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Journal of the Optical Society of America B

Near-resonant light propagation in an absorptive spatially anisotropic ultracold gas

Jonathan R. Gilbert, Colin Roberts, Jacob Roberts

Published Mar 7, 2018• 6 citations
Publisher

Abstract

A collection of ultracold atoms has an index of refraction that varies depending on the spatial density distribution of the gas. For many confined gases commonly found in ultracold atom experiments, there can be a substantial gradient in the spatial density distribution and hence the index of refraction. In addition, these gases can have a smaller spatial extent than that of the cross section of a laser beam that illuminates them. With a sufficient gradient in density under these conditions, the resulting index variation leads to frequency-dependent focusing or defocusing of incident near-resonant light as demonstrated by the calculations in this work. Nonintuitive intensity pattern formations result within the gas from the combination of refraction and diffraction of the incident light, including focusing effects that substantially increase the intensity of the light as compared to its incident value despite the nominally absorptive nature of the gas.

DOI
10.1364/josab.35.000718

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A collection of ultracold atoms has an index of refraction that varies depending on the spatial density distribution of the gas. For many confined gases commonly found in ultracold atom experiments, there can be a substantial gradient in the spatial density distribution and hence the index of refraction. In addition, these gases can have a smaller spatial extent than that of the cross section of a laser beam that illuminates them. With a sufficient gradient in density under these conditions, the resulting index variation leads to frequency-dependent focusing or defocusing of incident near-resonant light as demonstrated by the calculations in this work. Nonintuitive intensity pattern formations result within the gas from the combination of refraction and diffraction of the incident light, including focusing effects that substantially increase the intensity of the light as compared to its incident value despite the nominally absorptive nature of the gas.
Accurate modeling of optical nonlinearities is crucial to describe macroscopic laser propagation in a medium, including sum frequency generation, spectral broadening due to self-phase modulation, various ionization processes and soliton formation. For incident laser light the response of the medium is given by the induced polarization of the microscopic system. The polarization is usually expanded in a Taylor series for the electric field amplitude, which is truncated after the first non-linear term being of third order for isotropic media. A third-order nonlinearity leads to the well-known optical Kerr effect, where the refractive index of the medium becomes intensity dependent via n = n0+ n2I. This leads to an inherent problem when modeling laser propagation in two or more spatial dimensions, linked to the formal divergence (n2> 0) of the refractive index for increasing intensity.
When a gas of ultracold atoms is suddenly illuminated by light that is nearly resonant with an atomic transition, the atoms cannot respond instantaneously. This noninstantaneous response means the gas is initially more transparent to the applied light than in steady state. The timescale associated with the development of light absorption is set by the atomic excited-state lifetime. Similarly, the index of refraction in the gas also requires time to reach a steady-state value, but the development of the associated phase response is expected to be slower than absorption effects. Faraday rotation is one manifestation of differing indices of refraction for orthogonal circular light polarization components. We have performed experiments measuring the time-dependent development of polarization rotation in an ultracold gas subjected to a magnetic field. Our measurements match theoretical predictions based on solving optical Bloch equations. We are able to identify how parameters such as steady-state optical thickness and applied magnetic field strength influence the development of Faraday rotation.
We derive the dispersion relation for the propagation of quasiresonant light with frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$ in an ultracold gas of bosonic atoms in the dilute regime, i.e., for an atomic density ${\mathrm{\ensuremath{\rho}}}_{0}$\ensuremath{\ll}(${\mathrm{\ensuremath{\omega}}}_{\mathit{L}}$/c)3{)}^{3})3. In our calculation, valid up to order 2 in density, two types of corrections to the Lorentz-Lorenz formula for the refractive index appear. The first one is due to the bosonic nature of the atoms and its contribution is related to the two-body correlation function. The second correction originates from multiple scattering of photons within pairs of close atoms, giving rise to the resonant van der Waals interaction. The temperature dependence of the refractive index gives a clear signature of quantum statistical effects, even if the degeneracy threshold for Bose-Einstein condensation is not reached.
The resonant nonlinear refraction of $4\text{--}5\text{\ensuremath{-}}\ensuremath{\mu}\text{m}$ light in CO and CO2\mathrm{C}{\mathrm{O}}_{2}CO2​ gas at a peak intensity of 15MW/cm215\phantom{\rule{0.16em}{0ex}}\mathrm{MW}/\mathrm{c}{\mathrm{m}}^{2}15MW/cm2 was demonstrated using time- and frequency-resolved measurements of self-focusing and self-defocusing. The nonlinearity of these molecular gases exhibits intensity-dependent sign reversals and a $<4\text{\ensuremath{-}}\mathrm{ns}$ response time. A change from self-focusing to self-defocusing or vice versa was observed to occur for Rabi frequencies that are comparable to the collisional linewidth. A density matrix model for the nonlinear susceptibility of a strongly driven two-level system provides a qualitative explanation for these results.
Abstract We experimentally and theoretically investigate in-medium propagation effects of off-resonant light in dense, spatially inhomogeneous ultracold atomic gases. Focussing on frequency modulation spectroscopy as the dispersive detection tool of atoms, we observe that the refractive gradient-index lenses presented by localised atomic ensembles can significantly modify the interpretation of the dispersive signal even for large probe detuning, owing to the collective response of the atoms. We identify criteria for distinguishing between thin and thick atomic lenses, leading to either diffraction-dominated and lensing dominated regimes for the outgoing probe beams. Our findings are consistent with experimental data and solutions of paraxial wave equation for light propagation. Our study provides important practical insights for dispersive, minimally intrusive optical detection and imaging schemes of ultracold atoms and will be valuable for choosing optimal parameter regimes in numerous applications.
It is interesting to observe that all optical materials with a positive refractive index have a value of index that is of order unity. Surprisingly, though, a deep understanding of the mechanisms that lead to this universal behavior seems to be lacking. Moreover, this observation is difficult to reconcile with the fact that a single isolated atom is known to have a giant optical response, as characterized by a resonant scattering cross section that far exceeds its physical size. Here, we theoretically and numerically investigate the evolution of the optical properties of an ensemble of ideal atoms as a function of density, starting from the dilute gas limit, including the effects of multiple scattering and near-field interactions. Interestingly, despite the giant response of an isolated atom, we find that the maximum index does not indefinitely grow with increasing density but rather reaches a limiting value of n ≈ 1.7. This limit arises purely from electrodynamics, as it occurs at densities far below those where chemical processes become important. We propose an explanation based upon strong-disorder renormalization group theory, in which the near-field interaction combined with random atomic positions results in an inhomogeneous broadening of atomic resonance frequencies. This mechanism ensures that, regardless of the physical atomic density, light at any given frequency only interacts with at most a few near-resonant atoms per cubic wavelength, thus limiting the maximum index attainable. Our work is a promising first step to understand the limits of the refractive index from a bottom-up, atomic physics perspective, and it also introduces the renormalization group as a powerful tool to understand the generally complex problem of multiple scattering of light overall.