Colloquium: Bulk Bogoliubov excitations in a Bose-Einstein condensate

Bogoliubov theory for excitations in Bose-Einstein condensates was formulated over 50 years ago to qualitatively explain strongly interacting superfluids. Quantitative experimental verification of this theory came with the long-awaited realization of gaseous, weakly interacting condensates. This Colloquium reviews recent experimental advances in the study of Bogoliubov bulk excitations in Bose-Einstein condensates, obtained using two-photon Bragg scattering.

Figure 1

The Bogoliubov quasiparticle energy, plotted as a function of $k\xi$: solid line, for a homogeneous condensate; dotted line, the local-density approximation (cf. Sec. 5C below); dashed line, the free-particle kinetic energy. The energy is normalized by the average density of the condensate, and the healing length is evaluated at the average density.

Figure 2

The measured Bogoliubov amplitudes. The measurement, indicated by circles, was performed for phonons with $k\xi =0.38$. The height of the left peak is proportional to $v_k^2$ and that of the right peak to $u_k^2$. The solid curve is a sum of two Gaussians. The dashed curve is the LDA result. From 74.

Figure 3

The density dependence of (a) the static structure factor, and (b) the quasiparticle energy, for a $k$ value corresponding to ${ϵ}_k^0=h\times 1.38\mathrm{kHz}$. The filled circles are the measured values. The solid curves are the LDA results. The dotted curve and open circles are the single-particle values for ${ϵ}_k^0=h\times 100\mathrm{kHz}$, as in Fig. 6a. From 61.

Figure 4

The condensate response to the Bragg pulse, calculated from a Gross-Pitaevski simulation, performed at the Universita Cattolica del Sacro Cuore, Brescia, Italy, by C. Tozzo and F. Dalfovo. The simulation was done for the experimental parameters of 63, at $k\xi =0.74$ (solid lines). Momentum was imparted along the axial direction of the elongated condensate. Different curves, with increasing multipeaked structure, correspond to Bragg pulses of different duration, 1, 2, 3, 4, 5, and 6 msec. These curves are compared with the local-density approximation (LDA) average (dashed curves) from Eq. (41). From 63.

Figure 5

The normalized transferred momentum $P(k,\omega )$. The filled and open circles are for trapped $(k\xi =0.57)$ and released (low-density) condensates, respectively. The lines are fits of the difference of two Gaussians. From 61.

Figure 6

Measurements in the single-particle regime with a $k$ value corresponding to ${ϵ}_k^0=h\times 100\mathrm{kHz}$. The data were taken with two different trap radial frequencies, ${\omega }_{\rho }=2\pi \times 195\mathrm{Hz}$ (circles) and ${\omega }_{\rho }=2\pi \times 95\mathrm{Hz}$ (triangles): (a) the linear dependence on density of Eq. (37); (b) the total width (solid curve), which is the quadrature sum of the LDA width of Eq. (40) (dashed curve) and the Doppler width of Eq. (42) (dotted curve); (c) the width after subtraction of the calculated LDA and finite-time contributions (leaving Doppler broadening contribution only). From 66.

Figure 7

Radial and axial density profiles of the released, excited condensate after 38 msec of flight. The profiles are generated by computerized tomography of the absorption images (see 6B). The excitations’ momenta correspond to (a) $k\xi =0.67$ and (c) $k\xi =0.1$. For the lowest momentum value (c), the released excitations form fringes in the released condensate. For (a) the condensate and the released excitations form two fairly distinct clouds. (b) and (d) are the result of a full Gross-Pitaevskii equation simulation with the same experimental parameters as in (a) and (c), respectively. From 36.

Figure 8

The measured excitation spectrum $\omega \left(k\right)$ of a trapped Bose-Einstein condensate (filled circles): solid curve, the Bogoliubov spectrum with no free parameters, in the LDA for $\mu =1.91\mathrm{kHz}$; dashed curve, the parabolic free-particle spectrum. For most points, the error bars are not visible on the scale of the figure. From 65.

Figure 9

The measured static structure factor and Bogoliabov structure factor. The filled circles are the measured static structure factor, multiplied by an overall constant of 2.3 (65). Error bars represent $1\sigma$ statistical uncertainty, as well as the estimated uncertainty in the two-photon Rabi frequency. The solid line is the Bogoliubov structure factor in the LDA for $\mu =1.91\mathrm{kHz}$. The open circles are computed from the measured excitation spectrum of Fig. 8 and the Feynman-Bijl relation, Eq. (29). For the open circles, the error bars are not visible on the scale of the figure. From 65.

Figure 10

The response of the condensate to long Bragg pulses (63): (a) $k\xi =0.34$ and ${\tau }_B=10\mathrm{msec}$; (b) $k\xi =0.74$ and ${\tau }_B=6\mathrm{msec}$. The measured values are indicated by filled circles. Solid curves, the result of Gross-Pitaevskii simulations with various Bragg beam strengths (note that the smaller curves, corresponding to weaker pulses, have been multiplied by a factor, indicated in the figure, in order to fit to the figure scale); dotted curves, the LDA predictions; dashed curves are guides to the eye. From 63.

Figure 11

Excitation frequencies vs $k$ for a trapped, elongated BEC: open circles, the frequencies of the normal modes with $n=0$,1,2, … nodes in the radial direction, as calculated from the Gross-Pitaevskii equation; filled circles, the positions of the maxima of the main peaks in the measured values of Fig. 10. From 63.

Figure 12

Excitation energies measured from absorption images at different momenta: filled circles, the slope of the filled circles of the inset, measured in the combined region of the two clouds; open circles, the slope of the open circles in the inset, measured in the region of the excitation cloud only; solid line, the Bogoliubov dispersion relation, calculated in the LDA; dashed curve, the free-particle spectrum. The inset shows the measured energies of the clouds as a function of the number of excitations, for $k\xi =0.96$. The solid lines in the inset are linear fits. From 50.

Figure 13

Cross section for Beliaev damping as a function of $k\xi$, the excitation momentum: solid line, calculated in the LDA; dashed line, calculated without the contribution of $A_{k,q}$, the many-body suppression factor. The experimental points from 37 show the measured cross section (scaled from arbitrary units). The inset shows the energy- and momentum-conserving surfaces for the Bogoliubov dispersion. The $k_{\Vert }$ and $k_{\perp }$ axes correspond to the parallel and orthogonal components of the scattered momentum $\mathbf{k}$, respectively. The manifolds are calculated for the experimental values of $k\xi =3.87$, 2.49, 1.63, and 1 (outermost to innermost).

Figure 14

Beliaev damping spectrum from the $N=M=5\times {10}^3,k=3ħ∕\xi$, and $q=k∕\sqrt{2}$, dressed-state manifold. The condensate is homogeneous, with $3\times {10}^5{}^{87}\mathrm{Rb}$ atoms and a density of $3\times {10}^{14}\mathrm{atoms}∕{\mathrm{cm}}^3$. The two curves are (solid) for $\delta =0$, and (dashed) $\delta =1.2\mathrm{kHz}$. From 49.