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Journal of Nonlinear Optical Physics & Materials

Transient spectrum of sin2-pulsed driven harmonic oscillator

Saad S. M. Hassan, R.A. Alharbey, H. Al-Zaki

Published Dec 1, 2014• 2 citations
Publisher

Abstract

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).

DOI
10.1142/s0218863514500520

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Journal of Physics B Atomic Molecular and Optical Physics•Jun 30, 2008•12 citations
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).
The model of a non-dissipative qubit (taken as a single 2-level atom) coupled to a short resonant laser pulse of sin 2 -shape is investigated through the transient fluorescent spectrum.Exact analytical operator solutions for the model Heisenberg equations are utilised to calculate the spectrum in terms of Bessel functions.Asymmetry and ringing in the 3-peak spectrum with strong Rabi frequency are pronounced with initial atomic coherent state as compared with the ground state case.This is due to interference of the sequential-pulse structure of the sin 2 -pulse with the initial dispersive coherence.For moderate Rabi frequency, ringing disappears with ' hole burning' dip shown in the dominant central Lorentzian peak.
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.
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.
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 fluorescent spectrum of a non-dissipative single 2-level atom driven by a resonant rectangular pulse is investigated analytically and computationally with the Haar wavelet window function. The shift parameter (k) of the mother wavelet function induces periodic effect. For weak pulse, a central dip structure -characteristic of the window transmission lines- occurs in the spectrum. With initial atomic coherent state and for strong pulse and large (k), interference of the strong Rabi oscillations induces asymmetrical dense ’ringing’ around the two Rabi side bands of the spectrum.
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.