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arXiv:cond-mat/0108451 · Journal of Physics B Atomic Molecular and Optical Physics

Theory of the cold collision frequency shift in 1S-2S spectroscopy of Bose-Einstein-condensed and non-condensed hydrogen

Crispin W. Gardiner, Ashton S. Bradley

Published Nov 23, 2001• 2 citations
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Abstract

We show that a correct formulation of the cold collision frequency shift for two photon spectroscopy of Bose-condensed and cold non-Bose-condensed hydrogen is consistent with experimental data. Our treatment includes transport and inhomogeneity into the theory of a non-condensed gas, which causes substantial changes in the cold collision frequency shift for the ordinary thermal gas, as a result of the very high frequency (3.9 kHz) of the transverse trap mode. For the condensed gas, we find substantial corrections arise from the inclusion of quasiparticles, whose number is very large because of the very low frequency (10.2 Hz) of the longitudinal trap mode. These two effects together account for the apparent absence of a `factor of two' between the two possibilities. Our treatment considers only the Doppler-free measurements, but could be extended to Doppler-sensitive measurements. For Bose-condensed hydrogen, we predict a characteristic `foot' extending into higher detunings than can arise from the condensate alone, as a result of a correct treatment of the statistics of thermal quasiparticles.

DOI
10.1088/0953-4075/34/23/312

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We show that a correct formulation of the cold collision frequency shift for two photon spectroscopy of Bose-condensed and cold non-Bose-condensed hydrogen is consistent with experimental data. Our treatment includes transport and inhomogeneity into the theory of a non-condensed gas, which causes substantial changes in the cold collision frequency shift for the ordinary thermal gas, as a result of the very high frequency (3.9 kHz) of the transverse trap mode. For the condensed gas, we find substantial corrections arise from the inclusion of quasiparticles, whose number is very large because of the very low frequency (10.2 Hz) of the longitudinal trap mode. These two effects together account for the apparent absence of a factor of two' between the two possibilities. Our treatment considers only the Doppler-free measurements, but could be extended to Doppler-sensitive measurements. For Bose-condensed hydrogen, we predict a characteristic foot' extending into higher detunings than can arise from the condensate alone, as a result of a correct treatment of the statistics of thermal quasiparticles.
We calculate the two-photon $1S\ensuremath{-}2S$ spectrum of an atomic hydrogen Bose-Einstein condensate in the regime where the cold collision frequency shift dominates the line shape. Wentzel-Kramers-Brillouin and static phase approximations are made to find the intensities for transitions from the condensate to motional eigenstates for 2S2S2S atoms. The excited-state wave functions are found using a mean field potential, which includes the effects of collisions with condensate atoms. Results agree well with experimental data. This formalism can be used to find condensate spectra for a wide range of excitation schemes.
We report observation of Bose-Einstein condensation (BEC) of a trapped, dilute gas of atomic hydrogen. The condensate and normal gas are studied by two-photon spectroscopy of the 1S1S1S- 2S2S2S transition. Interactions among the atoms produce a shift of the resonance frequency proportional to density. The condensate is clearly distinguished by its large frequency shift. The peak condensate density is $4.8\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\times\else\texttimes\fi{}{10}^{15}{\mathrm{cm}}^{\ensuremath{-}3}$, corresponding to a condensate population of 109{10}^{9}109 atoms. The BEC transition occurs at about $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}50\ensuremath{\mu}\mathrm{K}$ and $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8\ifmmode\times\else\texttimes\fi{}{10}^{14}{\mathrm{cm}}^{\ensuremath{-}3}$.
We have performed precision microwave spectroscopy on ultracold 87Rb{}^{87}\mathrm{Rb}87Rb confined in a magnetic trap, both above and below the Bose-condensation transition. The cold collision frequency shifts for both normal and condensed clouds were measured, which allowed the intrastate and interstate density correlations (characterized by sometimes controversial ``factors of 2'') to be determined. Additionally, temporal coherence of the normal cloud was studied, and the importance of mean-field and velocity-changing collisions in preserving coherence is discussed.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
Atomic hydrogen has been loaded into a 1-K deep magnetic trap through the use of a dilution refrigerator. Evaporative cooling has been used to cool the trapped hydrogen into the submillikelvin temperature regime. These techniques have produced a sample with density of 8 × 1013 cm-3 at a temperature of 100 μK. At this density a transition to a Bose-Einstein condensate is predicted to occur at a temperature of 30 μK. These results have been obtained with a magnetic trap that has been constructed to accommodate upcoming experiments based on laser spectroscopy of the 1S-2S two photon transition in the trapped hydrogen. A frequency doubled narrow linewidth cw dye laser system has been constructed to excite the 1S-2S transition. The excitation will be detected by the observation of electric field-induced Lyman-alpha fluorescence. The optical detection system should yield improved sample density and temperature diagnostic capabilities that will facilitate a closer approach to the quantum degenerate regime. Additionally the cold, confined hydrogen gas offers the possibility of improving the level of resolution of Doppler-free hydrogen 1S-2S laser spectroscopy by several orders of magnitude.
We have observed the cold collision frequency shift of the 1S1S1S- 2S2S2S transition in trapped spin-polarized atomic hydrogen. We find $\ensuremath{\Delta}{\ensuremath{\nu}}_{1S\ensuremath{-}2S}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}3.8\ifmmode\pm\else\textpm\fi{}0.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}n\mathrm{Hz}{\mathrm{cm}}^{3}$, where nnn is the sample density. From this we derive the 1S1S1S- 2S2S2S sss-wave triplet scattering length, ${a}_{1S\ensuremath{-}2S}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}1.4\ifmmode\pm\else\textpm\fi{}0.3\mathrm{nm}$, which is in fair agreement with a recent calculation. The shift provides a valuable probe of the distribution of densities in a trapped sample.