltonian in such a way as to make it a member of a Lie algebra. The corresponding time-evolution operator behaves like a squeezing operator due to the temporal dependence of the oscillator’s frequency. We analyze the probability amplitude, the auto-correlation function, and the Husimi distribution function for different Hamiltonian parameters. We find very good agreement between our approximate results and converged numerical calculations.
We study the time evolution of driven quantum systems using analytic, algebraic, and numerical methods. First, we obtain analytic solutions for driven free and oscillator systems by shifting the coordinate and phase of the undriven wave function. We also factorize the quantum evolution operator using the generators of the Lie algebra comprising the Hamiltonian. We obtain coupled ODE's for the time evolution of the Lie algebra parameters. These parameters allow us to find physical properties of oscillator dynamics. In particular we find phase-space trajectories and transition probabilities. We then search for chaotic behavior in the Lie algebra parameters as a signature for dynamical chaos in the quantum system. We plot the trajectories, transition probabilities, and Lyapunov exponents for a wide range of the following physical parameters: strength and duration of the driving force, frequency difference, and anharmonicity of the oscillator. We identify conditions for the appearance of chaos in the system.
For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.
A variety of dynamics in this world can be approximately treated by a model: a driven and damped parametric oscillator. An intensive investigate of this time-dependent model from both classical and quantum mechanical points of view provides a dynamical insight to resolve the classical and quantum dynamics in order to understand the more complicated time-dependent evolutions not only happens in the macroscopic classical scale for the synchronized behaviors but also in the coherent microscopic scale for a quantum state control. By using a Lie transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamical behaviors of the driven and damped parametric oscillator to give a universal physical mechanism for the classical parametric control on the quantum state evolution in a dissipation system.
We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set-mass, frequency, driving strength, and parametric pumping-is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola-Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schrödinger equation becomes numerically unstable in the laboratory frame.