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Physical Review A

Momentum-dependent s -wave and d -wave interactions in atomic Bose-Einstein condensates

Moumita Gupta, Krishna Rai Dastidar

Published Sep 17, 2013• 0 citations
Publisher

Abstract

We investigate the role of momentum-dependent interactions to determine the ground-state properties in the Bose-Einstein condensate for large scattering lengths ($a$) such that $ka$ $\ensuremath{\ge}1$, even for small momentum ($p$) where $p=\ensuremath{\hbar}k$ and $k$ is the wave number. The results for momentum-dependent and momentum-independent interactions differ significantly, even for moderate values of $a$, and the effect of higher partial-wave interaction increases with the increase in $a$. 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 $N$ at different magnetic fields. We present here the results for those values of $N$ for which the theoretical column densities are comparable with the experimental results. It is shown that the column density for ${}^{85}$Rb atoms at 100 $\text{nK}$ with momentum-dependent interaction is in fairly good agreement with the experimental values for different values of $N$ at two different values of magnetic fields ($B=157.2$ and 156.4 G).

DOI
10.1103/physreva.88.033619

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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≥1\ensuremath{\ge}1\ensuremath≥1, even for small momentum (ppp) where p=\ensuremathℏkp=\ensuremath{\hbar}kp=\ensuremathℏk 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 have investigated the role of momentum dependent s-wave and higher partial wave strong interactions to determine the ground state properties and the column densities in the Bose-Einstein condensate (BEC) for large scattering length (a) such that ka >>1 even for small values of momentum where the momentum p=(h/2pi)k and k is the wave number. Since the scattering length is large we have included the first correction (Lee-Huang-Yang correction) both for the k-dependent (s-wave + higher partial wave) interactions and k-independent contact interactions (s-wave). We have derived the time-independent equations from the corresponding energy functionals and found that the ground state properties and the column densities differ significantly for these two types of interactions even for moderate values of scattering length (a = 3000 a_0) in BEC of cylindrically trapped 85Rb atoms at 100 nK. The effect of higher partial wave (d-wave) increases with increase in a and it is > 20% for peak density at a= 8700 a_0 which can be experimentally detected. Dependence of column density on particle number density has been studied. Column densities have been compared with experimental results.
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 discusses a series of studies that investigate the effects of interaction – essentially the s-wave scattering – in the various properties of Bose-Einstein condensates (BEC). The phonon wavefunction in a BEC was measured using Bragg spectroscopy and compared with the well-known Bogoliubov theory. Phonons were first excited in a BEC of 3× 10 condensed Na atoms via small-angle two-photon Bragg scattering. Large angle Bragg scattering was then used to probe the momentum distribution. We found reasonable agreement with the theory. With the same technique of Bragg diffraction, we studied the four-wave mixing process for matter waves. The BEC was split into two strong source waves and a weak seed wave. The s-wave scattering coherently mixed pairs of atoms from the sources into the seed and its conjugate wave, creating a pair-correlated atomic beams with “squeezed” number difference. A Feshbach resonance was used to produce ultracold Na2 molecules with initial phase-space density in excess of 20. Starting from an atomic BEC, a magnetic field ramp shifted a bound state from above the threshold of the unbound continuum to below, creating a molecular population with almost zero center-of-mass motion. A reverse field ramp dissociated the cold molecules into free atom pairs carrying kinetic energy dependent on the ramp speed. This dependence provided a measure of the coupling strength between the bound state and the continuum. Condensates were loaded into optical lattices formed with retro-reflected single frequency lasers. Quantum phase transition from the superfluid state to Mott-insulator state was observed in a three dimensional lattice. The increased interaction and flattened dispersion relation led to strongly enhanced quantum depletion in the superfluid state. Thesis Supervisor: Wolfgang Ketterle Title: John D. MacAurthur Professor of Physics
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.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the sss-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
We have measured the relative strength ${\ensuremath{\epsilon}}_{dd}$ of the magnetic dipole-dipole interaction compared with the contact interaction in a dipolar chromium Bose-Einstein condensate. We analyze the asymptotic velocities of expansion of the condensate with different orientations of the atomic magnetic moments. By comparing the experimental results with numerical solutions of the hydrodynamic equations for dipolar condensates, we obtain ${\ensuremath{\epsilon}}_{dd}=0.159\ifmmode\pm\else\textpm\fi{}0.034$. We use this result to determine the sss-wave scattering length $a=(5.08\ifmmode\pm\else\textpm\fi{}1.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9})\text{ }\mathrm{m}=(96\ifmmode\pm\else\textpm\fi{}20)\text{ }{a}_{0}$ of 52Cr^{52}\mathrm{Cr}52Cr. This is fully consistent with our previous measurements on the basis of Feshbach resonances and therefore confirms the validity of the theoretical approach used to describe the dipolar Bose-Einstein condensate.