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arXiv:0912.3336

Elementary Excitations in Bose-Einstein Condensates at Large Scattering Lengths

Risto Sarjonen, M. Saarela, Ferran Mazzanti

Published Dec 17, 2009• 0 citations
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Abstract

We present a theoretical analysis of excitation modes in Bose-Einstein condensates in ultracold alkali-metal gases for large scattering lengths and momenta where corrections to the mean field approximation become important. We assume that the effective interaction in the metastable, single channel, gaseous phase has a well defined Fourier transform that scales with the scattering length. Based on this we show that for increasing scattering lengths or equivalently increasing densities the system becomes less correlated and that at large values of the scattering length Bragg scattering measures directly the Fourier transform of the effective two-body potential. We construct model potentials which fit the recently measured line shifts in $^{85}$Rb by Papp et al. (Phys. Rev. Lett. {\bf 101}, 135301 (2008)), and show that they fix the low momentum expansion of the effective range function. We find excellent agreement with the experimental data when the effective range is $\ll 1$ and the coefficient of the $k^4$-term is $-7.5 \pm 0.5$ in scattering length units. The resolution in Bragg scattering experiments so far does not reveal details of the frequency dependence in the dynamic structure function $S(k,ω)$ and we show that the Feynman spectrum determines the measured line shifts. We propose the possibility of a transition to a novel density wave state.

DOI
10.48550/arxiv.0912.3336

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We present a theoretical analysis of excitation modes in Bose-Einstein condensates in ultracold alkali-metal gases for large scattering lengths and momenta where corrections to the mean field approximation become important. We assume that the effective interaction in the metastable, single channel, gaseous phase has a well defined Fourier transform that scales with the scattering length. Based on this we show that for increasing scattering lengths or equivalently increasing densities the system becomes less correlated and that at large values of the scattering length Bragg scattering measures directly the Fourier transform of the effective two-body potential. We construct model potentials which fit the recently measured line shifts in 85^{85}85Rb by Papp et al. (Phys. Rev. Lett. {\bf 101}, 135301 (2008)), and show that they fix the low momentum expansion of the effective range function. We find excellent agreement with the experimental data when the effective range is ≪1\ll 1≪1 and the coefficient of the k4k^4k4-term is −7.5±0.5-7.5 \pm 0.5−7.5±0.5 in scattering length units. The resolution in Bragg scattering experiments so far does not reveal details of the frequency dependence in the dynamic structure function S(k,ω)S(k,ω)S(k,ω) and we show that the Feynman spectrum determines the measured line shifts. We propose the possibility of a transition to a novel density wave state.
We present a theoretical analysis of excitation modes in Bose-Einstein condensates of ultracold alkali-metal gases for large scattering lengths, showing clear deviations from the Bogoliubov prediction as seen by Papp et al. [Phys. Rev. Lett. 101, 135301 (2008)]. We construct the atom-atom interaction by deriving the TTT matrix of such systems from two coupled (open and closed) channels assuming that the Feshbach resonance dominates the latter. We calculate molecular bound-state energies as a function of the magnetic field and compare with available experiments. The sss-wave phase shifts determine the local effective interaction with long-ranged repulsion and short-ranged attraction. We show that it becomes a universal function at large scattering lengths. Finally, we use this interaction to characterize the ground-state and elementary excitations of 85{}^{85}85Rb, 87{}^{87}87Rb, and 23{}^{23}23Na gases. Good agreement with line shift experiments in 85{}^{85}85Rb is achieved. We find that, at large scattering lengths, Bragg scattering experiments could directly measure the momentum dependence of the effective two-body potential.
This thesis studies bosonic and fermionic quantum gases. In the first part, a variational many-body theory is used to explain a Bragg scattering experiment carried out on a bosonic quantum gas [S.B. Papp et al. Bragg Spectroscopy of a Strongly Interacting 85Rb Bose-Einstein Condensate. Phys. Rev. Lett. 101, 135301 (2008)]. A key feature in the experiment was the use of a Feshbach resonance, which made large values of the scattering length accessible. Due to the large values of the scattering length, existing models such as the Beliaev model could not be used to explain the observations, and therefore the experiment was a particularly interesting one to analyze. In our first approach, we constructed ad hoc potentials that fitted the observed excitation spectrum, and later we improved our approach by using T-matrix formalism to describe the Feshbach resonant system. All in all, the phenomenological model we developed fits the observed excitation spectrum and yields correct molecular Feshbach resonance state energies in certain cases. The second part of this thesis studies fermionic quantum gases. We focus on studying a gas of spin-1/2 particles confined to a spin-dependent optical lattice. The lattice geometry is such that the up-spin component is loaded in a honeycomb lattice, and the down-spin component is confined to the underlying triangular lattice. We considered attractive on-site and nearest-neighbor interactions, and formulated the nearest-neighbor interaction term in such a way that it takes into account the possibility of spontaneous time-reversal symmetry breaking. Furthermore, we took into account the possibility of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which breaks spatial symmetry. Within a mean-field approximation, we showed that the FFLO state is the ground state of the system in many instances. In addition, we found out that the system spontaneously breaks time-reversal symmetry if the nearest-neighbor interaction strength is large. Due to the time-reversal symmetry breaking, the system has topologically non-trivial phases characterized by nonzero Chern numbers. Finally, there were also cases where the time-reversal symmetry was broken in the FFLO phase, and thus we found a phase where spatial and time-reversal symmetries are simultaneously broken.
We extend classical-field methods [Blakie et al., Adv. Phys. 57, 363 (2008)] to provide a description of atom-molecule systems. We use a model of Bose-Einstein condensation of atoms close to a Feshbach resonance, in which the tunable scattering length of the atoms is described using a system of coupled atom and molecule fields [Holland et al., Phys. Rev. Lett. 86, 1915 (2001)]. We formulate the basic theoretical methods for a coupled atom-molecule system, including the determination of the phenomenological parameters in our system, the Thomas-Fermi description of Bose-Einstein condensate, the Bogoliubov--de Gennes equations, and the Bogoliubov excitation spectrum for a homogenous condensed system. We apply this formalism to the special case of Bragg scattering from a uniform condensate and find that for moderate and large scattering lengths, there is a dramatic difference in the shift of the peak of the Bragg spectra, compared to that based on a structureless atom model. The result is compatible with the experimental results of Papp et al. [Phys. Rev. Lett 101, 135301 (2008)] for Bragg scattering from a nonuniform condensate.
We formulate the basic theoretical methods for Bose-Einstein Condensation of atoms close to a Feshbach resonance, in which the tunable scattering length of the atoms is described using a system of coupled atom and molecule fields. These include the Thomas-Fermi description of the condensate profile, the c-field equations, and the Bogoliubov-de Gennes equations, and the Bogoliubov excitation spectrum for a homogenous condensed system. We apply this formalism to the special case of Bragg scattering from a uniform condensate, and find that for moderate and large scattering lengths, there is a dramatic difference in the shift of the peak of the Bragg spectra, compared to that based on a structureless atom model. The result is compatible with the experimental results of Papp et al. [S. B. Papp et al., Phys. Rev. Lett., 101(13):135301, Sep 2008].
We investigate the role of momentum-dependent interactions to determine the ground-state properties in the Bose-Einstein condensate for large scattering lengths (aaa) such that kakaka $\ensuremath{\ge}1$, even for small momentum (ppp) where $p=\ensuremath{\hbar}k$ and kkk is the wave number. The results for momentum-dependent and momentum-independent interactions differ significantly, even for moderate values of aaa, and the effect of higher partial-wave interaction increases with the increase in aaa. We have made an attempt to compare the theoretical column density with experimental results at different magnetic fields [Cornish et al., Phys. Rev. Lett. 85, 1795 (2000)]. Since the initial number of atoms changes while swapping the magnetic field and the actual value of the number of atoms at different magnetic fields is not known, we studied the dependence of column densities on the value of NNN at different magnetic fields. We present here the results for those values of NNN for which the theoretical column densities are comparable with the experimental results. It is shown that the column density for 85{}^{85}85Rb atoms at 100 nK\text{nK}nK with momentum-dependent interaction is in fairly good agreement with the experimental values for different values of NNN at two different values of magnetic fields (B=157.2B=157.2B=157.2 and 156.4 G).
We present a theoretical model for Bragg scattering from a Bose-Einstein condensate (BEC) in the vicinity of a magnetic Feshbach resonance, using a two c-field formalism, one c-field for the atom and the other for a molecule formed of two atoms. We use this model to numerically simulate a recent experiment [S. B. Papp et al., Phys. Rev. Lett., 101(13):135301, Sep 2008] investigating the effects of strong interactions on the Bragg spectrum from a 85Rb BEC. Results from these simulations are in very good quantitative agreement with the experimental results, confirming the importance of the resonance bound state in the dynamics of the condensate for fast experiments like Bragg scattering.