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International Journal of Apllied Mathematics

DRIVEN HARMONIC OSCILLATOR BY TRAIN OF CHIRPED GAUSSIAN PULSES

Saad S. M. Hassan, Rania A. Alharbey, T. Jarad, S. Almaatooq

Published Feb 20, 2020• 10 citations
Publisher

Abstract

Exact analytical operator solutions of the interacting model of a single quantized (non-dissipative) harmonic oscillator (HO) with a train of n-chirped Gaussian pulses are derived in terms of the error function of complex argument.Explicit expressions are then calculated and examined computationally for the average photon number of the HO and the emitted spectrum.The chirp parameter (c) induces non-sinusoidal oscillations that lead to: (i) 'step-like plateau' in the dynamics of the average photon number with both n, τ R (repetition time) large, and, (ii) a 'hole burning' profile and asymmetrical ringing in the spectrum, depends on the initial state of the HO.

DOI
10.12732/ijam.v33i1.6

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Exact analytical operator solutions of the interacting model of a single quantized (non-dissipative) harmonic oscillator (HO) with a train of n-chirped Gaussian pulses are derived in terms of the error function of complex argument.Explicit expressions are then calculated and examined computationally for the average photon number of the HO and the emitted spectrum.The chirp parameter (c) induces non-sinusoidal oscillations that lead to: (i) 'step-like plateau' in the dynamics of the average photon number with both n, τ R (repetition time) large, and, (ii) a 'hole burning' profile and asymmetrical ringing in the spectrum, depends on the initial state of the HO.
The interacting model of a single quantised harmonic oscillator (HO) with a chirped Gaussian (laser) pulse is studied analytically and computationally. Specifically, exact formulae for the averaged photon number (related to the HO energy) and the transient scattered radiation spectrum are derived. The chirp and detuning parameters together with the initial state of the HO have essential roles for the oscillatory behaviour of the average photon number and for the asymmetry and additional oscillatory peaks in the scattered spectrum.
The simplest non-dissipative 2-level atom system, a qubit, excited by a train of resonant n-Gaussian laser pulses is investigated. This concerns examination of the averaged atomic variables, the intensity-intensity correlation function, and the transient fluorescent radiation. Analytical formulas for the above expressions are obtained. Computational results show that the transient spectra with the initial ground and coherent atomic states exhibit asymmetric Mollow structure with dip structure, dense oscillation, and narrowing, and depends on the pulse number (n), the repetition time (τR), and the observed time.
Exact analytical results are derived for the coupled system of nondissipative single mode quantized harmonic oscillator (HO) and arbitrary pulse shape. Specifically, for a sin2-pulse shape, the transient fluorescent spectrum is obtained in the general case of initial coherent state |α〉 of the HO. The dominant central Lorentzian is surrounded by weak oscillations due to the larger number of sequential pulses which get amplified asymmetrically in the nonresonant case as a result of balancing and interference processes between the initial excitation (|α2|) and the strength of the exciting pulse. Further pronounced oscillations is noticed with "hole burning" structure in the central peak due to nonzero phase of the initial amplitude (α).
Abstract We investigate the transient and stationary physical spectra of scattered radiation from a single quantized harmonic oscillator (HO) of frequency ω o , which is resonantly driven by an intense driving field of real (amplitude) strength Ω ∼ ω o and damped in a Markovian broadband radiation reservoir. The influence of such an intense field combined with manipulative spectral reservoir parameters results in field strength enhancement and a phase shift gain: Ω → Ω 1 + γ ′ 2 ( ω o ) e i γ ′ ( ω o ) , where γ ′ ( ω o ) = ( d γ ( ω ) / d ω ) ω = ω o is the spectral rate of variation of the damping constant at resonance. This in turn leads to induced coherence in the average amplitude of the HO. Two cases of spectral distribution of the radiation reservoir are considered: the Ohmic power law and the Lorentz-Drude law. The spectral components of the scattered radiation in the transient and stationary regimes are identified analytically and computationally, and comparison is made with the non-intense driving field case (Ω ≪ ω o ). The stationary spectrum is a symmetric Lorentzian of weight dependent on the reservoir spectral parameters and of enhanced value compared with the non-intense field case. The transient spectrum is composed of different lineshapes (Lorentzian, narrowed-Lorentzian, and dispersive) of oscillatory decaying weights. Interference of these transient-spectral components is initiated and caused by both types of non-zero coherence, namely the initial coherence of the HO and the induced coherence in the intense field case, and induces asymmetric side oscillations of amplitude dependent on the reservoir spectral parameters. The present results for the fluorescent spectra of intensely-driven HO and its dependence on the environmental radiation reservoir spectral structure are adequately fitted for intensely driven collective Rydberg atomic systems, as a ‘single site’ atomic sample, O(μm), and behaves as a single quantized HO.
The transient spectrum of scattered radiation from a qubit (represented by a spin-1/2 system or a two-level atom) driven by a resonant rectangular pulse is calculated analytically without adopting the rotating wave approximation (RWA). Operator solutions for the non-autonomous model of Heisenberg equations of motion are obtained within an iterative approach to O (λ) where λ is the system parameter representing the ratio of the Rabi frequency of the driving pulse to its circular frequency. The spectrum in the case of an initial ground-state atom and for increasing pulse area shows asymmetric profiles outside the RWA compared with a symmetric profile within the RWA. For the qubit this asymmetry is much pronounced with splitting of side peaks, which develops to ‘ringing’ for larger pulse area (within and outside the RWA). As shown analytically, this is attributed to the strong Rabi oscillations and interference between various amplitude spectral components initiated by the finite atomic coherent dispersion.
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.