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International Journal of Pure and Apllied Mathematics

TRANSIENT SPECTRUM OF sin^2-PULSED-DRIVEN QUBIT

R.A. Alharbey

Published Mar 13, 2015• 2 citations
Publisher

Abstract

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.

DOI
10.12732/ijpam.v100i2.3

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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 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.
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 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 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.
A study is presented for the time-dependent resonance-fluorescence spectrum of two-level atoms irradiated by hyperbolic-secant pulsed lasers, allowing for an arbitrary spontaneous-emission rate from the excited level to the ground level. Novel features of the temporally transient spectrum have been observed. The spectrum is dominated by a central peak which oscillates in time and is accompanied by multiple side peaks. Asymptotically (i.e., after the pulse has passed) the side peaks disappear, leaving a tail at the line center which displays a Lorentzian shape of width equal to the sum of the spontaneous-emission rate and the width of the spectrometer considered and of magnitude proportional to the instantaneous intensity of the pulse. The exact behavior of the spectrum is dictated by the area of the engaged pulse and by how the atom is prepared initially. In particular, it is found that the spectrum can be asymmetric even when the laser is tuned to exact resonance if the atoms are initially prepared in a certain class of proper coherent states, irrespective of how the field amplitude may vary in time. An analysis of the spectral symmetry without resorting to any explicit evaluation of the spectrum itself is also presented. Furthermore, the associated optical Bloch equations are solved analytically in terms of hypergeometric functions in a similar manner to that done previously for the undamped two-level Schr\"odinger equation. Specific numerical results are presented for pulses of area equal to \ensuremath{\pi}, 2\ensuremath{\pi}, 3\ensuremath{\pi}, 4\ensuremath{\pi}, and 5\ensuremath{\pi} for both symmetric and asymmetric cases.
The non-autonomous Bloch equations modelling a driven 2-level atom in the presence of an off-resonant broadband squeezed vacuum (SV) field is treated analytically. This concerns iterative solutions valid for large SV detuning parameter but for arbitrary strength of the laser field. Computational results are presented for the averaged atomic variables for various data and compared with the resonant SV field case. The iterated analytical results for nonzero SV detuning are compared with the (exact) numerical solutions of the Bloch equations, hence we have an insight about the range of other system parameters (other than the Rabi frequency) for which the iterated solutions are valid to O(10−2) or less. The main purpose of deriving these analytical results is to calculate analytically the transient fluorescent spectrum. For an initially ground-state atom, both the SV phase and detuning parameters induce pronounced asymmetrical spectrum in the strong field case.