Momentum-Resolved Observation of Thermal and Quantum Depletion in a Bose Gas

We report on the single-atom-resolved measurement of the distribution of momenta $\hslash k$ in a weakly interacting Bose gas after a 330 ms time of flight. We investigate it for various temperatures and clearly separate two contributions to the depletion of the condensate by their $k$ dependence. The first one is the thermal depletion. The second contribution falls off as $k^{-4}$, and its magnitude increases with the in-trap condensate density as predicted by the Bogoliubov theory at zero temperature. These observations suggest associating it with the quantum depletion. How this contribution can survive the expansion of the released interacting condensate is an intriguing open question.

Figure 1

(a) Sketch of cloud expansion and detection by a microchannel plate detector, yielding the 3D asymptotic momentum distribution (far-field regime). The initially cigar-shaped Helium condensate (black) undergoes anisotropic expansion, inverting its aspect ratio. Quantum depletion and/or thermally excited atoms (grays) populate momentum states beyond those associated with the condensate, and are expected to have a spherical symmetry. (b) Measured 3D distribution of atoms $n_{\infty }(\mathbf{k})$ after a 330 ms time of flight. The central dense part corresponds to the condensate while the isolated dots are excited particles outside of the condensate wave function. Also shown are the 2D projections, highlighting the condensate anisotropy.

Figure 2

1D momentum profiles obtained from cuts of the 3D data $n_{\infty }(\mathbf{k})$. (a) Profiles along the radial (blue) and longitudinal (purple) directions. In linear scale, the tails are not visible. (b) Log-log scale plot of the radial profile. The solid line is the scaling solution for the condensate in the Thomas-Fermi approximation (region $\mathbf{I}$: $k<1.7\mu {\mathrm{m}}^{-1}$); the dotted line is a Bose distribution fitting the thermal wings (region $\mathbf{II}$: $1.7<k<4.3\mu {\mathrm{m}}^{-1}$); the dashed line is a $k^{-4}$ power law fitting the high-momentum tails (region $\mathbf{III}$: $k>4.3\mu {\mathrm{m}}^{-1}$). Solid lines are a smoothed average of the density data, and the lightly shaded band is the running standard deviation. The dark count rate corresponds to a level less than $\sim {10}^{-2}\mu {\mathrm{m}}^3$, which is 1 order of magnitude below the lowest data point. The plotted range of momenta is limited by the physical size of the detector.

Figure 3

Measured $k^4{n}_{\infty }(k)$ plotted as a function of $k$ and at various temperatures. Solid blue lines are a smoothed average of the density data, and the lightly shaded blue band shows the running standard deviation. The dashed line is a fit using Eq. (4) which involves two terms: the thermal depletion (also shown as a dotted line) and the quantum depletion revealed by the $k^{-4}$ scaling at large $k$. The fitting procedure yields the temperature $T_a$, the thermal atom number $N_{\mathrm{th}}$ and the asymptotic constant ${\mathcal{C}}_{\infty }$. Noted in each subplot are $T_a$, the ratio of the thermal energy and the chemical potential $k_B{T}_a/\mu$ and $f_{\mathrm{th}}=N_{\mathrm{th}}/N$.

Figure 4

Contact constant ${\mathcal{C}}_{\infty }/N_0$ per condensed particle plotted as a function of the condensate density $n_0$. The geometric trapping frequency $\overline{\omega }/2\pi$ and the ratio $k_B{T}_a/\mu$ are indicated. The dashed line is the Bogoliubov prediction in the LDA, ${\mathcal{C}}_{\mathrm{LDA}}$ (see text), and the solid line is $6.5\times {\mathcal{C}}_{\mathrm{LDA}}$.