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Physical Review A

Refractive index of a dilute Bose gas

O. Morice, Yvan Castin, Jean Dalibard

Published May 1, 1995• 238 citations
Publisher

Abstract

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${)}^{3}$. 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.

DOI
10.1103/physreva.51.3896

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We derive the dispersion relation for the propagation of quasiresonant light with frequency \ensuremathωL{\mathrm{\ensuremath{\omega}}}_{\mathit{L}}\ensuremathωL​ in an ultracold gas of bosonic atoms in the dilute regime, i.e., for an atomic density \ensuremathρ0{\mathrm{\ensuremath{\rho}}}_{0}\ensuremathρ0​\ensuremath{\ll}(\ensuremathωL{\mathrm{\ensuremath{\omega}}}_{\mathit{L}}\ensuremathωL​/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.
We study the index of refraction of an ultracold bosonic gas in the dilute regime. Using phase-contrast imaging with light detuned from resonance by several tens of linewidths, we image a single cloud of ultracold atoms for 100 consecutive shots, which enables the study of the scattering rate as a function of temperature and density using only a single cloud. We observe that the scattering rate is increased below the critical temperature for Bose-Einstein condensation by a factor of 3 compared to the single-atom scattering rate. We show that current atom-light interaction models to second order of the density show a similar increase, where the magnitude of the effect depends on the model that is used to calculate the pair-correlation function. This confirms that the effect of quantum statistics on the index of refraction is dominant in this regime.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length aaa which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density nnn, and find for the critical temperature Tc{T}_{c}Tc​ a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
The well-known Bogoliubov expression for the spectrum of a weakly interacting dilute Bose gas becomes inadequate when the density or interactions strength is increased. The corrections to the spectrum due to stronger interactions were first considered by Beliaev (1958 Sov. Phys.—JETP 7 289). We revisit Beliaev's theory and consider its application to a dilute gas with van der Waals interactions, where the scattering length may be tuned via a Fano–Feshbach resonance. We numerically evaluate Beliaev's expression for the excitation spectrum in the intermediate-momentum regime, and we also examine the consequences of the momentum dependence of the two-body scattering amplitude. These results are relevant to the interpretation of a recent Bragg spectroscopy experiment of a strongly interacting Bose gas.
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 nearfield 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.
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.
Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of ρa3\sqrt{ρa^3}ρa3​, where ρρρ is the number density and aaa is the S-wave scattering length. We calculate the ground state energy density to second order in ρa3\sqrt{ρa^3}ρa3​. The coefficient of the ρa3ρa^3ρa3 correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on aaa, but also on an extra parameter that describes the low energy 3→33\to 33→3 scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is ρaℓV2ρa \ell_V^2ρaℓV2​, and ℓV\ell_VℓV​ is the length scale set by the van der Waals potential.