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Journal of the Optical Society of America B

Optical nutation in an optically thick and inhomogeneously broadened atomic ensemble

Devin Vega, Pamela B. Rambow, Ming Tian

Published May 23, 2018• 2 citations
Publisher

Abstract

Optical nutation has been used as an important spectroscopic tool to study the interaction between atoms and coherent light. It is based on the fact that the input pulse area needed to reach the first nutation peak is a constant value that only depends on the transverse mode of the optical beam. The interaction strength in terms of Rabi frequency can be measured directly from the time to reach the first nutation peak. However, the assumption of constant pulse area is only valid in the limit of the optically thin media, while optically thick media are of great interest for applications in classical and quantum information storage and processing. This paper studies the behavior of optical nutation in optically thick media through theoretical simulation and experiments in rare-earth atomic ensembles doped in a crystal lattice. A simple formula for the input pulse area as a function of the absorption length has been found, which covers optically thick media and is consistent with the value given by the low absorption limit.

DOI
10.1364/josab.35.001374

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Optical nutation has been used as an important spectroscopic tool to study the interaction between atoms and coherent light. It is based on the fact that the input pulse area needed to reach the first nutation peak is a constant value that only depends on the transverse mode of the optical beam. The interaction strength in terms of Rabi frequency can be measured directly from the time to reach the first nutation peak. However, the assumption of constant pulse area is only valid in the limit of the optically thin media, while optically thick media are of great interest for applications in classical and quantum information storage and processing. This paper studies the behavior of optical nutation in optically thick media through theoretical simulation and experiments in rare-earth atomic ensembles doped in a crystal lattice. A simple formula for the input pulse area as a function of the absorption length has been found, which covers optically thick media and is consistent with the value given by the low absorption limit.
The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by \ensuremath{\alpha}, the propagation length is L, and the transverse relaxation time is T2{\mathit{T}}_{2}T2​, the apparent decay rate 2/Ta{\mathit{T}}_{\mathit{a}}Ta​ of a time resolved four-wave mixing signal is given by 2/Ta{\mathit{T}}_{\mathit{a}}Ta​=(2/T2{\mathit{T}}_{2}T2​)(1+\ensuremath{\alpha}L/2).
Using a coherent radiation-semiconductor interaction model based upon the time dependent perturbation technique, the occurrence of optical nutation has been analytically investigated in direct-gap semiconductors such as GaAs, GaSb, InAs, and Hg1-xCdxTe, duly irradiated by near resonant laser beams. The present approach is much more simplified and straightforward than the conventional Maxwell-Bloch approach used so far for the study of coherent optical transient effects in molecular and atomic systems. The dispersion and absorption expressed in terms of the real and imaginary parts of the crystal optical susceptibility\chi_{r}and\chi_{i}exhibit temporally damped ringing behavior of Neumann and Bessel types, respectively. These processes have been identified as dispersive and absorptive optical nutation. The theory is developed for arbitrary laser intensity in presence of damping due to nonoptical processes. To understand the mechanisms of dispersive and absorptive optical nutation, we have restricted only to low-power resonant band-to-band transitions neglecting excitonic and damping effects. While studying analytically the transient behavior of\chi_{r}and\chi_{i}for a given pump field strength, it is noticed that the absorptive component exhibit remarkable qualitative agreement with the experimental observations of optical nutation in13CH3F.
Ramsey resonance is a high-resolution technique used in spectroscopy, precise measurement of time and frequency, and the creation of modern clocks. The Ramsey experiments are typically done in optically dilute samples of atoms to improve homogeneity and avoid backaction of atoms on excitation pulses. In contrast to latter belief, we predict and experimentally show that in optically thick resonant media, the backaction can lead to the highly enhanced narrowing of Ramsey resonance. The linewidth narrowing and corresponding precision of the frequency measurement scale exponentially with an increase in optical depth of a sample and can reach the limits set by homogeneous broadening. We show that this effect is caused by a nonlinear interference of multiple echoes formed inside the atomic medium, which is experimentally confirmed with ^{167}Er^{3+} ions in Y_{2}SiO_{5} crystal. Our findings open new opportunities for sensitivity enhancement of the Ramsey technique, as well as providing researchers with an additional method to investigate fundamental issues of coherent nonlinear interaction of light pulses with resonant atomic ensembles and numerous applications.
The coupled Maxwell-Schrodinger equations are used to describe propagation of a pulse in a nonlinear absorbing medium. The absorber is assumed to be comprised of a homogeneously broadened system of two-level atoms that are exactly resonant with the field. Numerical calculations have been performed for incident pulses of hyperbolic-secant shape and of duration that is long compared to the atomic phase-memory time. Selected values of input intensity are considered. For intensities below a certain level the calculations nearly agree with rate-equation predictions, but as the incident intensity is increased we find that optical nutation comes into play. The nutation effect leads to pulse breakup as the packet penetrates into the medium. An analytical calculation, for the case of an applied optical field of step-function form, has been performed in order to provide additional insight about the intensity threshold for nutation.
Spectroscopic studies and the efficiency of optical coherent transient devices can be adversely affected by interference arising from several distinct Rabi frequencies associated with crystallographically equivalent but orientationally inequivalent crystal sites. General symmetry principles have been used to find light field directions that guarantee both optimum single-Rabi-frequency interaction and the most efficient coherent transient generation in arbitrary crystals. This theoretical analysis has been applied experimentally to a wide range of rare earth crystals, many of which are important for technological applications. Site interference in optical nutation provides a simple illustration of this effect. Optimum single-Rabi-frequency nutation signal was obtained for the 3H6(1){}^{3}{H}_{6}(1)3H6​(1) to 3H4(1){}^{3}{H}_{4}(1)3H4​(1) transition of 0.1% Tm3+:Y3Al5O12{\mathrm{Tm}}^{3+}:{\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}Tm3+:Y3​Al5​O12​ when the light $\stackrel{\ensuremath{\rightarrow}}{E}$ vector is along special crystal directions 〈111〉〈111〉〈111〉 or 〈001〉.〈001〉.〈001〉. Quantitative comparison of nutation frequencies for light polarized along different directions allowed the determination of the optical transition dipole moment to be along 〈110〉〈110〉〈110〉 for this transition. The transition dipole moment derived from the nutation signal agrees within 15% with that obtained from absorption experiments. Other applications of nutation measurements are discussed, including a procedure for determining site occupancy in crystals with crystallographically inequivalent sites.
The properties of optically dense materials are influenced by interactions between elementary optical excitations (oscillators). Since such interactions are absent in the dilute limit, the resulting properties are unique to optically dense materials. While linear optical experiments can probe these effects, for example the Lorentz-Lorenz resonance shift, they are often more apparent in nonlinear experiments that are sensitive to coherence. Direct gap semiconductors are typically optically dense close to the fundamental gap and have been extensively studied using ultrafast coherent spectroscopy over the last ten years. However, their coherent optical properties are very complex because of many-body interactions among the extended excitations (electron-hole pairs or excitons). Dense atomic vapors have also been studied, but typically using frequency domain techniques. We present the results of using ultrafast techniques to study both semiconductors and dense atomic vapors. This reveals the similarities and differences of the two systems, yielding insight into the characteristics of each individually.