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Physica Scripta

Fluorescent spectra of an intensely-driven harmonic oscillator

Shoukry S. Hassan, Mohamed M. Hassan, O. M. Frege

Published Feb 26, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/adbabd

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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.
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 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 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 analyze the time-dependent physical spectrum of a driven Jaynes-Cummings model in which both the two-level system and the quantized cavity mode are subject to coherent classical driving. The time-dependent Hamiltonian is mapped, via well-defined unitary transformations, onto an effective stationary Jaynes-Cummings form. Within this framework, we derive closed-form expressions for the two-time correlation functions of both the atomic and field operators. These correlation functions are subsequently used to evaluate the time-dependent physical spectrum according to the Eberly-Wódkiewicz definition, which properly accounts for finite spectral resolution and transient emission dynamics. We show that the external driving leads to substantial modifications of the atomic spectral response, including controllable frequency shifts and asymmetric line shapes. Importantly, we identify a regime in which the driving parameters are chosen such that the coherent displacement induced in the cavity field exactly cancels out the initial coherent amplitude. In this limit, the system dynamics reduce to that of an effectively vacuum-initialized Jaynes-Cummings model, and the standard vacuum Rabi splitting is recovered. This behavior provides a clear and physically transparent interpretation of the spectral features as arising from coherent field displacement rather than from modifications of the underlying atom-cavity coupling.
We evaluate the power spectrum of light scattered by a weakly nonlinear Morse oscillator driven by a strong single-mode classical electromagnetic field, taking radiation damping into account as the only relaxation process. The spectrum is calculated in the small-noise limit under a near-resonant condition (the resonance is considered in terms of an effective detuning parameter). We show that the single-peak structure that appears in the weak-field limit splits into a symmetric doublet centering around the driving field frequency in the presence of an intense field when the Rabi frequency far exceeds the natural damping rate.
The steady-state populations, the fluorescent intensity-intensity correlation function, and the resonance fluorescence and absorption spectra of a two-level atom bichromatically driven by a strong, coherent field and a weak, stochastic field with tunable carrier frequency and wide bandwidth are investigated analytically. Surprisingly, the population is inverted if the stochastic field is appropriately detuned from the coherent field, which is assumed resonant with the atomic transition. The resonance fluorescence spectrum still displays a three-peaked structure, but with linewidths and heights that depend upon the frequency and bandwidth of the stochastic field as well as the intensities of both fields. Asymmetric fluorescence spectra are found when the coherent field is detuned from the atomic resonance frequency. The intensity-intensity correlation function varies dramatically with the carrier frequency of the stochastic field. For appropriate choices of the parameters, the fluorescent emission of the atom is antibunched for all time. The absorption spectral features also deviate significantly from those of the monochromatically driven atom. By appropriately tuning the stochastic field, one finds that a sharp peak (absorption) or hole (gain) can occur at line center and the probe beam can be amplified at either the lower-frequency Rabi sideband or the higher-frequency sideband and the inversionless gain can be enhanced.